INTERNATI01\AL SERIES IN PURE AND APPLIED PHYSICS G. P. HARNWELL, CONSULTING EDITOR
METHODS OF THEORETICAL PHYSICS
INTERNATIONAL SERIES IN PURE AND APPLIED PHYSICS G. P. HARNWELL, CONSULTING EDITOR ADVISORY EDITORIAL COMMITTEE: E. U. Condon, George R. Harrison Elmer Hutchisson, K. K . Darrow
Allis and Herlin Thermodynamics and Statistical Mechanics Cady Piezoelectricity Clark Applied X-rays Edwards Analytic and Vector Mechanics Finkelnburg Atomic Physics Gurney Introduction to Statistical Mechanics Hall Introduction to Electron Microscopy Hardy and Perrin The Principles of Optics Harnwell Electricity and Electromagnetism Harnwell and Livingood Experimental Atomic Physics Houston Principles of Mathematical Physics Houston Principles of Quantum Mechanics Hughes and DuBridge Photoelectric Phenomena Hund High-frequency Measurements Kemble The Fundamental Principles of Quantum Mechanics Kennard Kinetic Theory of Gases Marshak Meson Physics Morse Vibration and Sound Morse and Feshbach Methods of Theoretical Physics M uskai Physical Principles of Oil Production Read Dislocations in Crystals Richtmyer and Kennard Introduction to Modern Physics Ruark and Urey Atoms, Molecules, and Quanta Schiff Quantum Mechanics Seitz The Modern Theory of Solids Slater Introduction to Chemical Physics Slater Microwave Transmission Slater Quantum Theory of Matter Slater and Frank Electromagnetism Slater and Frank Introduction to Theoretical Physics Slater and Frank Mechanics Smythe Static and Dynamic Electricity Squire Low' Temperature Physics Stratton Electromagnetic Theory Thorndike Mesons : A Summary of Experimental Facts White Introduction to Atomic Spectra The late F. K. Richtmyer was Consulting Editor of the series from its inception in 1929 to his death in 1939. Lee A. DuBridge was Consulting Editor of the series from 1939 to 1946.
METHODS OF THEORETICAL PHYSICS Philip M. Morse PROFESSOR OF PHYSICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Herman Feshbach ASSOCIATE PROFESSOR OF PHYSICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY
P ART
I:
CHAPTERS
New York
1
TO
8
Toronto London
McGRAW-HILL BOOK COMPANY, INC. 1953
METHODS OF THEORETICAL PHYSICS Copyright, 1953, by the McGraw-Hill Book Company, Inc. Printed in the United States of America. All rights reserved. This book , or parts thereof, may not be reproduced in any form without permission of the publishers. Library of Congress Catalog Card Number: 52-11515
THE MAPLE PRESS COMPANY , YORK , PA.
Preface
This treatise is the outgrowth of a course which has been given by one or the other of the authors for the past sixteen years. The book itself has been in the process of production for more than half this time, though with numerous interruptions, major and minor. Not the least difficult problem in its development has been to arrive at a general philosophy concerning the subject matter to be included and its order of presentation. Theoretical physics today covers a vast area; to expound it fully would overflow a five-foot shelf and would be far beyond the authors' ability and interest. But not all of this area has recently been subjected to intensive exploration; the portions in which the most noticeable advances have been made in the past twenty years are mostly concerned with fields rather than with particles, with wave functions, fields of force, electromagnetic and acoustic potentials, all of which are solutions of partial differential equations and are specified by boundary conditions. The present treatise concent rates its attention on this general area. Fifty years ago it might have been entitled" Partial Differential Equations of Physics " or "Boundary Value Problems." Today, because of the spread of the field concept and techniques, it is perhaps not inappropriate to use a more general title. Even this restricted field cannot be covered in two volumes of course. A discussion of the physical concept s and experimental procedures in all the branches of physics which use fields for their de~cription would itself result in an overlong shelf, duplicating the subject matter of many excellent texts and, by its prolixity, disguising the fundamental unity of the subject. For .the unity of field theory lies in its techniques of analysis, the mathematical tools it uses to obtain answers. These techniques are essentially, the same, whether the field under study corresponds to a neutral meson, a radar signal, a sound wave, or a cloud of diffusing neutrons. The present treatise, therefore, is primarily concerned with an exposition of the mathematical tools which have proved most useful in the study of the many field constructs in physics, together with v
vi
Preface
a series of examples, showing how the tools are used to solve various physical problems. Only enough of the underlying physics is given to make the examples understandable. This is not to say that the work is a text on mathematics, however. The physicist, using mathematics as a tool , can also use his physical knowledge to supplement equations in a way in which pure mathematicians dare not (and should not) proceed. He can freely use the construct of the point charge, for example; the mathematician must struggle to clarify the analytic vagaries of the Dirac delta function . The physicist often starts with the solution of the partial differential equation already ' described and measured ; the mathematician often must develop a very specialized network of theorems and lemmas to show exactly when a given equation has a unique solution. The derivations given in the present work will, we hope, be understandable and satisfactory to physicists and engineers, for whom the work is written; they will not often seem rigorous to the mathematician . Within these twofold limits, on the amount of physics and the rigor of the mathematics, however, it is hoped that the treatise is reasonably complete and self-contained. The knowledge of physics assumed is that expected of a first-year graduate student in physics, and the mathematical background assumed is that attained by one who has taken a first course in differential equations or advanced calculus. The further mathematics needed, those parts of vector and tensor analysis, of the theory of linear differential equations and of integral equations, which are germane to the major subject, are all treated in the text. The material is built up in a fairly self-contained manner, so that only seldom is it necessary to use the phrase "it can be shown," so frustrating to the reader. Even in the earlier discussion of the basic mathematical techniques an attempt is made to relate the equations and the procedures to the physical properties of the fields, which are the central subject of study. In many cases derivations are given twice, once in a semi-intuitive manner, to bring out the physical concepts, a second time with all the symbols and equations, to provide as mu ch rigor as seems necessary. Often part of the derivation is repeated in a later chapter, from a different point of view, to obviate excessive back reference; this was considered desirable, though it has increased the size of the treatise. An effort has been made to avoid trivial and special-case examples of solutions. As a result, of course , the examples included' often require long and complicat ed explanations to bring out all the items of interest, but this treatise is supposed to explain how difficult problems can .be solved; it cannot be illustrated by easy ones. The variational technique applied to diffraction problems, the iteration methods used in calculating the scattering of waves from irregular boundaries, the computation of convergent series for eigenstates perturbed by strong interaction poten-
vii
Preface
tials are all techniques which show their power only when used on problems not otherwise soluble . Another general prin ciple has also tended to lengthen the discussions : The authors have tried, as often as possible, to attack problems "head on," rather than by "backing into them." They have tried to show how to find the solution of a new and strange equation rather than to set down a list of results which someone has found to be solutions of interesting problems. A certain number of "backing-up" examples, where one pulls a solution out of the air , so to speak, and then proceeds to show it is indeed the solution, could not be avoided. Usually such examples have saved space and typographic complications; very many of them would have produced a state of frustration or of fatalism in the student. It is hoped that the work will also prove to be reasonably self-contained in regard to numerical tables and lists of useful formulas. The tables at the end of each chapter summarize the major results of that chapter and collect , in an easily accessible spot, the properties of the functions most often used. Rather than scattering the references among the text, these also are collected at the end of each chapter in order to make them somewhat easier to find again. These only include titles of the books and articles which the authors feel will be useful to the reader in supplementing the material discussed in the chapter ; they are not intended to indi cate priority or the high points in historical development. The development of the subject matter of this book has been extensive and has involved the cont ributions of many famous persons. Techniques have been rediscovered and renamed nearly every time a new branch of physics has been opened up . A thorough treatment of the bibliography would require hundreds of pages, much of it dull reading. We have chosen our references to help the reader understand the subject, not to give each contributor in the field his" due credit." Frankly, we have put down those referen ces we are familiar with and which we have found ~M.
,
An attempt has been made to coordinate the choice of symbols for the various functions defined and used. Where usage is fairly consistent in the literature, as with Bessel functions, this has been followed. Where there have been several alternate symbols extant, the one which seemed to fit most logically with the rest and resulted in least duplication was chosen, as lnts been done with the Mathieu functions . In a few cases functions were renormalized to make them less awkward to use; these were given new symbols, as was done with the Gegenbauer polynomials. The relation between the notation used and any alternate notation appearing with reasonable frequency in physics literature is given in the Appendix, together with a general glossary of symbols used . The numerical table in the Appendix should be adequate for the majority of the calculations related to the subject matter. It was con-
viii
Preface
sidered preferable to include a number of tables of limited range and accuracy rather than to have a few tables each with a large number of entries and significant figures. Most of the fun ctions used in actual physical problems are tabulated, though many of the auxiliary functions, such as the gamma and the elliptic functions, and some fun ctions with too many independent parameters, such as the hypergeometric functions, are not represented. A few fun ctions, such as the parabolic and the spheroidal wave fun ctions, should have been included, but complete basic tables have not yet been published. Several of the figures in this work, which have to do with three dimensions, are drawn for stereoscopic viewing . They may be viewed by any of the usual stereoscopic viewers or, without any paraphernalia, by relaxing one's eye-focusing muscles and allowing one eye to look at one drawing, the other at the other. Those who have neither equipment nor sufficient ocular decoupling may consider these figures as ordinary perspective drawings unnecessarily duplicated. If not benefited, they will be at least not worse off by the duplication. The authors have been helped in their task by many persons. The hundreds of graduate students who have taken the related course since 1935 have, wittingly or unwittingly, helped to fix the order of presentation and the choice of appropriate examples. They have removed nearly all the proof errors from the offset edition of class notes on which this treatise is based; they have not yet had time to remove the inevitable errors from this edition. Any reader can assist in this by notifying the authors when such errors are discovered. Assistance has also been more specific. The proof of Cauchy's theorem, given on page 364 , was suggested by R. Boas. Parts of the manuscript and proof have been read by Professors J. A. Stratton and N. H . Frank; Doctors Harold Levine, K. U. Ingard, Walter Hauser, Robert and Jane Pease, S. Rubinow ; and F . M . Young, M. C. Newstein, L. Sartori, J. Little, E. Lomon, and F. J. Corbat6. They are to be thanked for numerous improvements and corrections; they should not be blamed for the errors and awkward phrasings which undoubtedly remain. We are also indebted to Professor Julian Schwinger for many stimulating discussions and suggestions. Philip 1.11. Morse Herman Feshbach May, 1953
Contents
v
PREFACE
PART I CHAPTER
1
1 Types of Fields
1.1 Scalar Fields Isotimic Surfaces.
4 The Laplacian.
1.2 Vector Fields Multiplication of Vectors. faces. Surface Integrals. Singularities of Fields.
1.3
8 Axial Vectors. Source Point.
Lines of Flow. Potential SurLine Integrals. Vortex Line.
Curvilinear Coordinates
21
Direction Cosines. Scale Factors. Curvature of Coordinate Lines. The Volume Element and Other Formulas. Rotation of Axes. Law of Transformation of Vectors. Contravariant and Covariant Vectors.
1.4
The Differential Operator V
31
The Gradient. Directional Derivative. Infinitesimal Rotation. The Divergence. Gauss' Theorem. A Solution of Poisson's Equation. The Curl. Vorticity Lines. Stokes' Theorem. The Vector Operator v.
1.5
Vector and Tensor Formalism
44
Covariant and Contravariant Vectors. Axial Vectors. Christoffel Symbols. Covariant Derivative. Tensor Notation for Divergence and Curl. Other Differential Operators. Other Second-order Operators. Vector as a Sum of Gradient and Curl.
1.6
Dyadics and Other Vector Operators
54
Dyadics. Dyadics as Vector Operators. Symmetric and Antisymmetric Dyadics. Rotation of Axes and Unitary Dyadics. Dyadic Fields. Deformation of Elastic Bodies. Types of Strain. Stresses in an Elastic Medium. Static Stress-Strain Relations for an Isotropic Elastic Body. Dyadic Operators. Complex Numbers and Quaternions as Operators. Abstract Vector Spaces. Eigenvectors and Eigenvalues. Operators in Quantum Theory. Direction Cosines and Probabilities. Probabilities and Uncertainties. Complex Vector Space. Generalized Dyadics. Hermitian ix
Contents
x
Operators. Examples of Unitary Operators. Transformation of Operators. Quantum Mechanical Operators. Spin Operators. Quaternions. Rotation Operators.
1.7
The Lorentz Transformation, Four-vectors, Spinors
93
Proper Time. The Lorentz Transformation. Four-dimensional Invariants. Four-vectors. Stress-Energy Tensor. Spin Space and Spacetime. Spinors and Four-vectors. Lorentz Transformation of Spinors. Space Rotation of Spinors. Spin Vectors and Tensors. Rotation Operator in Spinor Form .
Problems Table of Useful Vector and Dyadic Equations Table of Properties of Curvilinear Coordinates Bibliography CHAPTER
2.1
2
Equations Governing Fields
The Flexible Siring
107 114 115 117 119 120
Forces on an Element of String. Poisson's Equation. Concentrated Force, Delta Function. The Wave Equation. Simple Harmonic Motion, Helmholtz Equation. Wav e Energy. Energy Flow. Power and Wave Impedance. Forced Motion of the String. Transient Response, Fourier Integral. Operator Equations for the String. Eigenvectors for the un it Shift Operator. Limiting Case of Continuous String. The Effect of Friction. Diffusion Equation. Klein-Gordon Equation. Forced Motion of the Elastically Braced String. Recapitulation.
2.2
Waves in an Elastic Medium
142
Longitudinal Waves. Transverse Waves. Wave Motion in Three Dimensions. Vector Waves. Integral Representations. Stress and Strain. Wave Energy and Impedance.
2.3
Motion of Fluids
151
Equation of Continuity. Solutions for Incompressible Fluids. Examples. Stresses in a Fluid. Bernouilli's Equation. The Wave Equation. Irrotational Flow of a Compressible Fluid. Subsonic and Supersonic Flow Velocity Potential, Linear Approximation. Mach Lines and Shock Waves
2.4
Diffusion and Other Percolative Fluid Motion
171
Flow of Liquid through a Porous Solid. Diffusion. Phase Space and the Distribution Function. Pressure and the Equation of State. Mean Free Path and Scattering Cross Section. Diffusion of Light, Integral Equation. Diffusion of Light, Differential Equation. Boundary Conditions. Effect of Nonuniform Scattering. First-order Approximation, the Diffusion Equation. Unit Solutions. Loss of Energy on Collision . Effect of External Force. Uniform Drift Due to Force Field . Slowing Down of Particles by Collisions. Recapitulation.
2.5
The Electromagnetic Field
200
The Electrostatic Field . The Magnetostatic Field. Dependence on Time. Maxwell 's Equations. Retardation and Relaxation. Lorentz
Contents
xi
Transformation. Gauge Transformations. Field of a Moving Charge. Force and Energy. Surfaces of Conductors and Dielectrics. Wave Transmission and Impedance. Proca Equation.
2.6
Quantum Mechanics
222
Photons and the Electromagnetic Field . Uncertainty Principle. Conjugate Variables and Poisson Brackets. The Fundamental Postulates of Quantum Theory. Independent Quantum Variables and Functions of Operators. Eigenvectors for Coordinates. Transformation Functions. Operator Equations for Transformation Functions. Transformation to Momentum Spa ce. Hamiltonian Function and Schroedinger Equation. The Harmonic Oscillator. Dependence on Time. Time as a Parameter. Time Dependent Hamiltonian. Particle in Ele ctromagnetic Field . Relativity and Spin . The Dirac Equation. Total Angular Momentum . Field-free Wave Function. Recapitulation.
Problems 267 Standard Forms for Some of the Partial Differential Equations of Theoretical Physics 271 Bibliography 273 CHAPTER
3.1
3
Fields and the Variational Principle
The Variational Integral and the Euler Equations The Euler Equations.
3.2
275 276
Auxiliary Conditions.
Hamilton's Principle and Classical Dynamics
280
Lagrange's Equations. Energy and the Hamiltonian. Impedance. Canonical Transformations. Poisson Brackets. The Action Integral. The Two-dimensional Oscillator. Charged Particle in Electromagnetic Field. Relativistic Particle. Dissipative Systems. Impedance and Admittance for D issipative Systems.
3.3
Scalar Fields
301
The Flexible String. The Wave Equation. Helmholtz Equation. Velocity Potential. Compressional Waves. Wave Impedance. Planewave Solution. Diffusion Equation . Schroedinger Equation. · KleinGordon Equation.
3.4
Vector Fields
318
General Field Properties. Isotropic Elastic Media. Plane-wave Solutions. Impedance. The Electromagnetic Field . Stress-energy Tensor. Field Momentum. Gauge Transformation. Impedance Dyadic. Planewave Solution. D irac Equation.
Problems Tabulation of Variational Method
337 341
Flexible String or Membrane. Compressible, Nonviscous Fluid. Diffusion Equation. Schroedinger Equation. Klein-Gordon Equation. Elasti c Wave Equation. Electromagnetic Equations. Proca Equations. Dirac Equation.
Bibliography
347
Contents
xii CHAPTER
4.1
4
348 349
Functions of a Complex Variable
Complex Numbers and Variables
The Exponential Rotation Operator. Vectors and Complex Numbers . The Two-dimensional Electrostatic Field . Contour Integrals.
4.2
Analytic Functions
356
Conformal Representation. Integration in the Complex Plane. Cauchy's Theorem. Some Useful Corollaries of Cauchy's Theorem . Cauchy's Integral Formula. Real and Imaginary Parts of Analytic Functions. Impedances. Poisson's Formula.
4.3
Derivatives of Analytic Functions, Taylor and Laurent ~~
3U
The Taylor Series. The Laurent Series. Isolated Singularities . Classification of Functions. Liouville's Theorem. Meromorphic Functions. Behavior of Power Series on Circle of Convergence. Analytic Continuation. Fundamental Theorems. Branch Points. Techniques of Analytic Continuation.
4.4 4.5
398
Multivalued Functions Branch Points and Branch Lines. Example.
Riemann Surfaces.
An Illustrative
Calculus of Residues; Gamma and Elliptic Functions
408
Integrals Involving Branch Points. Inversion of Series. Summation of Series. Integral Representation of Functions. Integral Related to the Error Function. Gamma Functions. Contour Integrals for Gamma Function. Infinite Product Representation for Gamma Functions. Derivatives of the Gamma Function. The Duplication Formula. Beta Functions. Periodic Functions. Fundamental Properties of Doubly Periodic Functions. Elliptic Functions of Second Order. Integral Representations for Elliptic Functions.
4.6
Asymptotic Series: Method of Steepest Descent
434
An Example.
Averaging Successive Terms. Integral Representations and Asymptotic Series. Choosing the Contour. First Term in the Expansion . The Rest of the Series.
4.7
Conformal Mapping
443
General Properties of the Transformation. Schwarz-Christoffel Transformation. Some Examples. The Method of Inversion.
4.8 Fourier Integrals
453
Relation to Fourier Series. Some Theorems on Integration. The Fourier Integral Theorem. Properties of the Fourier Transform . Asymptotic Values of the Transform . General Formulation. Faltung. Poisson Sum Formula. The Laplace Transform. Mellin Transform .
Problems 471 Tabulation of Properties of Functions of Complex Variable 480 Euler's Algorithm Relating Series. Asymptotic Series for Integral Representation. Fourier Transforms. Laplace Transforms. Mellin Transforms.
Contents
xiii
Tables of Special Functions of General Use The Gamma Function.
Elliptic Functions.
Bibliography CHAPTER
5.1
. ~ 486
Theta FUnctions.
5 Ordinary Differential Equations Separable Coordinates
490 492 494
Boundary Surfaces and Coordinate Systems. Two-dimensional Separable Coordinates. Separable Coordinates for Laplace's Equation in Two Dimensions. Separation of the Wave Equation. Rectangular and Paraboli c Coordinates. Polar and Elliptic Coordinates. Scale Factors and Coordinate Geometry. Separation Constants and Boundary Conditions. Separation in Three Dimensions. The Staekel Determinant. Confocal Quadric Surfaces. Degenerate Forms of Ellipsoidal Coordinates. Confluence of Singularities. Separation Constants. Laplace Equation in Three Dimensions. Modulation Factor. Confocal Cy clides ,
5.2
General Properties, Series Solutions
523
The Wronskian. Independent Solutions. Integration Factors and Adjoint Equations. Solution of the Inhomogeneous Equation. Series Solutions about Ordinary Points. Singular Points. Indicial Equation . Classification of Equations, Standard Forms. Two Regular Singular Points. Three Regular Singular Points. Recursion Formulas. The Hypergeometric Equation. Functions Expressible by Hypergeometric Series. Analytic Continuation of Hypergeometric Series. Gegenbauer Functions. One Regular and One Irregular Singular Point. Asymptotic Series. Two Regular, One Irregular Singular Point. Continued Fractions. The Hill Determinant. Mathieu Functions. Mathieu Functions of the Second Kind. More on Recursion Formulas. Functional Series .
5.3
Integral Representations
577
Some Simple Examples. General Equations for the Integrand. The Euler Transform. Euler Transform for the Hypergeometric Function. Analytic Continuation of the Hypergeometric Series. Legendre Functions. Legendre Functions of the Second Kind. Gegenbauer Polynomials. The Confluent Hypergeometric Function. -T he Laplace Transform. Asymptotic Expansion. Stokes' Phenomenon. Solutions of the Th ird Kind. The Solution of the Second Kind. Bessel Functions. Hankel Functions. Neumann Functions. Asymptotic Formulas for Large Order. The Coulomb Wave Function. Mathieu Functions. The Laplace Transform and the Separated Wave Equation. More on Mathieu Functions. Spheroidal Wave Functions. Kernels Which Are Functions of zi .
Problems Table of Separable Coordinates in Three Dimensions
646 655
Rectangular. Circular Cylinder. Elliptic Cylinder. Parabolic Cylinder. Spherical. Conical. Parabolic. Prolate Spheroidal. Oblate Spheroidal. Ellipsoidal. Paraboloidal. Bispherical. Toroidal Coordinate Systems.
Contents
xiv
Second-order Differential Equations and Their Solutions
667
One Regular Singular Point. One Irregular Singular Point. Two Regular Singular Points. Three Regular Singular Points. One Regular, One Irregular Singular Points. Two Irregular Singular Points.
Bibliography 6
CHAPTER
6.1
674
Boundary Conditions and Eigenfunctions
Types of Equations and of Boundary Conditions
676 676
Types of Boundary Conditions. Cauchy's Problem and Characteristic Curves. Hyperbolic Equations. Cau chy Conditions and Hyperbolic Equations. Waves in Several Dimensions. Elliptic Equations and Complex Variables. Parabolic Equations.
6.2
DijJerence Equations and Boundary Conditions
692
First-order Linear D ifference Equations. Difference Equations for Several Dimensions. The Elliptic Equation and Dirichlet Conditions. E igenfun ctions. Green's Functions. The Elliptic Equation and Cauchy Conditions. The Hyperbolic Difference Equation. The Parabolic Difference Equation.
6.3
Eigenfunctions and Their Use
709
Fourier Series. The Green's Function. Eigenfunctions. Types of Boundary Conditions. Abstract Vector Space . Sturm-Liouville Problem . Degeneracy. Series of Eigenfunctions. Factorization of the Sturm-Liouville Equation. Eigenfunctions and the Variational Principle. Completeness of a Set of Eigenfunctions. Asymptotic Formulas. Compar ison with Fourier Series. The Gibbs ' Phenomenon. Generating Functions, Legendre Polynomials. E igenfunctions in Several Dimensions. Separability of Separation Constants. Density of Eigenvalues. Continuous Distribution of Eigenvalues. Eigenfunctions for the Schroedinger Equation. Discrete and Continuous Eigenvalues. Differentiation and Integration as Operators. The Eigenvalue Problem in Abstract Vector Space .
Problems Table of Useful Eigenfunctions and Their Properties Gegenbauer Polynomials. Laguerre Polynomials.
7.1
7
Green's Functions
Source Points and Boundary Points
781
Hermite Polynomials.
Eigenfunctions by the Factorization Method Bibliography CHAPTER
773
788
790 791 793
Formulation in Abstract Vector Space . Boundary Conditions and Surface Charges . A Simple Example. Relation between Volume and Surface Green 's Functions. The General Solution. Green's Functions and Generating Functions.
Contents 7.2
xv
Green's Functions for Steady Waves
803
Green 's Theorem. Green's Function for the Helmholtz Equation. Solution of the Inhomogeneous Equation. General Properties of the Green 's Function. The Effect of Boundary Conditions. Methods of Images. Series of Images. Other Expansions. Expansion of Green 's Function in Eigenfunctions. Expansions for the Infinite Domain. Polar Coordinates. A General Technique. A General Formula. Green's Functions and Eigenfunctions.
7.3
Green's Function for the Scalar Wave Equation
834
The Reciprocity Relation. Form of the Green's Function. Field of a Moving Source . Two-dimensional Solution. One-dimensional Solutions. Initial Conditions. Huygens' Principle. Boundaries in the Finite Region . Eigenfunction Expansions. Transient Motion of Circular Membrane. Klein-Gordon Equation.
7.4
Green's Function for Diffusion
857
Causality and Re cipro city. Inhomogeneous Boundary Conditions. Green's Function for Infinite Domain. Finite Boundaries. Eigenfunction Solutions. Maximum Velocity of Heat Transmission.
7.5
Green's Function in Abstract Vector Form
869
Generalization of Green's Theorem, Adjoint Operators. Effect of Boundary Conditions. More on Adjoint Differential Operators. Adjoint Integral Operators. Generalization to Abstract Vector Space. Adjoint, Conjugate, and Hermitian Operators. Green 's Function and Green's Operator. Reciprocity Relation. Expansion of Green's Operator for Hermitian Case . Non-Hermitian Operators, Biorthogonal Functions.
Problems Table of Green's Funct ions
886 890
General Properties. Green's Function for the Wave Equation. Function for the Diffusion Equation.
Bibliography CHAPTER
8.1
8
Green 's
894
Integral Equations
896
Integral Equations of Physics, Their Classification
896
Example from Acoustics. An Example from Wave Mechanics. Boundary Conditions and Integral Equations. Equations for Eigenfunctions. Eigenfunctions and Their Integral Equations. Types of Integral Equations ; Fredholm Equations. Volterra Equations.
8.2
General Properties of Integral Equations
907
Kernels of Integral Equations, Transformation to Definite Kernels. Properties of the Symmetric, Definite Kernel. Kernels and Green's Functions for the Inhomogeneous Equation. Semi-definite and Indefinite Kernels. Kernels Not Real or Definite. Volterra Integral Equation. Singular Kernels.
8.3
Solution of Fredholm Equations of the First Kind Series Solutions for Fredholm Equations. Orthogonalization. Biorthogonal Series.
925
Determining the Coefficients. Integral Equations of the
Contents
xvi
First Kind and Generating Functions. Use of Gegenbauer Polynomials. Integral Equations of the First Kind and Green's Functions. Transforms and Integral Equations of the First Kind. Differential Equations and Integral Equations of the First Kind. The Moment Problem. Recapitulation.
8.4
Solution of Integral Equations of the Second Kind
949
Expansions of the First Class . Expansions of the Second Class. Expansions of the Third Class . Other Classes . Inhomogeneous Fredholm Equation of the Second Kind.
8.5
Fourier Transforms and Integral Equations
960
The Fourier Transform and Kernels of Form v(x - xo). The Hankel Transform . The Kernel v(x - xo) in the Infinite Domain. The Homogeneous Equation. An Example. Branch Points. The Kernel v(x + xo) in the Infinite Domain. An Example. Applications of the Laplace Transform . Volterra Integral Equation, Limits (z, 00). Mellin Transform. The Method of Weiner and Hopf. Illustrations of the Method. The Milne Problem. A General Method for Factorization. Milne Problem, Continued. Inhomogeneous Weiner-Hopf Equation.
Tables of Integral Equations and Their Solutions
992
Types of Equations. Types of Kernels. Green's Function for the Inhomogeneous Equation. Solutions of Fredholm Equations of the First Kind. Solutions of Volterra Equations of the First Kind. Solutions of Fredholm Equations of the Second Kind. Solutions of Volterra Equations of the Second Kind.
996
Bibliography Index
PART II CHAPTER
9.1
9 Approximate Methods
999
Perturbation Methods
1001
The Usual Perturbation Formula. Convergence of Series. Multidimensional Problems. An Example. Feenb erg Perturbation Formula. Secular Determinant. An Example. Fredholm Perturbation Formula. An Example. Variation-Iteration Method. Convergence of the Method. An Example. Improved Perturbation Formulas. Nonorthogonal Functions.
9.2
1038
Boundary Perturbations
Perturbation of Boundary Conditions, f Small. Perturbation of Boundary Conditions, f Large. An Example. Formulas for Small w. Long, Narrow Rectangle. Perturbation of Boundary Shape. Evaluation of Integrals. Convergence. .I m proving the Convergence. Perturbation of Boundaries for Dirichlet Conditions. A Special Class of Boundary Perturbation.
9.3
Perturbation Methods for Scattering and Diffraction Boundary Conditions for Scattering.
Scattering Cross Section.
1064 Scatter-
Contents
xvii
ing from Spherically Symmetric Region-Phase Shifts. Integral Equation for Scattering. Integral Equation for One-dimensional Problem. Integral Equation for Three Dimensions. Born Approximation. Higher Born Approximations. Fredholm Series. An Example. A Threedimensional Example. Long-wavelength Approximation. Long-wavelength Approximation for the Schroedinger Equation. Convergence. Short-wavelength Approximation; WKBJ Method. Relation to the Integral Equation. Case of Well-separated Classical Turning Points. WKBJ Method for Bound Systems. Penetration through a Barrier. WKBJ Method for Radial Equations. WKBJ Phase Shifts. Case of Closely Spaced Classical Turning Points. Short-wavelength Approximation in Three Dimensions.
9.4
Variational Methods
1106
Variational Principle for Eigenvalue Problems. Variational Principles for Resonant Frequencies and Energy Levels. Vibration of a Circular Membrane. Nonlinear Variational Parameters. Rayleigh-Ritz Method. Application to Perturbation Theory. Integral Equation and Corresponding Variational Principle. An Example. Variational Principle for Phase Shifts. Variational Principle for the Phase Shift Based on an Integral Equation. Variational Principle for the Transmission Amplitude. Variational Principle for Three-dimensional Scattering Problems. Variational Principles for Surface Perturbations. Variational Principle Based on the Integral Equation for Boundary Perturbations. A Variational Principle for Radiation Problems. Variation-iteration Method. An Extrapolation Method. Lower Bounds for Xo. Comparison Method for Lower Bounds. An Example. .c Not Positive-definite. Variational Principles for the Higher Eigenvalues. Method of Minimized .Iterat ions.
Problems Tabulation of Approximate Methods
1158 1162
Bound States ; Volume Perturbation. Iterative-perturbation Series. Feenberg Series. Fredholm Formula. Variational Principles for Bound States. Variation-iteration Method. Perturbation of Boundary Condit ions. Perturbation of Boundary Shape. Perturbation Formulas for Scattering. Variational Principles for Scattering. Scattering from Spherically Symmetric Object.
CHAPTER
Bibliography
1170
10 Solutions of Laplace's and Poisson's Equations
1173
10.1 Solutions in Two Dimensions
1175
Cartesian Coordinates, Rectangular Prism Heated on One Side. Green 's Function. Polar Coordinates, Cylinders Placed in Fields. Flow of Viscous Liquids. Green's Function in Polar Coordinates. Internal Heating of Cylinders. Potential Near a Slotted Cylinder. Elliptic Coordinates. Viscous Flow through a Slit . Elliptic Cylinders in Uniform Fields . Green's Function in Bipolar Coordinates.
Contents
xviii
10.2
Complex Variables and the Two-dimensional Laplace Equation 1215 Fields, Boundary Conditions, and Analytic Functions. Some Elementary Solutions. Transformation of Solutions. Circulation and Lift. Fields Due to Distributions of Line Sources. Grid Potentials and Amplification Factors. Linear Arrays of Source Lines . Two-dimensional Array of Line Sources . Periodic Distribution of Images. Potentials about Prisms. Parallel Plate Condenser. Variable Condenser. Other Rectangular Shapes.
10.3
Solutions for Three Dimensions
1252
Integral Form for the Green's Function. Solutions in Rectangular Coordinates. Solutions in Circular Cylindrical Coordinates. Integral Representation for the Eigenfunction. Green's Function for Interior Field. Solutions for Spherical Coordinates. Fields of Charged Disks and from Currents in Wire Loops. Fields of Charged Spherical Caps . Integral Representation of Solutions. The Green's Function Expansion . Dipoles, Quadrupoles, and Multipoles. Spherical Shell with Hole . Pro late Spheroidal Coordinates. Integral Representation of Spheroidal Solutions. Green 's Function for Prolate Spheroids. Oblate Spheroids. Integral Representations and Green's Functions. Parabolic Coordinates. Bispherical Coordinates. Toroidal Coordinates. Ellipsoidal Coordinates.
Problems Trigonometric and Hyperbolic Functions
1309 1320
Trigonometric Functions. Hyperbolic Functions. Generating Functions Relating Hyperbolic and Trigonometric Functions.
Bessel Functions
1321
General Formulas Relating Bessel Functions. Series Relations. Hyperbolic Bessel Functions. Definite Integrals Involving Bessel Functions.
Legendre Functions Zonal Harmonics. Legendre Functions of the Second Kind. of Imaginary Argument. Toroidal Harmonics.
Bibliography CHAPTER
11 The Wave Equation
11.1 Wave Motion on One Space Dimension
1325 Functions
1330
1331 1332
Fourier Transforms. String with Friction. Laplace Transform. String with Friction. String with Elastic Support. String with Nonrigid Supports. Reflection from a Frictional Support. Sound Waves in a Tube. Tube with a Varying Cross Section . Acoustic Circuit Elements. Freewave Representations. Movable Supports.
11.2
Waves in Two Dimensions
1360
Fourier Transforms and Green 's Functions. Rectangular Coordinates. Other Boundary Conditions. Variable Boundary Admittance. Polar Coordinates. Waves inside a Circular Boundary. Radiation from a Circular Boundary.• Scattering of Plane Wave from Cylinder. Scattered and Reflected Wave . Short and Long Wavelength Limits. Scattering of
Contents
xix
Plane Wave from Knife Edge. Fresnel Diffraction from Knife Edge. Scattering from a Cylinder with Slit. Slotted Cylinder, Neumann Conditions. Waves in Parabolic Coordinates. Eigenfunctions for Interior Problems. Waves outside Parabolic Boundaries. Green's Function and Plane Wave Expansions. Elliptic Coordinates. The Radial Solutions. Approximations for Small Values of hand m. Approximations for h Small and m Large . Expansions for h Large . Waves inside an Elliptic Boundary. Green's Functions and Plane Wave Expansions. Radiation from a Vibrating Strip. Radiation from a Strip Carrying Current. Scattering of Waves from Strips. Diffraction through a Slit, Babinet's Principle.
11.3
Wav,es in Three Space Dimensions
1432
Green's Function for Free Spa ce. Rectangular Enclosure. D istortion of Standing Wave by Strip. Computing the Eigenvalue. Transmission through Du cts . Acoustic Transients in a Rectangular Duct. Constriction in Rectangular Duct. Wave Transmission around Corner. Membrane in Circular Pipe . Radiation from Tube Termination. Transmission in Elastic Tubes. Spherical Coordinates. Spherical Bessel Functions. Green's Function and Plane Wave Expansion. Waves inside a Sphere. Vibrations of a Hollow, Flexible Sphere. Vibrating String in Sphere. Resonance Frequencies of the System. Radiation from Sphere . Dipole Source . Radiation from Collection of Sources. Radiation from Piston in Sphere. Scattering of Plane Wave from Sphere. Scattering from Sphere with Complex Index of Refraction. Scattering from Helmholtz Resonator. Scattering from Ensemble of Scatterers. Scattering of Sound from Air Bubbles in Water. Spheroidal Coordinates. The Radial Functions. Green 's Function and Other Expansions. Oblate Spheroidal Coordinates.
11.4
Integral and Variational Techniques
1513
Iris Diaphragm in Pipe . A Variational Principle. Calculating the Transmission Factor. Hole in Infinite Plane. Reflection in Lined Duct. Fourier Transform of the Integral Equation. Factoring the Transformed Equation. Radiation from the End of a Circular Pipe . Formulas for Power Radiated and Reflected . The Fourier Transform of the Integral Equation. Factoring the Transformed Equation. Radiation from Vibrating Source. Angle Distribution of Radiated Wave . Applying the Boundary Conditions. Scattering of Waves, Variational Principle. Angle-distribution Function and Total Cross Section . Scattering from Spheres. Scattering from a Strip. Scattering of Short Waves.
Problems Cylindrical Bessel Fun ctions Amplitudes and Phase Angles.
Weber Fun ctions
1555 1563
Asymptotic Values, Roots.
1565
Addition Theorems.
Mathieu Functions
1568
Eigenfunction Solutions. Corresponding Radial Solutions. Second Solutions. Series Expansions. Amplitudes and Phase Angles.
Contents
xx
1573
Spherical Bessel Functions Series Expansions. Definite Integrals. Asymptotic Values. Roots.
Amplitude and Phase Angles .
1576
Spheroidal Functions Angle Functions.
Radial Functions.
AdditionTheorems,
Short Table of Laplace Transforms Bibliography CHAPTF;R
12.1
12
Diffusion, Wave Mechanics
Solutions of the Diffusion Equation
1579 1582 1584 1584
Transient Surface Heating of a Slab . Green's Functions and Image Sources. Radiative Heating. Transient Internal Heating of a Slab . Diffusion and Absorption of Particles. Fission and Diffusion. Laplace Transform Solution. The Slowing-down of Particles. Fission, Diffusion, and Slowing-down. General Case, Diffusion Approximation. Heating of a Sphere.
12.2
Distribution Functions for Diffusion Problems
1606
Uniform Space Distribution. Approximations for Forward Scattering. General Considerations, Steady-state Case . Integral Relations between the Variables. Calculating the Diffuse Scattering. Calculating the Diffuse Emission. Solution by Laplace Transforms. The Variational Calculation of Density. Loss of Energy on Collision. Uniform Space Distribution. Age Theory.
12.3
Solutions of Schroedinger's Equation
1638
Definitions. The Harmonic Oscillator. Short-range Forces. The Effect of a Perturbation. Approximate Formulas for General Perturbation . Momentum Wave Functions. Bound and Free States. Existence of Bound States. Reflection and Transmission. Penetration through a Potential Barrier. Central Force Fields, Angular Momentum. Central Force Fields, the Radial Equation. Coulomb Potential. Inverse Cube Force. Coulomb Field in Parabolic Coordinates. Rutherford Scattering . Other Soluble Central Force Systems. Perturbations of Degenerate Systems. The Stark Effect. Momentum Eigenfunctions. Scattering from Central Fields. Ramsauer and Other Resonance Effects. Approximate Calculations for Slow Incident Particles. The Born Approximation. Phase Angles by Born Approximation. The Variational Method . Vari at ion-iteration Method. Variational Principles for Scattering. Variationiteration Method for Scattering. Variational Principle for the Angle Distribution. Two Particles, One-dimensional Case . The Green's Function . Surface Waves. The Effects of Interaction between the Particles. Meaning of the Results. Coupled Harmonic Oscillators. Central Force Fields, Several Particles, Angular Momentum. Inversion and Parity. Symmetrizing for Two-particle Systems. Bound, Free, and "Surface" States. Green's Function for the Two-particle System. Bound States. Variational Calculation. Scattering of Electron from Hydrogen Atom . Elastic and Inelastic Scattering. Exchange of Particles. Recapitulation.
Contents
xxi
Problems Jacobi Polynomials General Relationships.
1745 1754 Special Cases .
1756 1757
Semi-cylindrical Functions Bibliography CHAPTER
13.1
13 Vector Fields
1759
Vector Boundary Conditions, Eigenfunctions and Green's Functions
1762
The Transverse Field in Curvilinear Coordinates. Vector Green's Theorems. The Green 's Function a Dyadic. Vector Eigenfunctions. Green's Function for the Vector Helmholtz Equation. Longitudinal and Transverse Green 's Dyadics. Green's Dyadic for Elastic Waves. Solutions of the Vector Laplace Equation. Green's Functions for the Wave Equation .
13.2
Static and Steady-state Solutions
1791
Two-dimensional Solutions. Polar Coordinates. Circular Cylinder Coordinates. Spherical Coordinates. Green 's Dyadic for Vector Laplace Equation. Magnetic Field Solutions. Viscous Flow about a Sphere. Elastic Distortion of a Sphere. Tractions Specified on Surface.
13.3
Vector Wave Solutions
1812
Reflection of Plane Waves from a Plane. Boundary Impedance. Elastic Wave Reflection. Waves in a Duct. Th~ Green 's Function. Generation of Waves by Wire . Losses along the Duct Walls . Reflection of Waves from End of Tube. Effect of Change of Duct Size. Reflection from Wire across Tube. Elastic Waves along a Bar. Torsional Forced Motions of a Rod. Nonstationary Viscous Flow in Tube. Ele ctromagnetic Resonators. Energy Loss at Walls, the Q of the Cavity. Excitation of Resonator by Driving Current. Excitation by Wave Guide. Resonance Frequencies of a Klystron Cavity. Scattering from Cylinders. Spherical Waves. Radiation from Dipoles and Multipoles. Standing Waves in a Spherical Enclosure. Vibrations of an Elastic Sphere. The Green's Function for Free Space. Radiation from a Vibrating Current Distribution. Radiation from a Half-wave Antenna. Radiation from a Current Loop . Scattering from a Sphere. Energy Absorption by the St;>here. Distortion of Field by Small Object. Recapitulation.
Problems Table of Spherical Vector Harmonics Special Cases . Relations.
Zonal
Bibliography Appendix GLOSSARY OF SYMBOLS USED
Vector Harmonics.
1891 1898 Integral and Differential
1901 1903 1904
Contents
xxii
1913
TABLES I.
II. III.
IV.
v. VI. VII. VIII. IX.
x. XI. XII. XIII . XIV.
xv. XVI. XVII .
Trigonometric and Hyperbolic Functions. Trigonometric and Hyperbolic Functions. Hyperbolic Tangent of Complex Quantity. Inverse Hyperbolic Tangent of Complex Quantity. Logarithmic and Inverse Hyperbolic Functions. Spherical Harmonic Functions. Legendre Functions for Large Arguments. Legendre Functions for Imaginary Arguments. Legendre Functions of Half-integral Degree. Bessel Functions for Cylindrical Coordinates. Hyperbolic Bessel Functions. Bessel Functions for Spherical Coordinates. Legendre Functions for Spherical Coordinates. Amplitudes and Phases for Cylindrical Bessel Functions. Amplitudes and Phases for Spherical Bessel Functions. Periodic Mathieu Functions. Normalizing Constants for Periodic Mathieu Functions Limiting Values of Radial Mathieu Functions.
1913 1914 1915 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1930 1934 . and
1937
BIBLIOGRAPHY
1938
Index
1939
CHAPTER 1
Types of Fields
Our task in this book is to discuss the mathematical techniques which are useful in the calculation and analysis of the various types of fields occurring in modern physical theory. Emphasis will be given primarily to the exposition of the interrelation between the equations and the physical properties of the fields, and at times details of mathematical rigor will be sacrificed when it might interfere with the clarification of the physical background. Mathematical rigor is important and cannot be neglected, but the theoretical physicist must first gain a thorough understanding of the physical implications of the' symbolic tools he is using before formal rigor can be of help. Other volumes are available which provide the rigor; this book will have fulfilled it's purpose if it provides physical insight into the manifold field equations which occur in modern theory, together with a comprehension of the physical meaning behind the various mathematical techniques employed for their solution. This first chapter will discuss the general properties of various fields and how these fields can be expressed in terms of various coordinate systems. The second chapter discusses the various types of partial differential equations which govern fields, and the third chapter treats of the relation between these equations and the fundamental variational principles developed by Hamilton and others for classic dynamics. The following few chapters will discuss the general mathematical tools which are needed to solve these equations, and the remainder of the work will be concerned with the detailed solution of individual equations. Practically all of modern physics deals with fields: potential fields, probability fields, electromagnetic fields, tensor fields, and spinor fields. Mathematically speaking, a field is a set of functions of the coordinates of a point in space . From the point of view of this book a field is some convenient mathematical idealization of a physical situation in which extension is an essential element, i.e., which cannot be analyzed in terms of the positions of a finite number of particles. The transverse displacement from equilibrium of a string under static forces is a very simple example of a one-dimensional field; the displacement y is different for 1
2
Types of Fields
[CR. 1
different parts of the string, so that y can be considered as a function of the distance x along the string. The density, temperature, and pressure in a fluid transmitting sound waves can be considered as functions of the three coordinates and of time. Fields of this sort are obviously only approximate idealizations of the physical situation, for they take no account of the atomic properties of matter. We might call them material fields.
Other fields are constructs to enable us to analyze the problem of action at a distance, in which the relative motion and position of one body affects that of another. Potential and force fields, electromagnetic and gravitational fields are examples of this type. They are considered as being "caused" by some piece of matter, and their value at some point in space is a measure of the effect of this piece of matter on some test body at the point in question. It has recently become apparent that many of these fields are also only approximate idealizations of the actual physical situation, since they take no account of various quantum rules associated with matter. In some cases the theory can be modified so as to take the quantum rules into account in a more or less satisfactory way. Finally, fields can be constructed to "explain" the quantum rules. Examples of these are the Schroedinger wave function and the spin or fields associated with the Dirac electron. In many cases the value of such a field at a point is closely related to a probability. For instance the square of the Schroedinger wave function is a measure of the probability that the elementary particle is present. Present quantum field theories suffer from many fundamental difficulties and so constitute one of the frontiers of theoretical physics. In most cases considered in this book fields are solutions of partial differential equations, usually second-order, linear equations, either homogeneous or inhomogeneous. The actual physical situation has often to be simplified for this to be so, and the simplification can be justified on various pragmatic grounds. For instance, only the" smoothedout density" of a gas is a solution of the wave equation, but this is usually sufficient for a study of sound waves, and the much more tedious calculation of the actual motions of the gas molecules would not add much to our knowledge of sound: This Procrustean tendency to force the physical situation to fit the requirements of a partial differential equation results in a field which is both more regular and more irregular than the" actual" conditions. A solution of a differential equation is more smoothly continuous over most of space and time than is the corresponding physical situation, but it usually is also provided with a finite number of mathematical discontinuities which are considerably more" sharp" than the" actual" conditions exhibit. If the simplification has not been too drastic, most
Types of Fields
3
of the quantities which can be computed from the field will correspond fairly closely to the measured values. In each case, however, certain discrepancies between calculated and measured values will turn up, due either to the "oversmooth" behavior of the field over most of its extent or to the presence of mathematical discontinuities and infinities in the computed field, which are not present in "real life." Sometimes these discrepancies are trivial, in that the inclusion of additional complexities in the computation of the field to obtain a better correlation with experiment will involve no conceptual modification in the theory; sometimes the discrepancies are far from trivial, and a modification of the theory to improve the correlation involves fundamental changes in concept and definitions. An important task of the theoretical physicist lies in distinguishing between trivial and nontrivial discrepancies between theory and experiment. One indication that' fields are often simplifications of physical reality is that fields often can be defined in terms of a limiting ratio of some sort. The density field of a fluid which is transmitting a sound wave is defined in terms of the "density at a given point," which is really the limiting ratio between the mass of fluid in a volume .surrounding the given point and the magnitude of the volume, as this volume is reduced to "zero." The electric intensity" at a point" is the limiting ratio between the force on a test charge at the point and the magnitude of the test charge as this magnitude goes to "zero." The value of the square of the Schroedinger wave function is the limiting ratio between the probability that the particle is in a given volume surrounding a point and the magnitude of the volume as the volume is shrunk to "zero," and so on. A careful definition of the displacement of a "point" of a vibrating string would -also utilize the notion of limiting ratio. These mathematical platitudes are stressed here because the technique of the limiting ratio must be used with caution when defining and calculating fields. In other words, the terms" zero" in the previous paragraph must be carefully defined in order to achieve results which correspond to "reality." For instance the volume which serves to define the density field for a fluid must be reduced several orders of magnitude smaller than the cube of the shortest wavelength of transmitted sound in order to arrive at a ratio which is a reasonably accurate solution of the wave equation. On the other hand, this volume must not be reduced to a size commensurate with atomic dimensions, or the resulting ratio will lose its desirable properties of smooth continuity and would not be a useful construct. As soon as this limitation is realized, it is not difficult to understand that the description of a sound wave in terms of a field which is a solution of the wave equation is likely to become inadequate if the "wavelength " becomes shorter than interatomic distances. In a similar manner we define. the electric field in terms of a test
4
Types of Fields
[cH.l
charge which is made small enough so that it will not affect the distribution of the charges" causing" the field. But if the magnitude of the test charge is reduced until it is the same order of magnitude as the electronic charge, we might expect the essential atomicity of charge to involve us in difficulties (although this is not necessarily so). In some cases the limiting ratio can be carried to magnitudes as small as we wish. The probability fields of wave mechanics are as "finegrained" as we can imagine at present.
1.1 Scalar Fields When the field under consideration turns out to be a simple number, a single function of space and time, we call it a scalar field. The displacement of a string or a membrane from equilibrium is a scalar field. The density, pressure, and temperature of a fluid, given in terms of the sort of limiting ratio discussed previously, are scalar fields. As mentioned earlier, the limiting volume cannot be allowed to become as small as atomic dimensions in computing the ratio, for the concepts of density, pressure, etc ., have little meaning for individual molecules. The ratios which define these fields must approach a "macroscopic limit" when the volume is small compared with the gross dimensions of the fluid but is still large compared with atomic size; otherwise there can be no physical meaning to the concept of scalar field here . All these scalar fields have the property of invariance under a transformation of space coordinates (we shall discuss invariance under a space-time transformation later in this chapter). The numerical value of the field at a point is the same no matter how the coordinates of the point are expressed. The form of mathematical expression for the field may vary with the coordinates. For instance, a field expressed in rectangular coordinates may have the form if; = y; in spherical coordinates it has the different form if; = r sin -a cos 'P, but in eith er coordinate system, at the point x = 10, y = 10, Z = 0 (r = y'2OO, -a = 45°, 'P = 0) it has the value if; = 10. This is to be contrasted to the behavior of the x component of the velocity of a fluid, where the direction of the x axis may change as the coordinates are changed. Therefore, the numerical value of the x component at a given point will change as the direction of the x axis is rotated. This property of invariance of a scalar will be important in later discussions and is to be contrastedalternatively to the invariance of form of certain equations under certain coordinate transformations. For some of the scalar fields mentioned here , such as density or temperature or electric potential, the definition of the field has been such as to make the property of invariance almost tautological. This is not always the
Scalar Fields
§l.l]
5
case with less familiar fields, however. In some cases, in fact, the property of invariance must be used as a touchstone to find the proper expression for the field. Isotimic Surfaces. The surfaces defined by the equation y; = constant, where y; is the scalar field, may be called isotimic surfaces (from Greek isotimos, of equal worth) . Isotimic surfaces are the obvious generalizations of the contour lines on a topographic map . In potential theory they are referred to as equipotentials; in heat conduction they are isothermals ; etc . They form a family of nonintersecting surfaces, p.=.8,
8=0' Z 8=30'
I
p.=0~ 8=90'
Fig. 1.1
Examples of isotimic surfaces, /.l = constant , where c cosh /.l = ! ,,/(r + C)2 + Z2 + ! V(r - C)2 + Z 2, of 8 = constant, where c cos 8 = ! V (r + C)2 + Z2 - ! V (r - C)2 + Z2, and of'" = a, where tan", = y/x .
which are often useful in forming part of a system of coordinates naturally suitable for the problem. For instance if the field is the well-known potential y; = (x 2 + y2 + z2)-i, the isotimic surfaces (in this case surfaces of constant potential) are concentric spheres of radius r = yx 2 + y2 + Z2 = constant; and the natural coordinates for the problem are the spherical ones, r, tJ, =.5,8=0
°
a positive definite quantity. The " lengt h" of e (written [el) will be taken as ye*. e. Once we have made these definitions, a number of important consequences may be derived. It follows, for example, from them that, if e* . e = 0, then lei = O. It may be easily verified that, if e and fare two vectors, e*·f
=~
Vectors in ordinary space have the property that their dot product is never larger than the product of their amplitudes :
AB 2:: A· B
82
Types of Fields
[cH.l
This is called the Schwarz inequality; it must also be true in abstract vector space, in generalized form, since it follows from the fact that the square of the "length" of a vector is never less than zero, i.e., is positive definite. For instance, if e and f are two vectors and a and b two complex numbers, (ae* - 6f*) . (ae - bf) ;:::: 0 By setting a = ' V(f* . f)(e* . f) and b = v(e* ' e)(f*· e) this can be reduced to v(e* . e)(f* . f) ;:::: V(f* . e)(e* . f)
or
lei · IfI ;:::: If* . e]
(1.6.33)
which is the generalized form of the Schwarz inequality we have been seeking. This inequality will be used later in deriving the Heisenberg uncertainty principle. Another inequality which has its analogue in ordinary vector space is the Bessel inequality; which states in essence that the sum of the lengths of two sides of a triangle is never less than the length of the third side :
lei + IfI ;:::: I(e + f)1
(1.6.34)
Generalized Dyadics. We now turn to the discussion of operators in complex vector spaces. We consider first the linear transformation which rotates the axes of the space into another set of orthogonal axes :
where e~ is a unit vector along the ith axis of the new coordinate system and en are the unit vectors in the original system. The relation between e~ and (e~)* must be the same as that for any two vectors in the unprimed system so that
\' 'Ynien - * (e;')* = ~ n
We can devise a dyadic operator which when operating on the vector en converts it into the new vector e~ . This dyadic is
Len ' ®ei = I ®=
'Yni •
e,
(1.6.35)
n,i
so that
en'Yni
= e~
n
The form of the dyadic implies the convention that we shall always place unstarred vectors to the right of an operator and starred vectors to the left. A useful property of the dyadic operator is that (e*®)f = e*(®f)
§1.6]
Dyadics and Other Operators
as
so that no parentheses are actually needed and we shall usually write it e*®f. Our generalizations have been carried out so that, as with ordinary dyadics, products of operators are also operators :
so that the (n,k) component of the operator (® . \I) is (®\I)nk =
[L 'YnjAjk] j
Returning now to the rotation operator ® defined in Eq. (1.6.33), we note that et® does not give (eD*. The proper operator to rotate e: shall be written ®* and defined so that e:®* =
(e~)*
The operator ®* is therefore related to the operator ® by the equation (®ei)* = e:®* An operator ®* which is related to an operator ® according to this equation is called the Hermitian adjoint of ®. Writing out both sides of the equation we obtain
Le~'Yni = L('Y*)ine~ n
n
so that
(1.6.36)
This equation states that the (i,n) component of the Hermitian adjoint dyadic ®* is obtained by taking the complex conjugate of the (n,i) component of ®. The notion of adjoint is the generalization ·of the conjugate dyadic defined on page 55. The Hermitian adjoint of a product of two operators ®\I is (®\I)* = \I*®*. The Hermitian adjoint of a complex number times an operator is the complex conjugate of the number times the adjoint of the operator. Hermitian Operators. An operator which is self-adjoint, so that ® = ®* or 'Ynm = 'Ymn is called a Hermitian operator. All classical symmetric dyadic operators are Hermitian, since their components are all real numbers. The operators in quantum mechanics which correspond to measurable quantities must also be Hermitian, for their eigenvalues must be real (after all, the results of actual measurements are real numbers) . To prove this we note that, if the eigenvalues an of an operator m: are real, then e*m:e is real for any e. For e can be expanded in terms of the eigenvectors
84
Types of Fields
[CR. 1
en, giving a series of real numbers for e*~e. Let e = f + bg; then + b(f*~g) is real. But this is possible only if (g*~f) is the complex conjugate of (f*~g) , that is, is (g*~*f) . Consequently g*(~l ~*)f = 0 for any f or g ; therefore ~ = ~*. The rotation operator ® defined in Eq. (1.6.35) is still more narrowly specialized. The components 'Ymn are direction cosines, so that one would expect the operator to have a "unit amplitude" somehow. Since a rotation of coordinates in vector space should not change the value of a scalar product, we should have b(g*m)
(e*®*) . (®f) = e* . f
for e and f arbitrary state vectors. ®*®
where 3' is the idemfactor
Lene:.
Therefore, it must be that
= 3' This implies that the adjoint of ®
n
is also its inverse : Such an operator, having" unit amplitude" as defined by these equations, is called a unitary operator. The rotation operator defined in Eq . (1.6.35) is a unitary operator [see also Eq. (1.6.12)). Most of the operators encountered in quantum mechanics are Hermitian ; their eigenvalues are real, but their effect on state vectors is both to rotate and to change in size. There are in addition several useful operators which are unitary; their eigenvalues will not all be real, but their effect on state vectors is to rotate without change of size. Examples of Unitary Operators. Important examples of unitary operators occur in wave propagation, quantum mechanics, and kinetic theory. For example, the description of the manner in which a junction between two wave guides (say of different cross section) reflects waves traveling in the ducts may be made by using a reflectance dyadic. This operator rotates, in the related abstract space, the eigenvector corresponding to the incident wave into a vector corresponding to the reflected wave . The unitary condition corresponds essentially to the requirement that the process be a conservative one. The equation ®* = ®-l may be related to the reciprocity theorem, i .e., to the reversi bility between source and detector. We shall, of course, go into these matters in greater detail in later chapters. A unitary operator may be constructed from an ordinary Hermitian operator ~ as follows:
+ i~) /(l - i~) + i~) so that ®®* is 3' and ® is unitary.
® = (1
The adjoint of ® is (1 - i~) /(l If ® is the reflectance dyadic, then the above equation yields the relation
§1.6]
Dyadics and Other Operators
85
which exists between the reflection coefficients constituting @ and the impedance coefficients constituting the impedance dyadic ~ . Another construction using a Hermitian operator ~ to obtain a unitary operator @ is @
=
ei~
(1.6.37)
where ei~ is defined by its power series 1 + i~ + -Hi~)2 + . . . . For example, for @ = (1 + i~) /(l - i~) , we have @ = ei~ , where .~ = 2 tan ? ~ . In terms of the physical example above, this last transformation corresponds to using the shift in phase upon reflection to describe the reflected wave rather than the reflection coefficient itself. Often a vector is a function of a parameter (we shall take time t as the typical parameter), the vector rotating as the parameter changes its value. The unitary operator corresponding to this rotation is useful in many problems. Its general form is not difficult to derive. We can call it 1)(t). Then by our definition 1)(tI)e(to)
=
e(t i
+
to)
where t l and to are specific values of the parameter t .
+ +
so that
1)(t2)1)(tI)e(tO) = e(t i to t2) 1)(t2)1)(th = 1)(t i
=
1)(ti
+ t 2)
Moreover,
+ t 2)e(t O)
For this equation to hold for all t2 and ts, 1) must be an exponential function of t. Since it must be unitary, it must be of the form 1) =
e~t
where S) is some unknown Hermitian operator. To determine the equation of rotational motion for vector e we apply 1) to e(t), changing t by an infinitesimal amount dt. Then 1) = 1 + is) dt so that . 1 ae (1 + S) dt)e(t) = e(t + dt) or S)e(t) = t at (1.6.38) We must, of course, find S) from the physics of the problem. For example, in quantum mechanics and where t is the time, S) is proportional to the Hamiltonian fun ction with the usual classical variables of momentum and position replaced by their corresponding operators. Transformation of Operators. In the preceding discussion we have dealt with the rotation of a vector as a parameter changes upon which it depends. It is also possible to simulate the effects of the changes in the parameter by keeping the vector fixed and rotate" space," i .e., to ascribe the appropriate parametric dependence to all the operators which occur. In other words, we change the meaning of the operators as the parameter changes, keeping the vectors fixed. We must find how this is to be done so as to give results equivalent to the first picture in 'which the vectors
86
Types of Fields
[CH. 1
rotated but the operators were fixed. Let the rotation of the vectors be given by the unitary operator ®: ®e = e' :
®f = f"; etc.
The appropriate equivalent change in ~ to ~/ is found by requiring that the relation between ~e/ and f' (first picture e and f change, ~ does not) be the same as the relation between ~/e and f (second picture e and f do not change, ~ transforms to ~/) . Analytically we write f'* .
~
• e' = f* .
•e
~/
Inserting the relation between f' and f and recalling that ®* = ®-I, we find f/* . ~ • e' = f* • (®-l~®) • e or ~/ = ®-l~® (1.6.39) We now investigate the effects of this unitary (sometimes called of~. For example, we shall first show that the eigenvalues of ~/ and ~ are the same . Let
canonical) transformation on the properties
= L'e . e = L'«
~/e
Hence
(®-l~®)
Multiplying through by ® we obtain ~(®e)
= L'(®e)
In words if e is an eigenvector of ~/ with eigenvalue L' , then ®e is an eigenvalue of ~ with the same eigenvalue. This preservation of eigenvalue is sometimes a very useful property, for if ~ is a difficult operator to manipulate, it may be possible to find a transformation ® which yields a simpler operator ~/ whose eigenvalues are the same as those of ~ . Because of the relation between an operator and its Hermitian adjoint, we can write (®-l~®)*
=
®*~*(®-l)*
=
®-l~ *®
Hence a Hermitian operator remains Hermitian after a unitary transformation. The relationship between operators is not changed by a unitary transformation. For example, if ~IDC
then
®-I~®®-IIDC®
=
in
= ®-lin®
or
~/IDC'
= in/
Two unitary transformations applied, one after the other, correspond to a transformation which is also unitary. If the two unitary operators are jJ®, then (jJ®) *jJ® = (®*jJ*)jJ® =
proving the point in question.
3
Dyadics and Other Operators
§1.6]
87
Finally we may consider the changes in an operator ~ under transformation by the unitary operator m(t) defined in Eq. (1.6.38). This will enable us to calculate the change in an operator due to a change in a parameter t. The transformed operator is e-i~ t~ei~ t
=
~'
It is clearly convenient to introduce the notation ~ = ~(O),
~(t)
for
~',
calling
so that ~(t)
=
e--i~t~(O) ei~t
The rate of change of ~ with t can be obtained by considering the difference ~(t + dt) - ~(t) = e"':i~dt~ ei~dt so that
~ :; = ~S)
-
S)~
(1.6.40)
Of course S) depends on the physics of the problem. For example, in many quantum mechanical problems S) is the Hamiltonian operator and t is time; the resulting equation is called the Heisenberg equation of motion. The equation has wide application; for an operator S) which is related to a rotation parameter t for state vectors of the form given in Eq. (1.6.38), the rate of change of any other operator ~ related to the same system is proportional to the commutator S)~ - ~S) . Quantum Mechanical Operators. Let us now apply some of these results to the operators occurring in quantum mechanics. We recall that the average result of measurements of a quantity (energy, momentum, etc.) represented by the operator p on a system in a state represented by the state vector e is pay = (e*pe) . Likewise, we point out again that the probability that a system in a state represented by the state vector e (in state e, for short) turns out to be also in state e' is given by the square of the absolute magnitude of the cosine between the state vectors, I(e*· e')12. Last section's discussion makes it possible to restate this last sentence in terms of unitary operators. Suppose that we find a unitary operator g which transforms an eigenvector e(a n ) for operator ~ with eigenvalue an to an eigenvector e'(b m ) for operator 58 with eigenvalue bn: g(a,b)e(a n)
= e'(b m )
Then, using Eq. (1.6.35), we see that the probability that a measurement of 58 gives the value b.; when ~ has the value an is le*(an )
•
e' (bm ) 12 = le*(an)g(a,b)e(anW = l'Ynn(a,b) 12
(1.6.41)
At this point we can utilize the Schwarz inequality [Eq. (1.6.33)] to show the relation between the quantum equations (1.6.32) and the Heisenberg un?ertainty principle. We have already shown that, if two
88
Types of Fields
[cnv l
operators do not commute, they cannot be simultaneously measurable; if one is measured accurately, the other cannot also be known accurately. From the physical point of view there is something inherent in the measurement of one which destroys our simultaneous accurate knowledge of the other. From the mathematical point of view, there is a reciprocal relationship between their uncertainties. As mentioned before, the uncertainty in measurement t.A of an operator m: is the root-mean-square deviation defined by the equations (t.A)2
=
e*[m: - a]2e;
a
=
e*m:e
for the state denoted by the vector e. To apply these general statements to the quantum relations (1.6.32), we define the operators
so that the rms value of the operator ~~ is the uncertainty in measurement of x in state e. By use of some of Eqs. (1.6.32), we can show that ~\lx ~~ -
~~ ~\l
= hj i
Taking the average value of this for state e, we have (~\lxe)* . (~~e) -
(~~e)* . (~\lxe)
= hji
The right-hand side of this equation is just twice the imaginary part of (~\lxe)*· (~~e), so that we can conclude that I(~\lxe)* · (~~e)1
?:: hj2
But the Schwarz inequality (1.6.33) says that the product of the amplitudes is never smaller than the amplitude of the dot product. Therefore, we finally obtain
which is the famous Heisenberg uncertainty principle. It states that simultaneous accurate measurement of position and momentum (in the same coordinate) is not possible, and that if simultaneous measurement is attempted, the resultant uncertainties in the results are limited by the Heisenberg principle. Spin Operators. The statements made in the previous section were exceedingly general, and there is need for an example to clarify some of the concepts. An example which will be useful in later discussions is that of electron spin . It is experimentally observed that there are only two allowed values of the component of magnetic moment of the electron in any given direction, which leads us to expect that the angular momentum of the electron is similarly limited in allowed values. The angular
89
Dyadics and Other Operators
§1.6]
momentum ID1 of a particle about the origin of coordinates is given in terms of its position from the origin and its momentum ~ as follows :
We next compute the commutator for the components of ID1, utilizing the quantum equations (1.6.32) :
ID1"ID1 y - ID1 yID1" = (~z3 - 3~z) (~~ y - ~~,,) = i hID1z; ID1 yID1z - ID1zID1 y = ihID1,, ; ID1zID1" -ID1"ID1z = ihID1 y
(1.6.42)
These equations indicate that we cannot know accurately all three components of the electron spin ; in fact if we know the value of M', accurately, we cannot know the values of M" and My. By utilizing these equations in pairs we can show that or and
ID1z(ID1" ± iID1 y) - (ID1" ± iID1 y)ID1 z = ± h(ID1" ID1z(IDe" + iID1 y) = (ID1" + iID1 y)(ID1z + h) ID1z(ID1" - iID1 y) = (ID1" - iID1 y)(ID1 z - h)
± iID1 y) (1.6.43)
Starting with a state vector am corresponding to a knowledge that the value of ID1 z is mh (that is, am is an eigenvector for ID1 z with eigenvalue mh) , we see that the vector (IDe" + iID1l1)am is an eigenvector for ID1 z with eigenvalue (m + l)h unless (ID1" + iID1l1)a mis zero and that (ID1" - i9,nIl)am is an eigenvector for ID1 z with eigenvalue (m - l)h unless (ID1" - i ID1y )a m is zero , for from Eqs. (1.6.43),
ID1z(ID1"
+ i9.ny)a
m
= (ID1" + iID1 y)[(ID1 z + h)am] = (m + l)h[(ID1" + iID1y)am] ; etc.
In the special case of electron spin, we will call the angular momentum E instead of ID1. In order to have only two allowed values of m, differing by h (as the above equations require) and to have these values symmetric with respect to direction along x, we must have the allowed values of ~z be +~h (with eigenvector a+) and - i h (with eigenvector a_), and that
Consequently the rules of operation of the spin operators on the two eigenvectors and the rules of multiplication of the spin operators are @5,,~
@5,.a@5,,~y @5z~"
= (hj2)a_; . @5y~ = (ih j2)a_; ~za+ = (hj2)~ = (~j2)a+; ~ya_ = -(ihj2)~ ; ~za- = -(hj2)a_ (1.6.44) = (th j2)@5z = -@5y@5,,; ~y@5z = (th j2)@5" = -@5z@5y = (ihj2)~y = -@5,,~.; (~,,)2 = (@5y )2 = (@5.)2 = (hj2)2
We have here a fairly simple generalization of the concept of operators. The" spin space" is a two-dimensional space, one direction corresponding
90
Types of Fields
[CR. 1
to a state in which the z component of spin is certainly h/2 and the perpendicular direction corresponding to a state where @5. is certainly - (h/2) (it should be clear by now that state space is a useful mathematical fiction which has nothing to do with "real" space) . A unit state vector in an intermediate direction corresponds to a state where we will find @5. sometimes positive and sometimes negative, the relative frequency of the two results depending on the square of the cosine of the angle between the unit vector and the two principal axes for @5 aThe operator 2@5./h reflects any state vector in the 450 line, i .e., interchanges plus and minus components. Therefore, the eigenvectors for @5. would be (l /v2)(a+ ± a.} , corresponding to a rotation by 450 of principal axes in spin space for a rotation of 900 of the quantizing direction in actual space (corresponding to changing the direction of the magnetic field which orients the electron from the z to the x axis, for instance). Therefore, if we know that @5. is h/2 , there is an even chance that @5. will have the value h/2 or - (h/2). On the other hand, the two eigenvectors for @5y are (1/V2)(8.+ ± ia_) , corresponding to an imaginary rotation in spin space (the square of the magnitude of a complex vector of this sort, we remember, is the scalar product of the complex conjugate of the vector with the vector itself). Quatemions. Viewed in the abstract, the operators i = 2@5./ih , i = 2@5y/ih, f = 2@5./ih, together with the unity operator 3a = a and a zero operator, exhibit a close formal relation to dyadic operators. They act on a vector (in this case a vector in two-dimensional spin space) to produce another vector. They can be added and multiplied to produce other operators of the same class. The multiplication table for these new operators is i 2 = i2 = f2 = -1 ; ii = f = -ji;
if = i = -fj ; fi = j = -if (1.6.45)
Curiously enough , operators with just these properties were studied by Hamilton, long before the development of quantum mechanics, in his efforts to generalize complex numbers. As was discussed on page 73, the quantity i = V -1 can be considered to correspond to the operation of rotation by 900 in the complex plane . For this simple operator, the only multiplication table is i 2 = -1, where 1 is the unity operator. Corresponding to any point (x,y) on the complex plane is a complex number z = x + iy, where the square of the distance from the origin is Izl2 = zz = (x - i y)(x + iy) = x 2 + y2. In order to reach the next comparable formal generalization one must use the three quantities i, i, and f defined in Eqs. (1.6.45) to produce a quaternion (see page 74) p= a
+ bi + cj + df
Dyadics and Other Operators
§1.6}
91
The square of the magnitude of this quantity is obtained by multiplying p by its conjugate
p*
= a - bi - cj - df;
Ipl2
=
pp*
= a2
+b +c +d 2
2
2
Therefore, the only quaternion with zero magnitude is one where In addition, the reciprocal to any quaternion is quickly found.
a, b, c, and d are all zero.
lip = p*/IW A quaternion can be related to the operation of rotation about an axis in three-dimensional space; the direction cosines of the axis are proportional to the constants b, c, and d, and the amount of the rotation is determined by the ratio of a 2 to b2 + c 2 + d 2 • Little further needs to be said about these quantities beyond what has been given on Ipage 74, et seq., for their interest is now chiefly historical. The spin operators are, however, closely related to them. Rotation Operators. Unitary operators of the type defined in Eq. (1.6.37) are very closely related to the general angular momentum operators (indeed, they are proportional) . Discussing them will enable us to use some more of the technique developed in the section on abstract spaces and at the same time shed some more light on the properties of angular momentum in quantum mechanics. Suppose that a vector e in abstract space depends parametrically upon the orientation of a vector r in ordinary space. If we should now rotate r about the x axis through an angle Oz, then e should rotate in abstract space . The operator which must be applied to e to yield the resultant vector is of the form given in Eq. (1.6.37). In the present case we write where now Mz is an operator. Similarly, one may define a 1)y and a 1). in terms of a Oy and an 0.. For most cases, since a rotation about the x axis of the vector r does not commute with a rotation about the y axis of this vector, :Dz:Dy ~ 1)y1)z. However, for an infinitesimal rotation of amounts (dO)z , (dO)y, (dO) . , it is easy to see that the rotations do commute and that the corresponding operator in abstract space is 1 + (ilh) (Mz ds; + My dOy + M. dO.) . Since the effect of this infinitesimal operator on vector e cannot depend upon the orientation of the x, y, z axes it follows that M z dO z + My dOy + M. dO. must be invariant under a rotation. Since (dOz,dOy,dO.) form the three components of a space vector, it follows that (Mz,My,M.) must also form the three components of a vector M and must transform, for example, as x, y, z in Eq. (1.3.8), for then the operator can be a simple scalar product, which is invariant under a rotation. Since M is a vector, it must transform like a vector. Specifically,
92
Types of Fields
[CR. 1
if the space axes are rotated by an angle de. about the z axis, the relation between the un primed and the primed components are IDl~
= IDl. ;
IDl~
= de. IDl.
+ IDly;
IDl~
= IDl. - de. IDly (1.6.46)
However, IDl. is an operator related to a rotation in vector space , related to the parameter e. in a manner analogous to .p and t in Eq. (1.6.38). To be specific, it is related to the rate of change of a state vector e by the equation 1 1 de h IDl.e = i de. Therefore, the rate of change of any other operator, related to the system. with respect to e. must be given by an equation of the form of (1.6.40). For instance, the rate of change of the operator IDl. with respect to the parameter e. is given by the equation 1 dIDl.
T de.
=
1
h (IDl.IDl. - IDl.IDl.)
But from Eq. (1.6.46), dIDl./de. = -IDly, we have
ihIDly = IDl.IDl. - IDl.IDl. which is identical with the last of Eqs. (1.6.42) for the angular momentum operators. ' The present derivation, however, has considered the operator vector IDl to be the one related to an infinitesimal rotation of axes in ordinary space . The results show that, as far as abstract vector space is concerned, this operator is identical with the angular momentum operator defined in Eqs. (1.6.42). The reason for the identity is, of course, that a measurement of the angular momentum of a system usually results in a space rotation of the system (unless the state corresponds to an eigenvector for IDl) just as a measurement of linear momentum p usually results in a change in position of the system. In terms of the formalism of abstract vector space the operator
performs the necessary reorientation of state vectors corresponding to a rotation of ordinary space by an amount e. about the z axis. Since, by Eqs. (1.6.43) the eigenvalues of IDl. are m.h, where m. is either an integer or a half integer (depending on whether ordinary angular momentum or spin is under study), when the operator acts on an eigenvector of IDl. with eigenvalue m.h, it changes neither its direction nor its magnitude
ei (!Dl.l hl 9' e(m.) = eim,9·e (m.) This whole subject of quantum mechanics and operator calculus will be discussed in more detail in the next chapter.
§1.7]
The Lorentz Transformation
93
1.7 The Lorentz Transformation, Four-vectors, Spinors Up to this point we have discussed vectors and other quantities in three dimensions, and some of the formulas and statements we have made regarding axial vectors, the curl operator, and vector multiplication are only correct for three dimensions. Since a great deal of the material in this volume deals only with the three space dimensions, such specialization is allowable, and the results are of considerable value. In many cases, however, a fourth dimension intrudes, the time dimension. In classical mechanics this added no further complication, for it was assumed that no physically possible operation could rotate the time axis into the space axis or vice versa, so that the time direction was supposed to be the same for all observers. If this had turned out to be true, the only realizable transformations would be in the three dimensions, time would be added as an afterthought, and the three-dimensional analyses discussed heretofore would be the only ones applicable to physics. Proper Time. Modern electrodynamics has indicated, however, and the theory of relativity has demonstrated that there are physically possible transformations which involve the time dimension and that, when the relative velocity of two observers is comparable to the speed of light, their time directions are measurably not parallel. This does not mean that time is just another space dimension, for, in the formulas, it differs from the three space dimensions by the imaginary factor i = y=I. It is found that, when an object is moved in space by amounts ax, dy, dz in a time dt, with respect to observer A, the time as measured by observer B, moving with the object, is dTB, where (1.7.1) where c is the velocity of light and dTB is called the proper time for observer = dT.!. is, of course, the proper time for observer A) . As long as the velocities dx/dt, etc., are small compared with the velocity c, the proper times dT.!., dTB differ little in value ; but if the relative velocity nearly equals c, the time intervals may differ considerably. Equation (1.7.1) is analogous to the three-dimensional equation for the total differential of distance along a path ds2 = dx? + d y 2 + dz 2, and the analogy may be made more obvious if Eq. (1.7.1) is written in the form (ic dTB)2 = dx 2 + dy2 + dz 2 + (ic dt)2 B (dt
The path followed by an object in space-time is called its world line, and the distance along it measures its proper time . The equation suggests that the proper times for two observers moving with respect to each other are related by an imaginary rotation in space-time, the amount of rotation being related to the relative velocity of the observers.
94
Types of Fields
[cnv l
The Lorentz Transformation. In order to begin the discussion as simply as possible, we shall assume that the relative velocity of the two observers is parallel to the x axis, which allows their y and z axes to remain parallel. We shall also assume that their relative velocity u is constant, so that their world lines are at a constant angle with respect to each other and the transformation from one space-time system to another is a simple (imaginary) rotation. Consideration of observers accelerating with respect to each other would involve us in the intricacies of general relativity, which will not be needed in the present book. The transformation corresponding to a rotation in the (x,iet) plane by an imaginary angle ia is
+
x = x' cosh ex. et' sinh ex. y=y'; z=z' et = x' sinh ex. et' cosh ex.
+
(1.7.2)
where (x,y,z,t) are the space-time coordinates relative to observer A and (x',y',z',t') those relative to B, moving at a relative velocity u, parallel to the x axis. In order that the time axis for B moves with relative velocity u with respect to A (or vice versa), we must have u
= c tanh ex.',
sinh ex.
=
u/e . - / - (u/e)2 ' cosh ex. = ,,===I=;==;=~ vI VI - (u/e)2
Consequently, we can write the transformation in the more usual form: x'
+ ut'
x = -Vr:I=-~(=u/: :;=e: ;: :;)2 y = y'; z = z' (ux'/ e 2) + t' t = --'-Vr:I;=-===;(u=/::;=e~)2
(1.7.3)
Incidentally this transformation also requires that, if observer B has velocity u = e tanh ex. with respect to A and observer C has velocity v = e tanh f3 with respect to B (in the same direction as u), then C has velocitye tanh (ex. + (3) = (u + v)/[I + (uv/e 2)] with respect to A . This set of equations, relating the space-time coordinates of two observers moving with constant relative velocity, is called a Lorentz transformation. It is a symmetrical transformation, in that the equations for x', t' in terms of x, t can be obtained from Eqs. (1.7.3) by interchange of primed and un primed quantities and reversal of the sign of u . This can be seen by solving for x' and t' in terms of x and t. The Lorentz transformation is a very specialized sort of a change of coordinates, for it is a simple (imaginary) rotation in a space-time plane, but it will suffice for most of our investigations into space-time in the volume. The equations for the general Lorentz transformation corresponding
The Lorentz Transformation
§1.7]
95
to relative velocity u = e tanh a in a direction given by ·t he spherical angles tJ, rp with respect to the x axis are
1)]x' + cos rp sin rp sin" tJ (cosh a - 1)y' cos rp cos tJ sin tJ (cosh a - 1)z' + cos rp sin tJ (sinh a)et' y = cos rp sin rp sin? tJ (cosh a - 1)x' + [1 + sin" rp sin- tJ (cosh a - 1)]y' + sin rp cos tJ sin tJ (cosh a - 1)z' + sin rp sin tJ (sinh a)et' z = cos rp cos tJ sin tJ (cosh a - 1)x' + sin rp cos tJ sin tJ (cosh a - 1)y' + [1 + cos" tJ (cosh a - 1)]z' + cos tJ (sinh a)et' ct' = cos rp sin tJ (sinh a)x' + sin rp sin tJ (sinh a)y' + cos tJ (sinh a)z' + (cosh a)et' x = [1
+ cos- rp sin" tJ (cosh a +
When rp = 0° and tJ = 90°, this reduces to the simple form given in Eqs. (1.7.2) . The scale factors h are all unity for this transformation, since it is a rotation without change of scale. Of course, the scale factors here involve the four dimensions 4
(h n)2 =
~ ., L.t (~xx~n)2 u
Xl,
X2, X3, X4
= X, y, z, 'I.e. t
m=l
Since the h's are all unity, there is no need to differentiate between contravariant, covariant, and "true" vector components. Four-dimensional Invariants. Just as with three dimensions, we shall find it remunerative to search for quantities which do not change when a Lorentz transformation is applied, i .e., which give the same result when measured by any observer traveling with any uniform velocity (less than that of light) . These are analogous to the scalar quantities which we discussed earlier and should, in many cases, correspond to measurable quantities, for according to the theory of relativity many physical quantities should give the same result when measured by different observers traveling at different velocities. Such quantities are said to be Lorentz invariant. The space-time length of a given portion of the world line of a nonaccelerated particle is a four-dimensional invariant. If its displacement in space to observer B is X' and its duration in time to the same observer is t', then the square of the proper lengths of its world line is •
S2
= (et')2 -
(X')2
To observer A, according to Eqs. (1.6.3), the square of the proper length IS
(et)2 - x 2 =
r
1 . (et')2 1 - (u/e)2
+ 2ux't' + (ux') -e
2
.:.- (X')2 - 2ux't' - (ut')2]
= (et')2 - (X')2
96
Types of Fields
[CR. 1
which is the same value. The square of the length of the same world line to an observer traveling with the particle would be his proper time, squared, multiplied by c2, which would also have the same value. Therefore, in relativity, the space-length of a line is not an invariant, nor is a time duration. The space-length of a line to observer A, moving with the line, is the distance between points on two world lines measured at the same time for observer A (X2 - Xl) ' To observer B moving at velocity u the space length of the same line is determined by measuring the distance between points on the world lines at the same time for B, that is, for t~ = t~. By Eqs. (1.7.3) we have X2 - Xl = (~ - xD / (u/c) 2, or the distance measured by observer B, moving with respect to the line, is x~ - x~ = (X2 - xlh.lI - (u /c)2, shorter than the length measured by A by the factor VI - (u /c) 2. Since the apparent size of objects changes with the relative velocity of the observer, the apparent density of matter is also not an invariant under a Lorentz transformation. Many other quantities which were scalars in three dimensions (i.e., for very small relative velocities) turn out not to be invariants in spacetime. The mass of a body, for instance, is not a four-dimensional invariant; the correct invariant is a combination of the mass and the kinetic energy of the body, corresponding to the relativistic correspondence between mass and energy. This will be proved after we have discussed vectors in four dimensions. Four-vectors. As might be expected from the foregoing discussion, we also encounter a few surprises when we expand our concept of a vector to four dimensions. As with a three-vector, a four-vector must have an invariant length, only now this length must be a proper length in space and time. The two points (x~,y~,z~,t~) and (x~,y~,z~,t~) as measured' by observer B define a four-vector F~ = x~ - x~ , . .. , F~ = c(t~ - t~) . To observer A , traveling with velocity u (in the X direction) with respect to B, the components of this vector are
vI -
. vI+_(uF~/c) (U/C)2'
F = F~ 1
= F"
F = F' . F 2
2,
3
3,
+ F~ vI _ (U/C)2
F 4 = (uFVc)
so that for this Lorentz transformation, the direction cosines are 1'11 1'12
= cosh a; 1'22 = 1'33 = 1; = 1'21 = 1'13 = 1'31 = 1'23 =
1'44
= cosh a ;
1'32
~
tanh
1'24
a =
=
1'14
1'42
=
= 1'41 = sinh a = 1'43 = 0 (1.7.4)
1'34
u /c
This transformation of components is typical of four-vectors. We note that the /I sum" of the squares of the four components F2 = F~ + F~ + Fj - F~ is invariant, as it must be. A very important vector is the four-dimensional generalization of
The Lorentz Transformation
§1.7]
97
the momentum of a particle which is traveling at a constant velocity u in the x direction with respect to observer A . To an observer B, traveling with the particle, the mass of the particle is mo and the proper time is T . With respect to observer A the particle travels a distance dx in time dt, where (dT)2 = (dt)2 - (dx/e) 2. The space component of the vector momentum cannot be mo(dx/dt) , for it would be difficult to find a corresponding time component such that the square of the resulting magnitude would be invariant. On the other hand, if we choose it to be mo(dx/dT), which transforms like dx, so that dx P" = mo- = dr
VI
mou dt ; PI = moe- = - (u/e)2 dr
VI
dx moc ; u= -dt - (u/e)2
where T is the proper time for the particle under study (and, therefore, for observer B), then the square of the magnitude of P is P; - P; = -(moe)2, which is invariant. With respect to observer C, traveling with velocity v = c tanh fJ in the x direction compared with A, the vector momentum transforms according to Eq. (1.7.4), P~ P~
= p" cosh fJ + PI sinh fJ = moe sinh (a + fJ) = mou'/VI - (u' / e)2 = p" sinh fJ + PI cosh fJ = moe cosh (a + fJ) = moc/Vi - (u' / C)2
where u = e tanh a is the velocity of the particle with respect to A and + fJ) is its velocity with respect to observer C. Thus the definition of the momentum four-vector is consistent with the rule of composition of velocities given earlier. Therefore, the four-vector corresponding to the momentum as measured by observer A , for a particle moving with speed u with respect to A, is
u' = e tanh (a
P" =
mo(dx/dt) VI - (u/e)2 ; moe
mo(dy/dt)
PI! 2
mo(dz/dt)
= VI _ (u/e)2 ; pz = VI _ (u/e) 2
u =
(~;y + (~~y + (~;y
(1.7.5)
where x, y, z, t are the coordinates of the particle according to observer A. The time component of the momentum is proportional to the total energy of the particle with respect to observer A ,
which is not invariant under a Lorentz transformation. This equation' also shows that the total energy can be separated into a rest energy and a kinetic energy only when u is small compared with c.
98
Types of Fields
[CR. I
Another four-vector is the space-time gradient of some scalar function of (x,Y,z,t), 01/r (which may be called quad 1/r) , where (01/rh
=
iN
iJx' etc. ;
1 iJ1/r (01/r)4 = - c iJt
Since the scale factors h are all unity and the Christoffel symbols therefore are all zero, these are also the components of the four-dimensional covariant derivative of 1/r. Consequently, the contracted second-order derivative
o 21/t =
iJ21/r iJx2
+
iJ21/r iJy2
+
iJ~
iJz 2
1 iJ21/r
-
C2 iJt2
(1.7.6)
is Lorentz invariant. The operator 0 2 is called the d' Alembertian. It is analogous, in a formal sense, to the three-dimensional Laplacian operator V 2• However, because of the presence of the negative sign before the time term, we shall see later that solutions of the Laplace equation V 21/r = 0 differ markedly from solutions of the equation 021/r = 0, which is the wave equation for waves traveling with the velocity of light, c. Stress-Energy Tensor. By analogy with the three-dimensional case, dyadics or tensors may be set up in four dimensions, having transformation properties with respect to a Lorentz transformation which can easily be determined by extending the discussion of the previous section. An interesting and useful example of such a tensor is obtained by extending the stress dyadic defined in Eq. (1.6.28) for an elastic medium to four dimensions. The dimensions of the nine stress components Tij are dynes per square centimeter or grams per centimeter per second per second , and they transform according to the rules for three dimensions. By analogy from the previous discussion of the momentum vector for a particle, we should expect that the time component of the four-tensor might be proportional to the energy density of the medium. In other words, this fourth component should be related to the total energy term pc 2 (where p is the mass density of the medium), which has dimensions grams per centimeter per second per second. Therefore, we assume that the stress-energy tensor P i; at a point (x,y,z,ct) in a medium has the following components for an observer A at rest with that part of the medium which is at (x,Y,z,ct) : Psi = Tn, etc.; P 12 = T Zl/ = P 21 = T yz, etc. P u = P 2 4 = P 34 = PH = P 42 = P 43 = 0 P 4 4 = c 2po
where
Po is
(1.7.7)
the density of the medium at (x ,y,z,t) as measured by observer
A. If these quantities are to be components of a true four-tensor, then an observer B traveling with velocity u in the x direction with respect
99
The Lorenlz Transformation
§1.7]
to the medium at (x,Y,z,ct) will measure the components of P to be those given by the general transformation rules : ~~
= b cosh
~~ =
b sinh a
a
+ ~4 sinh a
+ ~4 cosh a ;
c tanh a =
U
The results of the transformation are : P~l P~3 P~3 P~4 P~4
= T""" cosh! a + poc 2 sinh" a ; P~2 = P~l = T Zl/ cosh a = P~l = T",z cosh a; P~2 = T yy; P~3 = Tzz = P~2 = T yz; P~4 = P~l = (T",,,, + poC 2) cosh a sinh a (1.7.8) = P~2 = T Zl/ sinh a ; P~4 = P~3 = T",z sinh a
= T""" sinh"
a
+ poc
2
cosh" a
The space components (P~lI P~3' etc .) turn out to be the components of stress which observer B would measure in the medium if we take into account the finite velocity of light, and the component P~4 turns out to be c2 times the effective density as measured by observer B. An examination of component P~4 shows that it can be considered to be proportional to the momentum flow density of the medium parallel to the x axis as measured by observer B. Correspondingly the components P~4' P~4 must be proportional to momentum flows in the Y and z directions as measured by observer B. We therefore arrive at an interesting and important result, which can be verified by further analysis or by experiment, that relative motion transforms a stress int o a momentum flow and vice versa. Moreover, since we can verify, in the system at rest with respect to the medium (observer A), that the contracted tensor equations
n
or, in terms of the T's,
aT""" ax
+ aT",y + aT",z = ay az
0
t
.
' e c.;
iat (cPo)
= 0
are true, then these equations should also be true for observers moving relative to the medium (or, what is the same thing, for a medium moving with respect to the observer) . For instance, if we define the momentum vector M with respect to observer B as being the vector with space components
M", My
+
= (1/c)(poC 2 T",,,,) cosh a = P~2/C ; M, = P~3/C
sinh
a = P~l/C
100
[CR. 1
Types of Fields
and the density with respect to observer B as being
then one of the transformed equations would become
l: (a~in)
+ aMy + aM. = 0 or aM" ax' ay' az'
+ ap= 0 at'
n
where the primed coordinates are those suitable for observer B. This equation is, of course, the equation of continuity for the medium, relating momentum density (or mass flow) to change in mass density p, The other three transformed equations turn out to be the equations of motion for the medium. Spin Space and Space-Time. One of the most interesting developments in modern theoretical physics is the demonstration that there is a close connection between the two-dimensional "state space " connected with the spin of the electron and the four-dimensional space-time describing the motion of the electron. In the previous section we have initiated the discussion of a spin space corresponding to the two possible states of spin of the electron and there pointed out that a change of direction of the spin in ordinary space by 180° (reversal of spin) corresponded to a change of the state vector in spin space by 90°. This is somewhat analogous to the relation between the graphical representation of (-1) and y-=I on the complex plane, and if one wished to become fanciful, one might consider that the relation between ordinary space and spin space was some sort of a "square-root" relation. Actually, it can be shown that the two-dimensional spin space is a "square-root space, " not of three-dimensional space, but of four-dimensional space-time. More specifically, we will find that the four components of a dyadic in spin space could be identified with the components of a four-vector in space-time. In order to show this we must consider the components of vectors in spin space, and even the unit vectors themselves, as complex quantities with complex conjugates (a and d for the components, which are numbers, and e and e* for state vectors) , so that ad and e· e* are real positive numbers. We start out with the two mutually perpendicular state vectors el and e2 (with their complex conjugates ei and en representing states where the electron spin is known to be pointed in a certain direction or known to be pointed in the opposite direction. For two complex vectors to be called unit, orthogonal vectors, we must have that el . ei = et . el = e2· ei = ei . e2 = 1 and that el · ei = ei . e2 = e2· ei = ei . el = O. The values of the complex quantities el· el, el· e2, ei . ei, etc ., are not required.
The Lorentz Transformation
§1.7]
101
Any vector in spin space can be expressed in terms of these two unit vectors : s = aIel + a2el; s* = dlei + d2e: and any dyadic in spin space can be expressed in the form @5
= cllelei
+ cl2ele: + c2le2ei + c22e2e:
(1.7.9)
A dyadic in spin space is called a spinor of second order. I t has the usual properties of dyadics in that it can operate on a state vector in spin space to produce another state vector: @5 •
s* .
s
@5
= (Cll + c21)alel + (C12 + c22)a2e2;
= dl(Cll + c12)ei
+ d2(C21 + c22)e:
To give physical content to these definitions, we must relate the behavior of the spinor components ai and the dyadic components Cij under a Lorentz transformation, with their behavior under a rotation of axes in spin space . For example, a Lorentz invariant quantity should also be an invariant to rotation in spin space. Following our preceding remarks we shall require that the four second-order spinor components Cij transform like the components of a four-vector in ordinary space-time. A dyadic in spin space is a vector in space-time; this is the consummation of our desire to have spin space be a "square-root space ." Spinors and Four-vectors. We still have not worked out the specific rules of transformation for spin space which will enable the Cij components of the spin or to transform like a four-vector. The most general transformation is given as follows :
+ a12e2 ;
ei' = allei + al2e i ei' = a2lei + a22 e i el = a22e~ - a12e~ ; ei = a22 ei' - a12 ei' e2 = -a2lei + alle~; e: = -a2Iei' + allei' e~ e~
= allel
= a2lel + a22e2;
(1.7.10)
where, in order that the scale be the same in the new coordinates as in the old, aUa22 - a12a21 = 1, alla22 - al2a21 = 1. Under this transformation in spin space the general spinor components undergo the following transformations : Cm" =
L
C:iaimaj"
(1.7.11)
ij
The safest way to carryon from here is to find a function of the c's which is invariant under transformation of the a's in spin space, which can then be made invariant under a Lorentz transformation. Utilizing the multiplication properties of the a's, we can soon show that one invariant is the quantity CllC22 - Cl2C21 (it can be shown by substituting and multiplying out, utilizing the fact that alla22 - al2a21 = 1, etc .). This
102
Types oj Fields
[cH.1
quantity can also be made a Lorentz invariant if the c's are related to components F n of a four-vector in such a way that 011022 - 012C2l = 02F~ - Fi - Fi - Fi, for such a combination of the F's is invariant. There . are a number of ways in which this can be accomplished, but the most straightforward way is as follows: C11 = oF4 + F I ; F 4 = C22 = oF4 - F l ; F 3 = C12 = F 2 - iF 3 ; F 2 = 02l = F2 + iF 3 ; F I =
(1/20)(C11 + 022) (i/2) (012 - 021) (1/2)(012 + C2l) (1/2) (Cn - 022)
(1.7.12)
Lorentz Transformation of Spinors. For instance, for an observer B (primed coordinate) moving with velocity u in the x direction with respect to the electron, at rest with respect to A (unprimed coordinates), the transformation for the F's is cF4 = cF~ cosh a + F~ sinh a, F I = F~ cosh a + cF~ sinh a, F 2 = F~, F 3 = F~ , u = c tanh a. The transformation for the spinor components c is, therefore, C11 = t;le a; C12 = 0'12 ; On = C~I; C22 = ~2e-a; ea = V (c + u) /(c - u)
(1.7.13)
and that for the corresponding direction cosines for rotation of the unit vectors in spin space is (1.7.14)
Therefore, any state vector in spin space, with respect to observer A, s = aIel + a2e2, becomes Sf = alea/2e~ + a2e-a/ 2e~ with respect to observer B, moving with velocity u = c tanh a with respect to A . The transformation equations for the c's and a's for a more general Lorentz transformation will naturally be more complicated than those given here, but they can be worked out by the same methods. Any pair of complex quantities which satisfy the transformation requirements for the components of a state vector in spin space is called a spinor of first order; a quartet of quantities which satisfy the transformation requirements for the c's of Eqs. (1.7.9) and (1.7.11) is called a spinor of the second order; and so on. Equations (1.7.12) give the relationship between the spin or components and the components of a four -vector for this simple Lorentz transformation. Space Rotation of Spinors. As a further example of the "squareroot" relationship between spinor components and vector components, we consider the case where the time coordinate is not altered but the space coordinates are rotated in accordance with the Eulerian angles discussed on page 28. Under this transformation the time component of any four-vector will remain constant and, therefore, by Eq. (1.7.12), Cll + C22 will stay constant. Transforming en + C22 in terms of the
103
The Lorentz Transformation
§1.7]
a's, we see that for a purely. space rotation, where Cll
we must have
_
_
alnalm + a2na2m
=
_
_
anla ml + an2am2
=
onm
=
{Oi 1 i
+ C22
=
c'u
n m} n, m = ¢
n = m
+ ~2 ' 1, 2
This result is not unexpected, for a consequence is that e~· e:' = 1, and we should expect that the "length" of a state-vector would be unchanged by a space rotation. Adding together all the limitations on the a's, we see that (1.7.15) We write down the expressions for the transformation for the F's f01" a space rotation [modifying Eq. (1.3.8) and letting F l be Fz , F 2 be Fz , and Fa be F y ] . F~
= [cos
+ [sin P cos 0 cos if; + cos P sin if;]Fa - sin 0 cos if; F l -[cos P cos 0 sin if; + sin P cos if;lF 2 - [sin P cos 0 sin if; - cos P cos if;]F a + sin 0 sin if; F l Fi = sin 0 cos P F2 + sin 0 sin P Fa + cos 0 F 1 i F~ = F 4
F~ =
and we insert these into Eq. (1.7.12) both for the primed and unprimed components to determine the transformation equations for the c's: c'l2 = -
sin (0/2) cos (O/2) eil/'(cn - C22)
+ cos- (O/2)e il/'Hlcl2 - sin? (O/2)eil/'-llc21; etc.
In terms of the direction cosines unprimed c's by the equation
a
this component is related to the
where we have inverted Eq . (1.7.11) and used Eq. (1.7.15). Comparing these last two equations, we find that the direction cosines a for spin-space rotation corresponding to a space rotation given by the Euler angles P, 0, if;, [see Eq . (1.3.8) and Fig. 1.6] are an al2
= cos (O/2)e- i /rtll/2 i a2l = - sin (O/2)eil/'-ll/2 = sin (O/2)e-il/'-ll/2; a 22 = cos (O/2)e i l/'Hl/2
(1.7.16)
where we have again used Eq. (1.7.15) to help untangle the equations. Therefore, under this rotation of the space coordinate system a state vector in spin space s = alei + a2e~ becomes s = [al cos (O/2)e- i'1t/2 + a2 sin (O/2)ei'1t/2]eil/2el + [-al sin (O/2)e-i'1t/2 + a2 cos (O/2)e i'1t/ 2]eil/2e 2 This last equation shows that rotation in spin space is governed by one-half the angles of rotation in ordinary space . A rotation of 1800
104
Types of Fields
[cB.1
(0 = 11", I = if; = 0) change s s = ale~ + a2e~ into s = a2el - ale2, which is a rotation of 90° in spin space. The transformations given by Eqs. (1.7 .14 ) and (1.7.16) are the cases of usual interest. Any other case can be handled in the same manner, by use of Eqs. (1.7.12). Although we began t his discussion wit h a rather vague requirement to satisfy, we ha ve developed the theory of a quantity which has definite t ransformati on properties under a general rotation of coordinates (including a Lorentz rotation) and yet is not a te nsor, according to our earlier discussions. This came as quite a surprise when spinors were first studied. Spin Vectors and Tensors. A quartet of simple dyadics in spin space can be found which behave like four unit vectors in space-time:
d( = elet + e2ei = 3' dl = elei + e2et d2 = i(e2et - elei) d3 = elet - e2ei
(1.7 .17)
These quantities operate on the spin vectors e as follows :
Comparison with Eq s. (1.6.44) shows that the quantities 01, 02, 03 are 2/h times the spin operators for the electron. They are called the Pauli spin operators. The quantity o( is, of course, the unity dyadic. We see also that io 1, i0 2, - i0 3 are just the Hamilton quaternion operators. We now can rewrite the spinor dyadic given in Eq. (1.7.9) as a fourvector, by using Eqs. (1.7. 12) and (1.7.17) : (1.7. 18)
where the "unit vectors " 0 are operators, operating on state vect ors in spin space, but the component s F are ordinary numbers transforming like components of an ordinary four-vector. Thus we finally see how operators in spin spa ce can act like vectors in space-time. The extension of this discussion to the inversion transformation r' ---t -r and its correlation with spin space requires that we consider e* and e as independent quantities, so that transformations between them are possible. We shall not go into this further , however, except to say that the vector 0 transforms like an axial vector. (See Prob. 1.34.) One can go on to form spinor forms which transform like dyadics in space time. For instance a spin or of fourth order,
+ F22 + F + F + ol(F 14 + F 41 + iF 23 - iF 32) + 02(F24 + F42 + iF 31 - iF + 03(F + F + iF 12 - F
lopo.F p• = 04(F u 1'.'
33
H )
13)
34
43
21 )
has component s F p • which behave like components of a dyadic in space-
105
The Lorentz Transformation
§1.7]
time. - A particularly important form is the contracted tensor formed by multiplying one spinor vector by its conjugate: (d4CF 4
+ dlFl + d 2F 2 + d aFa)(d4CF4 =
d 1F 1 - d 2F 2 d4(C 2F~ -
Fi -
daF a) F~
-
FD
(1.7.19)
giving the square of the magnitude of the four-vector. This relation will be of use when we come to discuss the Dirac theory of the electron. Rotation Operator in Spinor Form. Reference to Hamilton's researches on quaternions (see page 75) has suggested a most interesting and useful spin or operator using the spin-vector direction cosines 01 as components :
m= =
OI11elei
L
+ 0I21elei + 0I12e2ei + 0I22e2ei
Rndn;
R 1
= ~ (01 12 +
(1
21) ;
n
(1.7.20)
The OI'S, according to Eq. (1.7.10), are the direction cosines relating the primed and unprimed unit vectors e in spin space. If they have the values given in Eq. (1.7.16), they correspond to a rotation of space axes by the Euler angles 8, 1/1, 1. As we have shown above, a spin or operator of the form of mhas the transformation properties of a vector, and this is emphasized by writing it in terms of its components R "along" the unit spin vectors d. However, vector (or spin or operator, whichever point of view you wish to emphasize) mis a peculiar one in that its components R 1 = i sin (8/2) sin [(I -1/1) /2]; R 2 = i sin (8/2) cos [(I - 1/1) /2] R; = i cos (8/2) sin [(I + 1/1) /2] ; R 4 = cos (8/2) cos [(I + 1/1) /2]
are themselves related to a particular transformation specified by the angles 8,1/1, 1. (This does not mean that the vector mcannot be expressed in terms of any rotated coordinates for any angles; it just means that it is especially related to one particular rotation for the angles 8, 1/1, 1.) As might be expected, the vector has a particular symmetry for this rotation, for if the unit vectors e' are related to the vectors e through the same angles [see Eqs. (1.7.10) and (1.7.16)], then it turns out that m has the same form in terms of the primed vectors as it does for the unprimed:
m=
OI11elei
+ OI21ele: + . .
= [0I110l22Q2 2 -
=
a22e~ei' -
= OI11e~ei'
0I21012 2a21 -
[0I110l22a12 -
0I120121a22
+ 0I220121a2de~ei' OI120121a12 + 0I220121all]e~er
0I210l22aU -
aI2e~e:' • • •
+ OI21e~er + . . .
as one can prove by utilizing the multiplication properties of the
OI'S .
106
[CR. 1
Types of Fields
However, 91 is also a spin-vector operator. As a matter of fact it operates on any spin vector to rotate it by just the amount which the transformation e ----+ e' produces. According to Eqs. (1.7 .10), where
91*
=
91· en = e~; 91*' e: = e:' alleiel + a2leie2 + al2e:el + a22e: e2
Another operator 91-1 performs the inverse operation e' ence to Eqs. (1.7 .10) indicates that the vector is
(1.7.21)
----+
e.
R.efer-
91-1 = a 22elei - a2lele: - al2e2ei + an e2e: = a22eiej' - a21eiet - al2e~et' + alle~et 91-1 . e~ = e n; (91-1)*. e:' = e:
so that
But since a22 = an, al2
-a2l, etc., we also can show that en' (91-1)* = e~ and e:· 91-1 = e:' e~ . 91* = e',' and e:'· 91 = e:
and that
=
(1.7.22)
which shows the close interrelation between the operator 91 and its inverse, 91-1• The particularly important property of the operators 91 is that, in addition to causing a rotation of vectors in spin space, they can also cause a related rotation of four-vectors in ordinary space. For instance, the spin or ~ = cnelei + Cl2ele: + c2le2ei + c22 e2e: (where the c's have any values) has the transformation properties of a four-vector [see Eq. (1.7.18)], with components F« [see Eqs. (1.7.12)]. The vector formed by operating" fore-and-aft" on ~ by 91: m
(J~
•
= ~
•
m-I
(J~
,
*, + ~12ele2, ,*, + C21e2el ,*,
=
cll~lel
=
F 1d 1
+ F 2d 2 + Fad a + F 4"'.
+ C22e'*' 2e 2
(1. 7.23)
is one with the same components F n, but these components are now with respect to the primed unit vectors, rotated with respect to the unprimed ones. Therefore, the fore-and-aft operation by 91 has effectively rotated the vector ~ by an amount given by the angles e, 1/1 and cfl. In keeping with the "square-root" relation between spin space and space-time, we operate on a spin vector once with 91 to produce a rotation, but we must operate twice on a four-vector in order to rotate it by the related amount. We note that we have here been dealing with rotations by finite-sized angles. If the rotation is an infinitesimal one, the Euler angles e and (cfl + 1/1) become small and the rotation can be represented by the infinitesimal vector Aw, its direction giving the axis of rotation and its magnitude giving the angle of rotation in radians. Consideration of the properties of the cross product shows that the operation of changing an
107
Problems
CH.1]
ordinary three-dimensional vector A into another A' by infinitesimal rotation is given by the equation A' = A +.1(,) X A
(1.7.24)
Inspection of the equations on page 103 for rotation in terms of the Euler angle shows that, when 0 and (I + ift) are small , (.1(,)h ---t
-
(I
+ ift);
(.1(,)h---t -0 sin ift; (.1(,))a ---t -0 cos ift
Inspection of the equations for the components of set of equations : R4
= 1; R 1 = -(ij2)(I Ra
= -
mresults in a related
+
ift); R 2 = -(ij2)0 sin ift; (ij2)0 cos ift
when 0 and (I + ift) are small . Consequently, for an infinitesimal rotation represented by the vector ~, the rotation spin or operator is (1.7.25)
These equations are sometimes useful in testing an unknown operator to see whether its components satisfy the transformation rules for fourvectors.
Problems for Chapter 1 1.1 The surfaces given by the equation (x 2
+ y2) cos" ift + Z2 cot- ift =
a 2; 0
< ift < 11"
for ift constant are equipotential surfaces. Express ift in terms of x, y, Z and compute the direction cosines of the normal to the ift surface at the point x, y, z. Show that ift is a solution of Laplace's equation. What is the shape of the surface if; = constant? ift = O? ift = 11"? 1.2 The surfaces given by the equation
h/x 2 +
y2 - ift]2
+ Z2
= ift2 -
a 2; a
< ift <
00
for if; constant, define a family of surfaces. What is the shape of the surface? What is the shape for the limiting cases ift = 0, ift = oo? Express ift in terms of z , y, z and compute the direction cosines of the normal to the ift surface at x , y, z. Is ift a solution of Laplace's equation? 1.3 The three components of a vector field are
F", = 2zx ; FlI = 2zy; F. = a2 + Z2 - x 2 - y2
108
[CR. 1
Types of Fields
Show that the equations for the flow lines may be integrated to obtain the flow functions I{J and JL, where x2
y
- = tan x
I{J '
+ y2 + Z2 + a 2 =
'2a
yx + y2 2
coth
JL
Show that a pseudopotential exists and is given by if;, where x2
+ y22+ Z2
- a2
az
cot if;
=
Show that the surfaces I{J, JL, if; constant are mutually orthogonal. 1.4 The three components of a vector field are F",
=
3xz;
F y = 3yz;
=
F.
2z2 - x 2 - y2
Integrate the equations for the flow lines to obtain the flow functions I{J
= tan -1
(~);
X--:2-~-:-2_:--:2,.-~=:-2_Z--:2"," ,i)
11 =
7(
and show that the pseudopotential is
z
if;
=
(x2
+ y2 + Z2)~
Is if; a solution of Laplace's equation? 1.6 Compute the net outflow integral, for the force fields of Probs. 1.3 and 1.4, over a sphere of radius r with center at the origin and also over the two hemispheres, one for; z < 0, the other for z > 0 (plus the plane surface at z = 0). Compute the net outflow integral, over the same three surfaces, for the vector field
F",
=
[x 2
x
+ y2 + (z -
. F =
a)2)1 '
y
[x 2
Y
+ y2 + (z -
.
a)2]i'
z - a
F. = '[x·2----=+-----,y2;;-+-,-----(;-z---a";""-;;)2:;"";)1
Compute the net circulation integral around the circle, in the
1.6
x , y plane, of radius r, centered at the origin, for the field
F", =
(x - a) (x - a)2 y2
+
x2
x
.
+ y2'
F = Y
x2
Y
+ y2
y (x - a)2
.
+ y2'
F. = 0
Compute the net circulation integral for the field of Prob. 1.3 for the circle defined by the equations ¢ = 0, JL = constant. 1.7 Parabolic coordinates are defined by the following equations: >..
= Y Y x2 +
y2
+ Z2 + z; ¢
Po
= tan- 1
= Y Y x 2 + y2 + (y jx)
Z2 - z;
CR.
1]
109
Problems
Describe (or sketch) the coordinate surfaces. Calculate the scale factors and the direction cosines for the system in terms of (x,y,z) . Express x, y, z in terms of X, IJ., tP, and thence obtain the scale factors and direction cosines in terms of X, IJ., tP. Write out expressions for curl F, V 21/1. Calculate, in terms of X, IJ., tP, the X, IJ., tP components of the following vector field : 2 2 Fz = x/yx + y2 + Z2 . FlI = y/yx + y2 + Z2 . 2 Z Y x + y2 + Z2' Z Y x 2 y2 + Z2'
+
+
+
1
F. = ----;;=;;==;====;;;:::::::====;;; yx 2 y2 Z2
+ +
In terms of X, IJ., tP, calculate the divergence of F. 1.8 The flow functions tP, IJ. and the pseudopotential 1/1, given in Prob. 1.3, form the toroidal coordinate system. Describe (or sketch) the surfaces. Calculate the scale factors as functions of x, y, z and also of IJ., 1/1, tP. Write out the expressions for curl F, div F, and V 2 U· Express the vector F given in this problem in terms of components along the toroidal coordinates, and calculate the direction of its velocity lines. 1.9 One family of coordinate surfaces, which may be used for a family of coordinates, is
for ~ constant. Show that an appropriate additional pair of families, to make a three-dimensional system, is 71 = j(x 2 + y2) + z; tP = tan- 1 (y/x) i .e., show that they are mutually orthogonal. These may be termed exponential coordinates. Why? Compute the scale factors and direction cosines for transformation of vector components. 1.10 The bispherical coordinate system is defined by the equations a sin
x
= cosh IJ.
~
-
cos tP cos ~;
a sin
y
= cosh IJ.
~
-
sin tP cos ~;
z
a sinh IJ. - cos ~
= cosh IJ.
Describe (or sketch) the surfaces, and give the effective range of IJ., ~, tP. Calculate the scale factors and direction cosines. Write out the expressions for the Laplacian and the gradient. Show that the curvature of the IJ. surfaces is a constant ; i.e., show that (l /h,,)(dal'/d~) = (l/hq,) . . (iJal'/dtP) is independent of ~ and tP and that therefore these surfaces are spheres. 1.11 Write out the expressions for the components of directional derivatives (a,, · V)A and (aq, · V)B in spherical coordinates and in the spheroidal coordinates x = a cosh IJ. cos ~ cos tP;
y = a cosh IJ. cos
~
sin tP ; z
=
a sinh IJ. sin ~
110
Types of Fields
[CH. i
1.12 A scalar function 1/t(h, ~2, ~3) in an orthogonal, curvilinear coordinate system h, b, ~3 may be made into a vector by multiplication by the unit vector al,' normal to the h coordinate surfaces. Another vector may be obtained by taking the curl A = curl (al1/t). Show that A is tangential to the h surfaces. What equation must 1/t satisfy, and what are the limitations on the scale factors h n in order that A satisfy the equation 1.13 By the use of the tensor notation, find the expression for V X (uVv) in general orthogonal curvilinear coordinates. 1.14 We can define the curvature of the ~n coordinate surfaces in the ~m direction as the component along am of the rate of change of an with respect to distance in the am direction. Express the two curvatures of the ~n surface in terms of the Christoffel symbols. 1.16 Work out the expressions for the Christoffel symbols and for the covariant derivative of the components fi = hiF i for the bispherical coordinates given in Prob. 1.10 and the parabolic coordinates given by z = 'Ap. cos cP ; y = 'Ap. sin cP;
z = j.('A 2
p.2)
-
1.16 Give explicit expressions for the components of the symmetric dyadic j.(VA + AV) for the spheroidal coordinates given in Prob. 1.11 and for the elliptic cylinder coordinates given by x = a cosh 'A cos cP ; y = a sinh 'A sin cP;
z
=z
Also give expressions for the Laplacian of a vector in both of these systems. 1.17 Find the principal axis for the strain dyadic
a-
C2~3 :r (r ~~)] 3
-
=
0
Show that a solution of this equation is 1/; = (1/r 3 ) cosh a . Give the x, y, Z, t components of the four-vector formed by taking the fourgradient of 1/;. Show that this is a true four-vector. 1.30 A particle of rest mass mo, traveling with velocity v in the x direction, strikes another particle of equal rest mass originally at rest (with respect to observer A). The two rebound, with no change in total energy momentum, the striking particle going off at an angle () with respect to the x axis (with respect to the observer) . Calculate the momentum energy four-vectors for both particles, before and after collision, both with respect to observer A (at rest with the struck particle before it is struck) and to observer B, at rest with respect to the center of gravity of the pair, and explain the differences. 1.31 A fluid is under a uniform isotropic pressure p according to observer A at rest with respect to it . Calculate the density, momentum density, and stress in the fluid , with respect to an observer B, moving at 0.8 the velocity of light with respect to the fluid. 1.32 Give the direction cosines a for the transformation of spin vectors for a combined Lorentz transformation (along the x axis) plus a space rotation. ' 1.33 An electron certainly has a spin in the positive x direction with respect to an observer at rest with respect to the electron. What are the probabilities of spin in plus and minus x directions for an observer B moving with velocity u in the x direction with respect to the electron? What is the probability that the electron has a spin in a direction at 45° with respect to the positive x axis, with respect to observer A? For observer B? 1.34 Let d be a three-component vector-spin operator with components Ul, U2, U3 . a. Show that, if A is a vector,
+
(d . A)d = A i(d X A) d(d . A) = A - i(d X A) (d X d) = 2id d X (d X A) = i(d X A) - 2A
b. Show that, if a is a unit vector and X is a constant, d2 dX2 exp (£Xd. a)
and therefore
=
-X2 exp (tAd · a)
exp (iXd. a) = cos X
+ i(d . a)
sin X
[cH.l
Types of Fields
114
Table of Useful Vector and Dyadic Equations A.B
=
A",B",
+ AyB y + AzCz;
A X B = i(AyB z - A.By)
+ j(AzB", -
(A x B) X C = (A . C)B - (B . C)A A X (B X C) = (A . C)B - (A. B)C A . (B X C) = (A X B) . C = (C x A) . B
A",Bz)
+ k(A",By -
AIIB",)
= C . (A x B) = B • (C X A)
= (B X C) • A
(A x B) . (C X D) = (A. C)(B. D) - (A. D)(B. C) (A X B) X (C X D) = [A. (B X D)]C - [A. (B X C)]D = [A. (C X D)]B - [B . (C X D)]A VU = grad u; V· F = div F; V X F = curl F V(uv) = uVv vVu ; V · (uA) = (Vu) . A uV • A uV X A V X (uA) = (Vu) X A V . (A X B) = B . (V X A) - A . (V X B) V· (V X F) = .0; V X (Vu) = 0; V· (Vu) = V 2u V X (V X F) = V(V. F) - V2F
+
+
+
If f(v . F) dv = IfF . dA; I If (V X F) dv = - I IF X dA IIf(V",) . (V1/!) dv = If",(v1/!) • dA - ffI",V21/! dv
where the triple integrals are over all the volume inside the closed surface A and the double integrals are over the surface of A (dA pointing outward). If (V _X F) . dA = IF . dr
where the double integral is over an area bounded by a closed contour C, and the single integral is along C in a clockwise direction when looking in the direction of dA . A vector field F(x,Y,z) can be expressed in terms of a scalar potential v and a vector potential A, F = grad 1/!
+ curl A ;
div A = 0
When F goes to zero at infinity, the expressions for 1/! and A in terms of F are
fff
1/! = -
div
F4~~Y"z') dx' dy' dz' ;
where R2 = (x - x')2 ~
+ (y
- y')2
+
·A =
fff
cu rl
~~dY"z') dx' dy' dz'
(z - Z')2.
= iA", + jAy + kA.;
I~I =
~.
~ * = iA: + jA: + kA: = A",i + Ayj + A.k + j. A + k · A.; (~ ) = i X A", + j X A + k X A A:B", + A:By + A:B. = i(A",' B) + j(A y' B) + k(A B) j-(A", + A:)B", + j-(Ay + A:)B y + j-(A. + A:)Bz - !(~) X B
i· A",
B = =
y
z
y
z •
..
Table of Properties of Curvilinear Coordinates
CH.1]
. 18 = A:B z + A;B y + AiB. : 18 = A: . B, + A: . By + Ai . Bz = VF = i of + . of + k of . FV = of i ~ ~
I~
115
. 181
+ of . + of k = (VF) * ~ J~~, ~ ~J ~ grad (A· B) = A· (VB) + B· (VA) + A X (V X B) + B X (V X A) curl (A X B) = B· (VA) - A· (VB) + A(V· B) - B(V. A) V • ~ = (oAz jox) + (oAyjoy) (oAz/Oz) = i div (A:) + j div (A;) + k div (Ai) V· (VF) = V2F; V· (Fv) = V(V. F) = V2F + V X V X F V· (~. B) = (V . ~) • B + I~' (VB)I
+
Table of Properties of Curvilinear Coordinates For orthogonal, curvilinear coordinates h, ~2, ~a with unit vectors al, a2, aa, line element ds 2 = h;'(d~n)2, and scale factors h n, where
L n
the differential operators become grad l/I = Vl/I =
2:
an
10l/l
i; es;
-
-
n
div A
= V. A = _1_ ~ ~(hlh2ha An) h 1h 2h a L.t O~n
.i.:
s;
n
curl A = V X A =
hlal
[o~m (hnA n) - O~n (hmAm)
l,m,n
I, m, n
= 1, 2,3 or 2, 3, 1 or 3, 1, 2 1 ~ 0 [h1h2ha Ol/l] d _ 2 _
di
gra l/I - V l/I - h 1h2h a L.t O~n ~ O~n
VA
= (VA).
IV
l
n
{VA).
+ (vA)a; (VA)a = i(curl A) X 3' = i-(VA + Av) = [a~m ~= + A· ,grad (In hm)] amam
2: m
~ + ~ L.t 1
mt = -T(iN/ax) at x = 1 = -iwpc[A+ei(",/c)l - A_e-i(",/c)l]e-i"'l = -iwpcA sinh [i(wl/c) + 1rao - i1r{1o]e- iwl = ZzU1e-iwt = Zl( -a1/;/at) at x = 1 - iwZI[A+ei(",/c)1 + A_e-i(",/c )/]e- iwt = -iwZIA cosh [i(wl/c) + 1rao - i1r{1o]e- iwt where we have used the definition of the transverse mechanical impedance of the support to obtain the last four forms. From these equations, we can obtain the complex ratio between the wave amplitudes A_, A+, and also the constants ao and (10 in terms! of the impedance Zl:
A_ = pc - Zl e2i(",/c)l. ao - t{1o . = -1 tan h1r pc + Zl ' A+
l(Jl) -
t. -2l
A
C
(2.1.14)
where A = c/v = 21rc/w is the wavelength of the waves on the string The ratio A_/ A+ is called the standing-wave ratio or, alternately, the reflection coefficient. If Zz is a pure imaginary, i .e., just reactive, IA_/ A+I = 1, so that the amplitudes of the reflected and incident waves are equal, as they should be, though, of course, the phase of the reflected wave will be different from that of the incident one. The relation between the reflection coefficient and Zz given in (2.1.14) is an example of the relation between the unitary reflection operator and the impedance operator discussed in the section on abstract vector spaces. From (2.1.14) we see that the boundary condition at x = 1 fixes the relative phases and amplitudes of the incident and reflected waves . Once this is known , the ratio Zo between the applied force and the velocity of the driving point (x =:= 0), which is the driving-point, im pedance for the string, can be obtained at once : - (A_/ A+) h [ ( 'R )] Z o = oc 11 + (A_/ A+) = oc tan 1r ao - t,....o
(2.1.15)
In other words, if the force is known, the string velocity at x = 0 can be calculated and also the expression A+, A_, A, and 1/;. For instance, if the driving force is j(w)e-iwt, the expression for the wave is .,.( ¥'
W,x,
t)
= j(w)e-i"'t cosh [i(wx/c) + 1rao . Z0 -tw
. R] cosh [1rao - t1r,....o
= !(w).e-i"'t [coth (1ra o _ -twpC
1
See the footnote on page 128.
i1r{1o]
i1r{1o) cos (wx)
c
+i
sin (wx)] C
(2.1.16)
§2.1]
The Flexible String
131
Transient Response, Fourier Integral. Just as with the Poisson equation discussed on page 121, a solution for several different forces acting simultaneously is the sum of the solutions for the forces acting separately. For instance, if forces of all frequencies are acting, as would be the case when we could express the total transverse force acting on the z = 0 end of the string in the form of an integral
F(t) =
J-.. . f(w)e-
iw l
dw
(2.1.17)
then the expression for the shape of the string as a function of x and t would be if; =
J-.. . if;(w,x,t) dw
(2.1.18)
where if;(w,x,t) is given in Eq. (2.1.16). In Chap. 4 we shall show that a very wide variety of functions of t can be expressed in terms of an integral of the type given in Eq. (2.1.17) (which is called a Fourier integral), and we shall show there how to compute few) if F(t) is known . Therefore, the integral of Eq . (2.1.18) is a general solution for the motion of the string under the action of nearly any sort of physically realizable force applied transversely to its end . This technique of solution is analogous to the Green's function technique touched on in the discussion of Eq. (2.1.18) and will also suggest similar methods for solving other equations discussed later in this chapter. One finds a solution for a particularly simple form of "force," which involves a parameter (point of application for the Poisson equation, frequency for the wave equation) . A very general form of force can then be built up by expressing it as an integral of the simple force over this parameter ; the resulting solution is a similar int egral of the simple solutions with respect to the same parameter. This is the general principle of the Green's function technique to be discussed in Chap. 7 and elsewhere in this book. Operator Equations for the String. Before leaving the problem of wave motion in a simple string, it will be of interest to outline an alternative approach to the problem which is related to the discussions of operators in abstract vector space given in Chap. 1 and later in this chapter. We start out by considering the string to be an assemblage of equal mass points connected by equal lengths of weightless string. At first we consider that there are only a finite number N of these masses (obviously a poor approximation for a uniform string), and then we approach the actual string by letting N go to infinity. Thus we can show the relation between the coupled oscillators discussed on page 77 and the flexible string.
132
ICH. 2
Equations Governing Fields
We approximate the string of uniform density stretched under tension T between rigid supports a distance I apart by N equally spaced mass points, each of mass pl/ N a distance I/ (N + 1) apart. A glance at Fig. 2.5
Yn
x=o-- -
------~~--t-li~-------
Fig. 2.6
x=l
N+I Displacements of mass points on elastic string.
shows that, if the displacement from equilibrium of the nth mass is Yn. the transverse force on this mass due to the displacements of its neighbors IS
(N
+ l)T{[(Yn+!
- Yn)/Ij
+ [(Yn-l
- Yn)/lll = (N
+ l)(T/I)(Yn+l + Yn-l -
2Yn)
(The last expression in parentheses is the analogue, for finite differences, of the second derivative.) Therefore, our set of simultaneous equations of motion for the N particles is 2
ddtYl 2
+ 2WoYl 2
2 -. . WoY2
d;~2 + 2W5Y2 = 2
W5(Yl
+ Yo)
ddtYn + 2WoYn 2 -_ Wo2(Yn-l 2
+ Yn+I \'
(2.1.19)
where w~ = N(N + 1)(T/pl 2) . We now consider the displacements Yn to be the components of a vector y in an abstract vector space of N dimensions with unit vectors, en along the coordinate axes. The parts of the equations on the righthand side represent the operation of a dyadic w~U which transforms the vector en into a vector with components along en-l and en+!. The dyadic U can be called the unit shift operator, for it shifts the index n by aunit up or down. It can be written in terms of the e's in the form
u=
ele2
+ e2(el + ea) + ... + en(en-l + en+l) + .. . + eNeN-l (2.1.20)
133
The Flexible Siring
§2.1]
Therefore, the equation for the vector y which represents the displacements of all the particles,
can be written in the form (d 2y /dt 2 )
+ 2w~y =
w~U • y
Eigenvectors for the Unit Shift Operator. The solution of the differential equation for y can be most easily effected by using the eigenvectors u, of the operator U. where u, is a unit vector along a principal axis of the operator U. Introducing u, for y into the equation for y we obtain the equation determining the time dependence of Un : (d 2u n/dt 2)
+ w~(2
- 1/n)un =
°
so that the time dependence of u, is e- i",.v'2-~nt . The space ("vector space") dependence of u, may be determined by solving the eigenvalue equation above. Let u, be expressed in terms of em by the expansion N
u,
=
L 'Ynmem m=l
the 'Y's being the direction cosines of the transformation. satisfy the equation 'Yn,m-l - 1/n'Yn,m + 'Yn,m+l =
Then the 'Y's
°
(2.1.21)
except for the first and last equations, for m = 1 and m = N, where the quantities 'YnO and 'Yn,N+l are naturally omitted. Even these two equations can be given this same form, however, if we just assume that the quantities 'YnO and 'Yn,N+l are always zero. The solution of Eqs. (2.1.21) is obtained by the use of the trigonometric formula cos a sin (ma) =
i
sin [em - I}«]
+i
sin [em
+ l)a]
for if we set 'Ynm = A sin [man] (the time dependence being understood) and n« = 2 cos an, all of the equations are satisfied. One of the additional requirements, that 'YnO = 0, is likewise complied with, and the remaining requirement, that 'Yn,N+l = 0, can be satisfied if we allow an to equal [n7r/(N + 1)]. Since
f .
0' (mn7r) . (mn'7r) '-' sm N + 1 sin N + 1 = { i(N m-l
+ 1);
n' n'
;>6 =
n n
134
[CH. 2
Equations Governing Fields
we can choose the value of the constant A so that the -r's are properly normalized direction cosines, and the u's will be unit vectors if the e's are . The final results are N
u, =
~N ~ 1 2: em sin (Nm~7r 1) exp [ -2iwet sin (2(Nn~ 1») ] m=l
U· u, = 2 cos [n7r/(N
+
1)] u,
(2.1.22)
Thus we have discovered an alternative set of N mutually orthogonal unit vectors in abstract vector space which point along the principal axes of the operator U (i.e ., which are eigenvectors for U). In terms of this new coordinate system the solution of the equation of motion for the vector y representing the N particles is straightforward : N
N
2: Ymem = y = 2: UnUn = m=l
n=l
~N ~ 1
N
m 2: Unem sin (N ; 1) n, m = l
exp {-2iwot sin
[2(Nn~ 1)J}
Therefore, N
Ym =
~N ~ 12: u, sin (Nm;l) exp {-2iwet sin [2(Nn~ l)J} n=l
(2 .1.23)
The allowed frequencies are wn / 27r, where W n = 2wo sin [n7r/2(N + 1)]. The component motions Un are called normal modes of the motion. If the particles are initially displaced by the amounts Y~ and initially all have zero velocities, then the values of the Un'S can be obtained by use of the last equation on page 133 :
ts -Un)t=o
=
u, = ~ N ~~
N
LY~
sin
(Nm~7r 1)
(2.1.24)
m=l
Thus the coefficients of the series for the y's can be obtained in terms of the initial values of the y's and the direction cosines of the transformation. Limiting Case of Continuous String. To go from a collection of M particles to a continuous string, we increase N to infinity, so that each "point" on the string is labeled by a different n . If the string were actually continuous, this would require N to be nondenumerably infinite, which would mean that the corresponding abstract vector space would have a nondenumerable infinity of mutually perpendicular directions. Such a vector space is rather difficult to imagine, though we may console ourselves that such niceties in distinguishing of types of infinities are rather academic here, since any actual string is only approximately
135
The Flexible Siring
§2.1]
continuous, and our present solutions are not valid for details of atomic size or smaller. We can also reassure ourselves that only a small subspa ce of the "supervector " space corresponds to physics, for as the distance between successive points goes to zero, continuity requires that Yn approach Yn+l in value. At any rate, for the continuous string we can discard the nondenumerable set of indices m and use the distance x of the point from one end as the label, i.e., set x = ml/(N + 1). Moreover since N is so large, the difference between Nand N + 1 is negligible. The index n labeling the different allowed modes of motion need not become infinite or continous, however, since we are usually interested in the lowest few (the first hundred or so!) allowed frequencies . Therefore, n will be retained as an integer and n/N will be a small quantity. To be specific, the transition is as follows:
Wn~ n1!'c/l;
y~~
c
= yT/p; em~ e(x) ; y =
L
r,«, =
n
L
l y(x)e(x) '"
Yne(x)
sin(n~x) e-int
n Ix
y(x) =
Lr,
sin
(n~x) e-i·t
n
The last equation is the usual Fourier series for the free oscillations of the uniform string between rigid supports. The function sin (n1!'x/l) e-int, giving the shape of the nth normal mode, is the transformation function, changing the denumerably infinite set of eigenvectors u, for the operator U to the nondenumerably infinite set of unit vectors e(x), each of which corresponds to a different point along the continuous string. The summation over all the points is symbolized by the summation sign ~> though it could also be expressed in terms of an integral x
over x. The limiting case of Eq. (2.1.24) is best expressed in terms of an integral, for instance. We have let Y n be the limiting value of U'; y2/N + 1, so that the equation for Yn in terms of the initial values of displacement y~ (when the initial velocity is zero) is
r, =
Ly~ N
J~n: {~
sin
(m~1!')};
where
m~ xf
m=l
The number of terms in the sum over m between x and x + dx is, therefore, (N/ l ) dx. Therefore, in the limit the sum for Y n becomes the integral
r,
=
. (n1!'x) 12 l«(I yO(x) sm "t: dx
136
Equations Governing Fields
[CH. 2
which is the usual integral for the Fourier series coefficients giving the amplitudes of the various normal modes. Finally, if the distance between supports is increased without limit (we place the origin at the mid-point of the string) , another series of limiting calculations, which will be explained in detail in Chap. 4, brings us to the general solution for the wave motion on an infinite string originally held at a displacement yO (z) and released at t = 0:
t.
y(x t) = -1 , 121r
eia(z-cll da
-00
f'"
yO(~)e-ia~ d~
(2.1.25)
-00
where the real part of this expression gives the actual displacement of point x at time t. Finally it is of interest to see what limiting form of the operator U takes on for the continuous string. To fit in with the equation for the vector y, we ask for the limiting expression for the operator W5[U - 2) on the vector y =
L
Ymem ~
m
Ly(~)e(x).
Before going to the limit, the
z
operator has the following effect on the vector components Ym : 2
wo[U - 2]· Y =
N(N
'" + 1)1' ~ \' [(Yn+l
pl2
- Yn) - (Yn - Yn-l)]en
m=l
As the distance between particles gets smaller and smaller, the difference (Yn+l - Yn) approaches the differential dy(x) , the distance between particles, l/(N + 1), being dx . Therefore, (N/ l)(Yn+ l - Yn) goes in the limit to ()y/{)x and the expression above becomes W5[U - 2]· y ~ c
2L()2:;~) e(x) z
so the equation of motion for y becomes in the limit
and therefore the equation for the transformation functions y(x) is {)2 y / {)t 2 =
c2 ({) 2 y jax 2 )
which is just the wave equation (2.1.9). Thus we have come back again to the partial differential equation for waves on the simple string via the roundabout route of abstract vector space. This excursion has been taken because similar ones will have to be taken later in this chapter for cases where alternative routes are not quite so direct or so simple .ii.-$ is the case of the string.
§2.1]
137
The Flexible String
The Effect of Friction. So far we have neglected the reaction of the surrounding medium (air or water) on the motion of the string. For small-amplitude motions this reaction is opposed to the motion of each element of length and is proportional to its velocity. The force on the element between x and x + dx is proportional to the velocity aif;jat of this element and is opposed to this velocity. The proportionality constant R is usually dependent on the frequency of oscillation of the string, but when the viscosity of the medium is great enough, it is independent of frequency. This last case is the simplest and will be taken up first. The equation of motion, when we take into account tension and friction of the medium but not stiffness or internal friction, is a2if; at2
aif;
+ 2k at
2if; _ . 2 a - c ax2 - 0,
_ R. 2 _ T k - Zp' c - p
The effect of friction is, of course, to damp out the free vibrations. If the string is held between two rigid supports a distance 1 apart, the shapes of the normal modes are not affected by the friction, being still sin (1f"nxjl) . However, the individual oscillations are damped out in time, for a solution of this equation is .s. = \' A sin (1f"nx) ." Z;": 1
e-kt- inl.
,
w2
n
= 7f"nC 1
_
k2
n
If k depends on frequency, it will have the value corresponding to W n for the nth normal mode, so that the different modes will damp out at different rates. On the other hand if the string is very long and is driven at one end by a sinusoidal force re-«, then the waves will be damped in space rather than in time. A solution is
Therefore, a has a positive imaginary part, which produces damping in the direction of wave motion. Diffusion Equation. In one limiting case, the viscous forces may completely predominate over the inertial effects, so that the equation becomes (2.1.26) This equation will be encountered many times in this book. Since it also represents the behavior of some solute diffusing through a solvent (where if; is the density of the solute), it is usually called the diffusion equation . As with the wave equation (2.1.9) the tendency is to straighten out the curvature ; however, here the velocity of any part of the string is proportional to hut opposite in sign to the curvature of the part, whereas
138
Equations Governing Fields
[cH.2
in the wave equation it is the acceleration that is proportional and opposite in sign to the curvature. In short, we are essentially dealing with an equilibrium condition. In the wave equation a curved portion continually increases in velocity until it is straightened out and only then starts slowing down, thus ensuring oscillatory motion. But with the diffusion equation the velocity of any portion comes to zero when this portion is finally straightened out, so there is no oscillatory motion. One would expect this behavior of a string of no mass in a viscous fluid, for the damping is more than critical. In the case of the wave equation the general solution -could be expressed as a ~ superposition of two waves in opposite 1=1 "-direction, f(x + ct) + F(x - ct), due to the symmetrical relation between x and t in the equation. In the diffusion equa1=2 tion this symmetry is not present, and there is no simple form for the general solution. Here also there is a difference 1=3 between the positive and negative time direction, due to the fact that the time derivative is a first derivative whereas there is a second derivative in the wave t=4~ -2 -I 0 +1 +2 +3 +4 equation. X AXIS For instance, if the string has a sinusFig. 2.6 Sequence giving soluoidal shape Aei"'%/c, then for the wave tions of diffusion equation after equation the time dependence is also initial shape as shown at top. sinusoidal, e-ic.>t. But for the diffusion equation the time-dependent term is e- J.lC
(2.1.30)
At high frequencies the wave impedance is real and nearly equal to the value oc for the simple string. As the frequency is diminished, however, th e wave impedance decreases and the wave velocity increases until, at w = J.lC = VK7P, the wave impedance is zero and the wave velocity is infinite. This is the resonance frequency for the mass of the string and the elasticity of the medium. Below this frequency the impedance is imaginary, similar to a stiffness reactance, and there is no true wave motion. . Recapitulation. We ha ve discussed the motions of a flexible string in detail for several reasons. In t he first pla ce our procedure in studying the string is a simple example of what will be our discussion of other equations for fields. We shall in each case discuss the various equations which result when one force after another is allowed to become predominant ; thus by the study of the various limiting cases we shall arrive at a fairly complete understanding of the most general case. Second, the motions of different sorts of strings are easily picturable representations, in the simplest terms, of the solutions of a number of important partial differential equations, which turn up in many contexts and which represent many physical phenomena. Many of the other manifestations of the same equations correspond to physical situations which are much harder to visualize. In the third place, the techniques of solution which have been touched upon here and related to the physical problem have useful application in many other cases, and reference to the simple picture of the string will help our understanding of the others. We shall now broaden the scope of our discussions and study a number of typical physical phenomena of considerable interest and importance in order to show what types of fields can be used for their picturization and what partial differential equations they mus t satisfy. ,
Equations Governing Fields
142
[CR. 2
2.2 Waves in an Elastic Medium For our first three-dimensional analysis of the interrelation between physical phenomena and differential equations for fields, we shall return to the problem of the behavior of an elastic medium (which was begun in Sec. 1.6) to take up the problem of wave motion in the medium. As in the case of the string, we assume that the displacements of the medium are small and that we are not concerned with translation or rotation of the medium as a whole. The displacement s(x,Y ,z;t) of an element dx dy dz of the medium at x , y, z and at time t is small, and its rotation due to the strain is also small. The inertial reaction of the element to an acceleration of s is (iJ 2sjat 2)p dx dy dz, where p is the density of the medium. In Sec 1.6 we defined the stress dyadic ~ = F%i + Fyj + F.k = iF% + jFy + kF. by saying that the force across a surface element dA of the medium is ~ . dA. For inst ance, the force across the face dy dz of the element, perpendicular to the x axis, is F%dy dz. Therefore, the net force on the element dx dy dz due to the difference in F%from one face dy dz to the opposite face of the element is dx (iJF%jiJx) dy dz, and the net force due to forces acting on all faces of the element is thus V·~ dx dy dz. But in Eq . (1.6.28) we showed that the stress dyadic is related to the strain dyadic @5 by the relation ~ = XI@513 + 2}.L@5, where }.L is the shear modulus of the medium and (X + %}.L) is its compression modulus (ratio of isotropic pressure to fractional rate of decrease of volume) . From Eq . (1.6.21) we have the relation between @5 and the displacement s given symbolically by @5 = Hvs + sV) . Putting all these equations together we finally arrive at the equation of motion for the medium under the influence of its own elastic restoring forces : p
~~ = =
V .
(X
[X3 div s
+ 2}.L)
+ }.LVS + }.LSVj
= (X
+
}.L) grad div s
grad div s - }.L curl curl s
+ }.L div grad s (2.2.1)
where we have used Eq . (1.5.12) to rearrange the vector operators. Longitudinal Waves. The form of the 'equation of motion suggests that at least part of the vector s may be expressed in terms of the gradient of a scalar potential 1/;, since the equation then simplifies considerably: When s = grad 1/;, the equation for 1/;, .
div grad
.1.
'I'
=
V 2. 1. 'I'
X + 2}.L = -c~1 -iJ21/; ' c2 = iJt2 ' c P
(2.2.2)
is just the wave equation for the scalar wave potential 1/;, the three-dimensional generalization of Eq . (2.1.9). The wave velocity Cc is greater
§2.2]
Waves in an Elastic Medium
143
the larger are the moduli A and p. (i.e., the stiffer is the medium) and is smaller the larger p is (i.e. the more dense is the medium). When the solution of Eq. (2.2.1) is a gradient of a scalar, the dyadic operator m = V(Vif;) is symmetric, rotation dyadic mis zero and m = @5, the pure strain dyadic. For such a solution there is no twisting of the medium, only stretching and squeezing. Waves of this sort are called longitudinal or compressional waves. They can be propagated even in liquid and gaseous media, where the shear modulus p. is zero. But a gradient of a scalar potential is certainly not the most general vector field possible for the strain displacement s, as was shown on page 53. The most general vector field requires three scalar functions of position to specify, one for each component, whereas the gradient of a scalar is specified by a single function, the potential. Consequently, two more scalar functions of position are needed to specify the most general solution of Eq. (2.2.1). Of course, we could set up equations for each of the rectangular components of s, but this would result in three equations, each containing the three components, which would have to be solved simultaneously-a cumbersome procedure. Transverse Waves. It would be much better to utilize some of the properties of the vector operator V to obtain the other solutions, as we did for the gradient of the potential : since the curl of a gradient is zero, one term in the equation dropped out and the wave equation for the scalar potential resulted. This result [plus the results of Eq. (1.5.15)] suggests that we try the curl of some vector, for the divergence of a curl is zero and therefore the divergence term would drop out . Accordingly we let another solution for s be curl A, and the resulting equation for A is (2.2.3) - curl 'curl A = c~(a2A/at2); c~ = p./p which is also a wave equation, as we shall later demonstrate. The wave velocity c. for this wave is smaller than the velocity for longitudinal waves, being proportional to the square root of the shear modulus p. instead of the combination A + 2p.. It suggests that this part of the solution is a shear wave, which indeed turns out to be the case. For with this type of displacement the dilation 8 = div s [see Eq. (1.6.23)] is zero, so there is no expansion or contraction, and therefore the strain must be a type of shear. We shall usually call this type of wave the transverse wave. • This separation of the general solution int o a longitudinal part, which is the gradient of a scalar potential if;, plus a transverse part, which is the curl of a vector potential A as suggested on page 53, is a neat one, for these two waves travel at different speeds and any other separation of the solution would result in waves of both velocities being part of both solutions, certainly a more clumsy procedure.
144
Equations Governing Fields
[cH.2
But at first sight the two solutions we have obtained appear to be redundant. We mentioned earlier that only three independent functions of position are needed to give the most general solution for the vector s, but here we appear to have four : one for the scalar potential and three for the components of the vector potential. This redundancy is only apparent, however, for we do not use all the degrees of freedom of the vector potential A. The process of taking the curl of A to obtain s discards a part of A, namely, the part which can be expressed as a gradient of a scalar, and uses only that part which has zero divergence. Therefore, the part of A which is used to contribute to the general solution for s involves only two independent fun ctions of position, and these, with the scalar potential, make up the requisite three. To put the whole argument in another form , any vector solution of Eq. (2.2.1) can be split into two parts : a longitudinal part having zero curl, which can always (see page 53) be represented as the gradient of a scalar potential, and a transverse part having zero divergence, which can always (see page 54) be represented as the curl of a vector potential. Equation (2.2.2) shows that, if the solution starts out as a longitudinal one, it will continue to be longitudinal or, if it starts out transverse, it will remain transverse as long as the quantities A and Jl. have everywhere the same values. If A or Jl. or both change abruptly at a boundary surface or change continuously in a region of space, then wave reflection will occur and the longitudinal and transverse waves may become intermingled. In the present case the longitudinal part corresponds to wave motion of one velocity and the transverse part corresponds to wave motion of another, lesser velocity. Actually there are two independent transverse parts. One of these may be taken to be the curl of some solution A of Eq. (2.2.3) [incidentally the curl of a solution of Eq. (2.2.3) is also a solution of Eq. (2.2.3), as may be quickly verified]; this will be called the first transverse solution . The other transverse part may be taken to be the curl of the first solution (which is proportional to the zero-divergence part of A itself, as may be quickly verified); this will be called the second transverse solution of Eq. (2.2.1) . Wave Motion in Three Dimensions. Waves on a simple string are only of two general types : ones which travel to the right, represented by the general function F(x - ct), and ones which travel to the left, represented by f(x + ct) . In three dimensions many more types of wave motion are possible. Confining ourselves for the moment to the scalar wave potential 'P, we, of course, can have a simple generalization of the one-dimensional wave, 'P = f(a . r - cet) (2.2.4) where r = xi + yj + zk and a is a unit vector in some arbitrary direction given by-the spherical angles (J and cf> (see Fig . 2.8) . The wave motion here is all in one direotionvandthe comments made in Sec. 2.1.concerning
§2.2]
Waves in an Elastic Medium
145
one-dimensional waves need no modification to apply here. Such waves are called plane waves for reasons which will shortly become apparent. Inherent in our thinking of three-dimensional waves is the motion of a wave front. Crests and troughs of a wave often maintain their identity as they move along, which can be represented by surfaces everywhere perpendicular to the direction of wave motion and which move with the wave velocity c. These surfaces are called surfaces of constant phase or simply phase surfaces. For the simple plane wave form f(a· r - ct)
Fig. 2.8
'"
/ / Unit propagation vector a and radius vector r.
the surfaces are the planes a ' r = constant, perpendicular to the unit vector a, which gives the direction of wave motion. If the wave is a sinusoidal one, represented by the complex exponential term if; = Aei(",/c) (a'r-c/) for all points on one of the surfaces, the wave function has the same value of the phase angle of the complex exponential (which is why the surfaces are called phase surfaces). We can ask whether there are other types of waves in three dimensions having crests and troughs which maintain their identity as the wave moves along. A bit of investigation will suffice to convince one that only plane waves of the form given in Eq . (2.2.4) maintain their shape and size completely unchanged as they travel. It is possible to have waves, other than plane, which keep their shape but not their size; these have the form (2.2.5) if; = A(x;Y,z)f[cp(x,y,z) - ct] The function f provides for the motion of the wave, and the surfaces cp = constant are the surfaces of constant phase; the factor A provides for the change in size of the wave from point to point. Substituting this form into the wave equation V2Tj; = (1/c 2 )( a2if;/Ot2 ) results in the equation fV2A + (f' / A) div [A 2 grad cp] + Af"[grad 2 cp - 1] = 0 where the primes indicate differentiation of f with respect to its argument. If fis to be any arbitrary function of its argument (cp - ct), the coefficients of f, f', and f" must each be equal to zero: V2A = 0;
[grad cp)2 = 1; div [A2 grad cp] = 0
(2.2.6)
The last equation is equivalent to stating that the vector A 2 grad cp is
146
Equations Governing Fields
[cH.2
equal to the curl of some vector, and the second equation states that grad If' is a unit vector. These are stringent limitations on A and 'P, and not many solutions can be found. In other words, not many types of waves maintain their shape as they move through space. One solution of these equations has spherical wave fronts, 'P = r = V X2 + y2 + Z2, and has an amplitude A = B f r which varies inversely with the distance r from the center of the wave (B is a constant). The solution 1/! = (Blr)f(r - ct) represents a spherical wave radiating out from the cent er r = O. Naturally there can also be an ingoing wave (Dlr)F(r + ct). Another way of analyzing the same general problem consists in determining what curvilinear coordinat e surfaces can be wave fronts. If the wave equation in some coordinate system has solutions which are functions of only one of the three coordinates, then a traveling-wave solution can be formed from these solutions which will have one set of the coordinate surfaces as its surfaces of constant phase. Suppose that we choose an orthogonal, curvilinear coordinate system ~l, ~2, ~a, with scale factors hI, h2, b«and unit vectors aI, a 2, as. According to Eq. (1.5.11) the wave equation for 1/! in terms of these coordinates is 21/! 1 \ ' a (h lh 2h a a1/!) 1 a h lh 2ha ~ a~n ~ a~n = & at2
(2.2.7)
n
To simplify matters we separate off the time dependence in the exponential factor e- iwl • If, in addition, we can separate the space part of 1/! into three factors, each dependent on only one coordinate, the equation is said to be separable and one or another of the three families of coordinate surfaces,can be the family of phase surfaces of a wave . In other words, if the equation
a (h2ha a1/!) 1 h lh 2h a ah JL; ah
+ (W)2 c 1/! --
0
will yield solutions which are functions of ~l alone, then the wave equation is separable for the coordinate h . If one solution of this equation can be found , two independent solutions Y(~l) and Y(~l) can be obtained (this will be proved in Chap. 5), and the combination y + iY = A(~I)ei(", /c) ..(EI ) will serve to give us an expression for a simple harmonic wave having the coordinate surfaces h = constant as the surfaces of constant phase of the wave:
(2.2.8)
§2.2]
Waves in an Elastic Medium
147
This form of wave is more specialized than that given in Eq. (2.2.5), since we have contented ourselves here with sinusoidal dependence on time . In exchange for the simplification the limiting requirements on A and If' are not so stringent as in Eq. (2.2.6). For instance grad If' need not be a unit vector, which corresponds to the statement that the surfaces of constant phase do not travel with the velocity c everywhere. The functions A and If' may depend on w, so that the shape of the wave may differ for differing frequencies. Nevertheless we again find that a form of wave front which allows wave propagation with reasonable permanence of wave form is not at all common ; as we shall see in Chap. 5, only a few coordinate systems have a separable equation. The wave equation, it turns out, is rather particular about the shape of the wave ' fronts it allows. Further discussion of these points is not profitable here. It has been sufficient to point out that there is a close connection between the property of a coordinate system of allowing separable solutions of the wave equation (solutions consisting of factors, each functions of only one coordinate) and the possibility for the corresponding coordinate surfaces to be surfaces of constant phase for some wave. In Chap . 5 we shall deal with the problem of separability in more detail. Vector Waves. We must now return to the shear waves which cannot be represented by a scalar wave function but which can be represented by a divergenceless vector potential, satisfying the equation curl curl A + (1/c)2(a 2A/at 2) = 0 These also can have plane wave solutions : A
= (B
X a)f(a· r - ct)
(2.2.9)
where B is any constant vector and, .therefore, (B X a) is a constant vector perpendicular to the unit vector a, which determines the direction of propagation of the wave. Since the magnitude of A is independent of position along a line in the direction of A (i.e.: since the gradient of f is perpendicular to B X a), the divergence of A is zero, as was required. The curl of A is a vector perpendicular both to A and to a, curl A = (grad 1) X (B X a) = [B - a(a · B)]f' and the curl of this vector is again parallel to A, curl curl A = - (B X a)f" = - (1/c 2 )(a2Ajf)t2 ) as, of course, it must be in order to satisfy the vector wave equation (2.2.3). The directions of A and curl A are both perpendicular to the direction a of propagation of the wave, which is the reason for calling these waves transverse.
148
Equations Governing Fields
[cH.2
There is also a vector potential representing a spherical vector wave, analogous to the scalar spherical wave (B/r)f(r - ct) mentioned earlier. If a, is a unit vector pointed along the radius r, a, a unit vector perpendicular to a, and to the axis of the spherical coordinates, and a" = a, X a, another unit vector perpendicular to both, then the vector A = (a,,/r)f(r ± ct) is a solution of the vector wave equation which is satisfactory except along the spherical axis iJ = O. For the outgoing wave, for inst ance, and
curl A = (alP/r)f'(r - ct) curl curl A = -(a,,/r)f"(r - ct)
=
-(1/C2)(02A/ot 2)
It is obvious that the vector curl A is also a solution of the vector wave equation, so that the most general outgoing spherical vector wave is (a,,/r)f(r - ct)
+
(alP/r)F(r - ct)
The more complex problem of the separability of the vector wave equation will be discussed later. Integral Representations. More general types of waves can be constructed by adding up plane waves in different directions. As shown in Fig. 2.8 the vector a(8,ep) is the unit propagation vector pointed in the direction defined by the spherical angles 8, ep and r is the radius vector of length r, with direction defined by the angles iJ and 'P. The most general sort of scalar wave can be represented by the integral
if!
=
f depf sin 8 d8 f[ep, 8; r· a(8,ep) - ct]
(2.2.10)
where f is a traveling wave of shape depending on the angles 8 and ep. The limits of int egration are usually from 0 to 211" for ep and from 0 to 11" for 8, but they may extend to imaginary or complex values [such as from 0 to (11"/2) + i oo for 8]. The most general vector wave function can be formed in a similar manner: (2.2.11) A = fd¢f sin 8 d8 F[ep, 8; r· a(8,ep) - ct] where F(ep,8;z) is a vector function of ep and 8 and z which is pointed in a direction perpendicular to a(fJ,ep) . Since every element in the integrand is a transverse wave, ·t he result must have zero divergence. One can also express more specialized waves in this same manner. In the very important case of simple harmonic waves, for instance, with time factor e- iwt , the expression for the scalar wave becomes
if!
=
fdepfY(ep,8)e i ( wt c)(T co.
+
o - ct )
sin 8d8
(2.2.12)
where r cos n = r[cos 8 cos iJ sin 8 sin iJ cos (ep - rp)] = r· a(8 ,ep) and Y(ep,8) is some function of the spherical angles. For the vector solution Y is a vector perpendicular to a for every value of fJ and ep. In future
149
Waves in an Elastic Medium
§2.21
chapters we shall find it extremely valuable to express all the solutions of the wave equation in such an integral form. Stress and Strain. To return to physics after this incursion into mathematics, it will be well to compute the stresses in the medium corresponding to the various types of wave motion. For a compressional (longitudinal) wave traveling to the right along the x axis, the scalar potential is if; = f(x - cet) and the displacement of the medium at point x, y, z at time tis s = grad if; = iI'(q; - cet); I'W = (d/d~)fW The strain dyadic is dyadic is ~
=
A3 div s
@5
= i[Vs
+ /-L(vs + sV)
+ sVl
(2.2.13)
= iiI"(x - cet), and the stress
= [(A + 2/-L)ii
+
+
AOj kk)lI"(x - ct) ; I"W = (d2 /de)fW (2.2.14)
In other words the force across a unit area perpendicular to ,t he z axis is in the z direction and of magnitude (A + 2/-L)f", whereas the force across a unit area parallel to the x axis is perpendicular to the area and equal to AI". The motion is entirely in the z direction, and the tractile forces are all normal; no shear is present. For a shear (transverse) wave traveling to the right along the z axis, with motion parallel to the z axis, the vector potential is A = jF(x - c,t) and the displacement of the medium at point z, y, z at time tis s = curl A = kF'(x - c,t ); F'W = (d/d~)FW (2.2.15) The strain dyadic is ~
= /-L(Vs
+ sV)
@5
= (ik
= /-L[ik
+ ki)F"(x
+ kiW" (x
- c,t), and the stress dyadic is
- cst);
F" (t) = (d2/ dt 2)F(t) (2.2 .16)
since div s is zero. In this case the force across a unit area perpendicular to the x axis is in the z direction and of magnitude /-LF" ; the force across one perpendicular to the z axis is in the z direction and also equal to /-LF" . There is no tractile force across a surface perpendicular to the y axis. This stress is, of course, a simple shear in the x, z plane. Wave Energy and Impedance. To find the potential energy stored in a certain volume of medium when its strain dyadic is @5 = i(Vs + sv) and its stress dyadic is ~ = A3/@5! + 2/-L@5, we first find the increase in potential energy when the displacement s of the medium at x , y, z is increased by the small amount os. The work done by the stress forces on the medium in the volume can be computed in terms of the scalar product of the tractive force (~. dA) across each element dA of the surface of the volume and the displacement os of the element : ow = Hos, (~. dA)
= JH[div
(~.
os)]dv
Equations Governing Fields
150
[cH.2
where the first integral is over the surface bounding the volume in question and the second integral is over the volume itself. We have made use of Gauss's theorem, Eq. (1.4.7), to derive the second integral from the first. However, a juggling of components shows that, for any vector A and dyadic 58, the following formula holds : div (58 ,A)
=
(V. 58) . A
+ 58 :(VA) ;
where 58:1)
=
LBmnD nm
158· 1)1
=
mn
Therefore, fff div
(~.
ss) dv =
fff[(v.~) .
os + ~ :(Vos)] dv
Since (V . ~) dv is the net force on the volume element dv, which is zero when the medium is in equilibrium (as it is when we measure potential energy) , the integrand of the potential-energy integral becomes ~:(Vos)
=
~:o[i(Vs
= [hJ~llo~1
+ sV)] = ~:o~ + 21L~:0~] = o[i~ :~l
= o[il~ · ~11
where , since ~ is symmetric, I~ · Vsl = I~· sVJ . This represents the increase of potential energy due to the increment of displacement ss. It is clear, therefore, that the total potential energy due to the displacement field s is given by the volume integral
The kinetic energy is, of course, the integral of ip(asjat) 2 over the same volume. The total energy density in the medium is, therefore, w = ip(asjat)2
+ il~ · ~I;
W = fffwdv
(2.2.18)
For the plane compressional and shear waves given in Eqs. (2.2.13) and (2.2.15) the energy densities turn out to be
The flow of energy across any given closed surface may be obtained by finding the rate of change of the total energy inside the surface. Using Eq . (2.2.1) in the process, we find that
Motion of Fluids
§2.3]
a:
= =
=
151
fff [p (:~). (a;:) + I~> ~~IJ fff [(:i)' v · ~ + I~· v G~)IJ fff [(:~) ~ ff [(:~) .~ ]. dv
dv
div
.
] dv =
dA
The last integral, being a surface integral, must equal the flow of energy in through the surface to cause the increase in W . With a minus sign in front of it, it is the net outflow of energy across the closed surface. Therefore, the vector representing the energy flow density in a medium carrying elastic waves is
s
= -(as/at) . ~
(2.2.20)
This is not a surprising result. The quantity as/at is the velocity of the particle of the medium at x , y, z, The tractile force across an element of surface dA u perpendicular to as/at is ~ . dA u , and the expression for power is force times velocity. Since the dimensions of ~ are force per unit area, the dimensions of S are power per unit area. For the plane longitudinal wave given in Eqs. (2.2.13) and (2.2.14) the transmitted power is S = i(}. + 2JL)cclf"(x - cet)J2
(2.2.21)
and for the transverse plane wave given in Eqs. (2.2.15) and (2.2.16) it is S = i/lc.[F" (x - czt)J2
(2.2.22)
The density of energy flow for a plane elastic wave is usually called the We see in each case that the magnitude of the intensity is the energy density times the wave velocity. In a plane wave the energy moves along with the velocity of the wave. In these cases we can consider the quantities ccf" and c.F" , the amplitudes of velocity of the medium, as being analogous to an electric-current density and the quantities (}. + 2JL)J" and /IF'', the amplitudes of the tractile forces, as being analogous to voltages. The product of the two gives power density. The ratio of the two would give a quantity which could be called the impedance of the medium for waves of the type considered. For compressional waves the impedance is (A + 2JL)/cc = pce, and for shear waves it is /l/c. = pc«.
intensity of the wave.
2.3 Motion of Fluids A fluid differs from an elastic solid in that it yields to a shearing stress. We cannot expect to relate the displacement of a fluid with the
152
Equations Governing Fields
[cH.2
stress tensor, for if the displacement. were kept constant, the shearing stress would vary with time, or ifthe shearing stress were kept constant, the displacement would vary with time . It requires a constant rate of shear to maintain a constant shearing force in a fluid. This indicates (if it were not clear already!) that it is more convenient to express the behavior of a fluid in terms of velocities rather than displacements. Two types of description can be used ; one which gives the velocity of each particle of the fluid at each instant of time and another which gives the fluid velocity at each point in space at each instant of time. In the first description the vector field follows the particles of fluid as they move around; in the second case the field is attached to a fixed coordinate system, the vector at a given point giving the velocity of that part of the fluid which is at that point at the time . The two types of description of the motion of a fluid correspond in a distant way to the atomic and to the continuum picture of a fluid. An actual fluid, of course, is a collection of molecules, each moving under the influence of forces. Some of the forces are internal, due to other molecules nearby; the nature of these forces determines the compressibility of the fluid. Other forces are external, due to bodies at some distance, such as gravitational or electrical forces, which act more or less equally on all molecules in a given 'Small region. In a thoroughgoing analysis of the first type of description we would start by labeling each molecule by its position in space at t = O. For a detailed analysis we should also have to know the initial velocity of each molecule before we could expect to determine in detail their subsequent motions. For many problems, however, it will suffice to know only the average position and velocity of the molecules in each element of volume (such as the one dx dy dz at xo, Yo, zo) with dimensions large compared with molecular size but small compared with the total extent of the fluid considered. When these averages are obtained, the internal forces cancel out (except in determining the relation between pressure and density) and leave only the external forces acting on the portion of fluid in the element. By this averaging procedure we obtain equations for the gross motions of the fluid which disregard its detailed discontinuities and correspond to a continuous, nongranular approximation to the actual fluid. The discussion in Sec. 2.4 will show how this transition, from an overdetailed molecular picture to a smoothed-out, average picture for the fluid, is performed. The second type of description usually starts immediately from the smoothed-out approximation. The average velocity of those fluid particles which are close to the fixed point x, y, z at time t is computed as a function of t, as though the fluid actually were continuous. We shall choose the second method of representation, for it corresponds more closely to the types of fields studied in other parts of this
§2.3]
M olion of FJg.ids
153
chapter. The vector v (x ,y,z,t) is the velocity of that portion of the fluid which happens to be at x, y , z at time t. The expression div v-is the net outflow of fluid from the" region around z, y, z"; that is, dx diFdz div v is the net outflow of fluid from the element dx dy dz. If div v is everywhere zero, the fluid is then said to be incompres sible. The vector w = !- curl v represents the circulation of fluid "around the point z , y, z"; it is called the vorticity vector of the fluid (see page 42). If w is everywhere zero, the flow of fluid is said to be irrotational (in which case the vector v can be expressed as the gradient of a scalar velocity potential). This brings us back to the discussion of vector fields given in Sec. 1.2. As a matter of fact we used there the example of fluid flow to help us picture a vector field, and a number of terms, such as vorticity, flow lines, and net outflow, were chosen to further the analogy. We can now return to this point of view to obtain quantitative measures for the fluid motion. For instance, the flow lines plot the, average paths of the various particles of fluid. The differential equation for these lines is dx /v:r: = dy /v y = dz /v.. The number of flow lines crossing a given surface, which is equal to the outflow integral Jv . dA across the surface, is also equal to the average flow of fluid across the surface, and so on. If there is no vorticity (i.e. , if curl v = 0) and a velocity potential exists, the flow lines are everywhere perpendicular to the equipotential surfaces and constitute a natural coordinat e system for the problem. Equation of Continuity. Two general properties of the velocity field for a fluid should be mentioned before we go into details. One has to do with the relation between net outflow and change of density of fluid . If v is the fluid velocity and p is the fluid density at x, y, z, t, then pv is the vector representing the flow of mass per square cent imet er and dx dy dz div (pv) is then the net outflow of mass from the volume element dx dy dz . Since matter is neither created nor destroyed in most of the cases considered, this net outflow of mass must equal the loss of mass p dx dy dz of the fluid in the element. In other words
ap/at =
- div (pv)
(2.3.1)
This equation is called the equ ation of continuity for the fluid. From this equation it is obvious that for a fluid of constant density p (incompressible fluid) the net outflow div v must be zero. In some problems it will be convenient to assume that fluid is being created (or destroyed) at some point or points. Such a point is called a source (or a sink) of fluid. Naturally the equation of continuity does not hold there. The other general property of the velocity field is related to the fact that the coordinat e system for the vector field does not move with the fluid. To find the rate of change of some property F(r,t) of the fluid
Equations Governing Fields
154
[cH.2
at or near a specified fluid particle (whose position is given by the radius vector r at time t) we cannot just compute the rate of change aF/ at of F at the point x, y, Z, for the particle does not usually stay at the one point. The change in F we are interested in is the difference between the value F(r,t) at point x, y, z, where the fluid particle is supposed to be at time t, and the value F(r + v dt, t + dt) at point x + v", dt, y + Vy dt, Z + v. dt, which is where the particle is at time t + dt. This difference, when expanded out and the first-order terms kept, turns out to be dF = [( aF/ at) + v . v F] dt. The rate of change of the property F of the fluid, which is denoted by the total derivative sign, is therefore given by the equation dF dt
= aF at
+v
»
vF
(2.3.2)
in terms of the time rate of change of F at point x , y, z (given by the partial derivative of F) and the space dependence of F near x, y, z (given by the VF term) . For instance, the acceleration of the part of the fluid which is "at " z, y, z at time t is dv dt
av
= at
+ v . Vv
av
= at
+ tV(v . v)
- v
x
curl v
av
= at + tV(v 2 )
-
2v X w
(2.3.3)
The second form of this expression is obtained by a reshuffling of vector components, and the third form is obtained by substituting the vorticity vector w for t curl v. According to the discussion of page 41, the magnitude of w equals the angular velocity of the portion of fluid "near" x, y, z, and the direction of w is that along which a right-hand screw would move if it were turning with the fluid. The rate of change of a scalar property of the fluid can also be computed. The rate of change of density of a given element of fluid, which happens to be "at" x, y, z at time t as it travels along, may also be calculated by the same method, with the following result: dp dt
dp
= at
+ v . grad p
(2.3.4)
But the equation of continuity has that ap/at = - div (pv), so that (dp/dt)
= - div (pv)
+v
»
grad p = -p div v
(2.3.5)
Solutions for Incompressible Fluids. When the fluid density p is everywhere constant, the equation determining v is just div v = o. The most general solution of this can be expressed in terms of a scalar and vector potential (as shown on page 53) . v = curl A
+ grad if;i
V 2if;
= div grad if; = 0
(2.3.6)
§2.3]
155
Motion of Fluids
The vector A can be any well-behaved ve ctor field which satisfies the boundary conditions. The equation for the velocity potential if; is called Laplace's equation. It will be discussed at great length later in the book. The flow lines , discussed on page 12, are, of course, perpendicular to the surfaces of constant velocity potential. When there is no vorticity, A = 0 and the velocity is completely determined by the scalar potential. If, in addition, the flow lines lie in 00
o
0
=+2.0
J 7
=+ 1.0 • 4>=0
'\4>=0 4>=21T
+-f
4> =- 2.0 4>=-3.0
~
Q
r0 I
Q
C\J
.,. .,." .,." I
00
I
9
0
.,."
I
.;
9 + " ~
a+
.,."
1:1",
'"
00
-eFig. 2.9 Potential '" and flow ~ lines for two-dimensional flow of incompressible fluid. Circulation is present, so there is a discontinuity in tit at cf> = o.
parallel planes, the velocity potential can be made a fun ction of only two coordinates, and the motion is called t wo-dimensional flow. This special case has numerous important applications in aerodynamics. Here the flow line s and the potential lines made an orthogonal set of curvilinear coordinates in the two dimensions. The equation for the flow lines is (see page 12) dxj v", = dyj vlI
Therefore if
VII
or
-
VII
dx
+
V",
dy
=0
= - (ajax) and v'" = ajay, we have that
cajax) dx
+ (ajay) dy
= 0 or (x,y) = constant
156
Equations Governing Fields
[cH.2
along a stream line. The function is called a stream function j it is related to the velocity potential 1/1 by the relations iJ/iJy
=
iJ1/I/iJx,
iJ/iJx = - (iJ1/I/iJy)
which are called Cauchy-Riemann equations and which will be discussed in much greater detail in Chap. 4 in connection with functions of a complex variable. We have mentioned earlier that the "density of flow lines" is a measure of the total flow and therefore of the velocity of the fluid . This can be quickly shown in the two-dimensional case, for the outflow integral Jv - dA between two stream lines (x,y) = 2 and (x,y) = 1 can be reduced to a line integral in the xy plane. The outflow integral concerned is between two planes parallel to the z, y plane a unit distance Y
y
z
Z Fig. 2.10
Flow integral for two-dimensional flow.
apart, and the element of area dA can be a thin strip of unit length and of width equal to ds, where ds is the element of length along the path from 1 to 2 in the x, y plane. The direction of dA is, of course, perpendicular to the direction of ds; in fact dA = ds X k where, of course, ds is always perpendicular to k. The flow integral is then
/2 v- dA =
h V. 2
(ds X k)
h = h (v =
2
(v X ds) . k
2
x
dy -
Vy
dx)
=
ft2 d = 2 - 1
In other words the total flow of fluid along the region enclosed between the planes z = 0 and z = 1 and the surfaces defined by the flow lines 1 and 2 is just the difference between the values 2 and 1 of the flow function . The usual boundary conditions in fluid flow are that the velocity is tangential to all bounding surfaces. When viscosity is important, we must require that the fluid immediately next to the surface move with
157
Motion of Fluids
§2.3]
the surface; i.e., if the surface is at rest, even the tangential component of the velocity must go to zero at the boundary. If viscosity is not large, however, we may safely assume that the fluid may slip along the surface without appreciable drag, so that a finite tangential component next to the surface is allowed. Examples. A few simple examples will perhaps clarify some of these statements. and definitions. The scalar potential and related velocity field, given by the equations if;
= -(Q /r); v = (Q/r 2 )a..
(2.3.7)
have been shown on page 17 to be due to a point source of fluid at the origin (r = 0) in a fluid of infinite extent. As indicated in Eq. (1.2.9),
z
z
x
x
Fig. 2.11
Flow lines from a point source.
the total normal outflow from the source is 411"Q, which is called the strength of the source. Since no vector potential enters here, there is no related vorticity vector w = i curl v veloclty v.'II/' (see page 153) and the flow is said to 10~11~;/~~~~~~~~~~~ be irrotational. ~ .. Another case represents the shearing flow which results when fluid is between two plane parallel surfaces (z = 0 and z = 1 for instance) one of which moves with respect to the other. If the surface at z = 0 is at Fig. 2.12 Flow velocity for fluid in rest and the surface at z = 1 is moving shear. in the x direction with unit velocity, the fluid between is said to be subjected to a unit shearing rate. The velocity field, which is at rest with respect to both surfaces and uniformly distributed between, is derivable from a vector potential : // // /(/////!jjl.
A = _j.Z2j ; V = zi
(2.3.8)
There is no normal outflow (div v = 0), but the vorticity vector w = curl v = ij is uniform over the region.
i
158
Equations Governing Fields
[cH.2
Another example of velocity field, exhibiting both vector and scalar potentials, is the following, expressed in cylindrical coordinates, r, 41, z: if; = { 0; 2..1.
wa'l';
r r
><
a; A = {0-!wr a ;
2a z
;
r
a
(2.3.9)
where w is the angular velocity of the fluid inside the cylinder r = a. The velocity vector is then wra . r v-' - { (wa2jr)a ; r
a
The vorticity vector w = ! curl v is waz (as is to be expected from its definition on page 41) for r < a and is zero for r > a. We note that for r > a the velocity field is that found outside a simple vortex line, as given in Eq. (1.2.11). Here we have made the vortex motion finite in Z
Fig. 2.13 Flow velocity, flow lines, and surface of zero pressure (free surface) for simple vortex.
extent (r < a) rather than concentrated in an infinitely narrow line, as was done in Chap. 1. Stresses in a Fluid. Before we can go much further in this analysis, we must study the internal stresses in the fluid. There is, of course, the pressure, which inay be due to gravitational or other 'forces on the fluid or may be due to a compression of the fluid or both. In addition, there are the frictional stresses due to rate of change of strain, proportional to the velocity vector v instead of the displacement vector s, as was the case with elastic solids. We saw on page 67 that the strain in an elastic solid could be represented by a symmetric dyadic @5 = !(Vs + sV). The rate of change of this strain is also a dyadic U = !(Vv
+ vV)
The expansion factor IU! = div v is, by the equation of continuity, proportional to the rate of change of density of the fluid (which is zero if the fluid is incompressible). The "remainder" of U, which can be r~pre sented by the dyadic
= !CVv + vV) - is' div v; !U.! = 0 corresponds to pure shearing rate and would represent the form of the rate of change of strain dyadic for incompressible fluids.
U.
Motion of Fluids
§2.3]
159
Now we must work out the form of the stress dyadic and how it depends on the rate of strain. When the fluid is not in motion, the only stress is the static pressure p, which is completely symmetrical;
so that the force across any element of area dA is -p dA (the negative sign indicating pressure, negative tension). When the fluid is expanding without shear (U. = 0), it is possible that there is a frictional effect to pure expansion and that the pressure is altered by the rate of expansion (this turns out to be the case with all fluids except monatomic gases). In such a case the stress would be .
st =
(-p
+ A div v)3
where A can be called the coefficient of expansive friction. If, in addition, there is a rate of shear of the fluid, there will be a proportional shearing stress, 27]U., where 7] is called the coefficient of viscosity. The total stress dyadic is therefore related to the pressure and to the rate of strain by the following equation:
st =
-p3
+ A3IU! + 27]U.
= - (p + 'Y div v)3 + 7](Vv + vV)
(2.3.10)
where 'Y = "h - A can be called the second viscosity coefficient. This equation is similar to Eq. (1.6.28) for the stresses in an elastic solid, except that velocity v now enters where displacement s occurred before (and, of course, the pressure term has been added). This difference is not trivial, however, for a force proportional to a velocity is a dissipative force whereas the stresses in Eq. (1.6.28) are conservative. One might, of course, have assumed that the constants 'Y and 7] were dyadics rather than scalars, but we have less reason to expect such complication here than we did in isotropic solids . We expect that_a fluid is isotropic, and experimental results seem to bear this out. Returning to our examples, we can use the expression for the dyadic i (Vv + vV) in spherical coordinates, given on page 117, to calculate the stress tensor, (2.3.11) for the flow from a simple source given in Eq. (2.3.7) . In other"words, the force across a surface element perpendicular to a radius vector is a compressional one of magnitude p + (47]Q/r 3 ) , whereas the compressional force across any surface perpendicular to the former element is p - (27]Q /r 3 ) . When there is viscosity (7] > 0), therefore, the force on an element of fluid is not isotropic, and for a large enough flow (Q large) "Or a small enough radius, the force "across" a radial flow line becomes a tension, whereas the force "along" the flow line is everywhere com-
160
Equations Governing Fields
[cH.2
pressional. This sort of force is, of course,' needed to change the shape of an element of fluid as it travels out radially from the source, for the element must spread out in all directions perpendicular to the radius and must correspondingly become thinner radially. If the fluid is viscous, it takes a nonisotropic force to produce such a deformation. The constant 'Y does not enter, for we are assuming that the fluid is incompressible and div v = O. For the unit shear case given in Eq. (2.3.8) , the stress tensor is ~
= -p3
+ '7(ki + ik)
(2.3.12)
Here the force on a unit area of stationary surface at z = 0 is just ~ . k = -pk + '7i. The component -pk normal to the surface is, of course, the pressure (the minus sign indicating force into the surface) . The component '7i parallel to the motion of the upper surface (at z = 1) is that due to the viscosity of the fluid; in fact we have set up just the conditions corresponding to the fundamental definition of the coefficient of viscosity '7 of a fluid ('7 is the magnitude of the tangential force per unit area for a unit rate of shear). In the last example, given in Eq . (2.3.9), we have for the stress
~ _ { -p3 ;
-
-p3 - (4'7wa2/r2)(a,.a(p
+ a(pa
r r) ;
r
a
(2.3.13)
In the portion of fluid for r < a the only stress is the isotropic pressure, which is not surprising, for this portion of the fluid is rotating as a rigid solid with angular velocity w. Outside this vortex core, for r > a, there is shear of the fluid, and the force on unit area perpendicular to r has a tangential component - (4'7wa2 /r2)a(p, representing the drag of the fluid outside the cylinder of radius r on the fluid inside the cylinder (or vice versa) . The force on an element of fluid at x, y, z is, as we have stated, (V • ~ + F) dx dy dz . This must equal the acceleration of the element av f at times its mass p dx dy dz. The resulting equation (which is obtained by the use of the formulas on page 115) p(av/at)
+ pV· Vv = F + V· [-(p + 'YV · v)3 + '7(Vv + vV)] = F - grad [p - (t'l + X) div v] + '7V2V (2.3.14) = F - grad [p - (t'l + A) div v] - '7 curl curl v
where 'Y = ·h - A, serves to calculate the pressure if the velocity is known or enables the transient and oscillatory motions to be computed. This equation, together with the equation of continuity, Eq. (2.3.1), and the equation of state, relating the pressure and the compression of the fluid, is fundamental to all the multiform problems encountered in fluid dynamics. The various forms are obtained by considering one
§2.3]
Molioti of Fluids
161
term after another in this equation negligibly small and combining the remaining terms. Bemouilli'sEquation. The simplest case is for ' the steady-state motion of an incompressible fluid, for then iIv/ at and div v are both zero. In addition we assume that the external force F can be obtained from a potential energy V, F = - grad V, and we use the vector relation j- grad v2 = v • Vv
+v
X curl v
We finally obtain 21) curl w - 2pv X w = - grad U U = V p j-pv2 ; W = j- curl v
+ +
(2.3.15)
The scalar quantity U can be considered to be the energy density of the moving fluid. The first term is the potential energy of position due to external forces; the second term is the kinetic energy density. If the fluid motion is irrotational, the vorticity vector w is zero and U is a constant everywhere for a given flow pattern. In this case we determine the fluid velocity, in terms of a velocity potential, from the boundary conditions and then calculate the pressure from the equation p = U - V - j-pv2
(2.3.16)
where U is a constant determined by the boundary conditions. This is called Bernouilli's equation for incompressible fluids (p = constant). We note that it is possible for the solution to require a large enough velocity, in certain regions , so that the pressure, computed from this equation, would turn out to be negative. In principle this cannot happen, for cavitation would result and the boundary conditions would be modified. A very large number of problems of practical interest can be computed with fair accuracy by assuming that the flow can be represented by a velocity potential (i .e., irrotational flow) which is a solution of Laplace's equation. The pressure at any point can then be computed from Bernouilli's equation. Many problems even in aerodynamics can be calculated in this manner, although air is far from being incompressible. Only when the velocity of an important portion of the air approaches the speed of sound does the approximation become invalid. The more complicated case of supersonic flow will be touched later in this section. As an example of irrotational, incompressional fluid motion, we return to the flow from a simple source, given in Eq. (2.3.7). If we neglect the gravitational potential, the pressure as a function of r is p", (pQ2/2r 4 ) , where P'" is the pressure an infinite distance from the source . We see that, if the actual size of the source is too small (r too small) , the pressure will be negative and cavitation will result. Finally we consider the case given in Eq. (2.3.9) of a vertical vortex of radius a. This time we shall take into :account the gravitational
162
[CH. 2
Equations Governing Fields
potential V = pgz. For r > a the vorticity w is zero, so that U is a constant. Suppose that the fluid has a free surface (p = 0) at z = 0 when it is at rest. The constant value of U is therefore set so that p = 0 is at z = 0 and r = 00; that is, pgz + p
+ (pw
2a 4
/2r 2 ) = 0;
r
>
a
For r < a the vorticity w is not zero but it has zero curl, so that grad U = 2pv X w = 2pw2 rar • Integrating for U and adjusting the constant of integration so that the pressure is continuous at r = a, we have piJz
+ p + (pw /2)(2a 2
2
-
r2 ) = 0;
r
a
In both of these cases, the viscosity has had no effect on the pressure, because the only term involving viscosity in the equation for the pressure for steady-state motion of incompressible fluids is one involving the curl of the vorticity w, and the examples have been simple enough so that curl w was zero. Other examples can be worked out for which curl w is not zero and the viscosity does have an effect on the pressure, but the most frequently encountered examples of this sort are cases where v and p change with time. The Wave Equation. The first examples to be considered of nonsteady motion will be for small-amplitude vibrations. In this case all terms in Eq. (2.3.14) involving the squares of v can be neglected, and we obtain the simpler equation p(av/at) = - grad (p
+ V) + (h + X) grad div v -
l'/
curl curl v (2.3.18)
where we have again set F = - grad V and where we do not now assume that the fluid is incompressible. In order to get any further we must discuss the relation between the pressure and the state of compression of the fluid. Flow of material out of any volume element will reduce the pressure in a compressible fluid; in fact for any elastic fluid , as long as the compression is small, the rate of change of p is proportional to the divergence of v, ap/at = - K div v. The constant K is called the compressibility modulus for the fluid under consideration. When the displacements are small, we can write this in terms of displacement s: p = - K div s, as/at = v. We have seen on page 53 that every vector field can be separated in a unique way into a part which is a gradient and a part which is a curl. Here we utilize this fact twice, once by setting the unknown velocity v equal to the gradient of a velocity potential'" plus the curl of a vector
,§2.3]
Motion of Fluids
163
potential A. Inserting this into Eq. (2.3.18), we equate the gradients and curls of each side separately. The equation for the curls is p(aAjat) = -1J curl curl A
(2.3.19)
This is not a vector wave equation, but a vector analogue of the diffusion equation mentioned on page 137 and to be discussed in Sec. 2.4. Since only the first time derivative of A occurs instead of the second derivative, solutions of this equation are not true waves, propagating with definite velocity and unchanged energy, but are critically damped disturbances, dying out in time and attenuated in spatial motion. They will be discussed more fully in Chap. 12. We note that the pressure is not affected by these waves. We note also that the equation for the vorticity w = i curl v is identical with that for A. Viscosity causes vorticity to diffuse away, a not unexpected result. Collecting the gradient terms on both sides of the equation derived from Eq . (2.3.18) and differentiating both sides with respect to time finally give us the equation for the longitudinal wave:
+ ~p (!TJ + A)V2 (a1/;) . at '
21/;
a 2 = c2V21/; at
c2 =
~p
(2.3.20)
When the compressional viscosity t1J + A is small , ordinary compressional waves are transmitted through the fluid with velocity c, and all the remarks we have made concerning compressional waves in elastic media are applicable here. If this is not zero a damping term is introduced. For instance, for simple harmonic waves, with time dependence given by the exponential e- iwt , the equation for the space dependence of 1/; is V2y;
+
2
2
c
-
w 1/; i(w j3p)(41J
+ 3A)
= 0
In other words the space dependence will have a complex exponential factor, representing a space damping of the wave. On the other hand if a standing wave has been set up , with space part satisfying the equation V 21/; + k21/; = 0, the equation for the time dependence of 1/; is ' 21/;
a2 at
+ ~ (!TJ + "A)k p
2
a1/; at
+ c k y; = 2 2
0
which is the equation for damped oscillators in time. Irrotational Flow of a Compressible Fluid. Our next example of the different sorts of fluid motion represented by Eq. (2.3.14) is that of the steady, irrotational flow of a fluid which is compressible. This is the case of importance in aerodynamics when the fluid velocity approaches the speed of compressional waves, C = v'KfP, discussed in the previous
164
Equations Governing Fields
[cH.2
subsection, Since the applications are nearly always to the flow of air, we may as well specialize our expression "for the compressibility K to the case of a gas. We cannot go far enough afield here to discuss the thermodynamics of a perfect gas in detail; many texts on this subject are available for reference. We need only to write down two equations relating the pressure p, density p, and temperature T of a gas during adiabatic expansion (expansion without loss of heat contained in the gas) : pjpo = (pj po)-r = (T j T o)-r/(-r- o
(2.3.21)
where the indices zero designate the pressure, density, and temperature at standard conditions (for instance, where the fluid is at rest) . Another way of writing this is to relate the pressure and density to the entropy S of the gas : pjp-r = Ae a S An adiabatic expansion is one for constant entropy S . The constant "I is the ratio of specific heats at constant pressure and constant volume (its value for air is 1.405). Taking the differential of this equation, for constant S, dpjp = 'Ydpjp, and using the equation of continuity, (2.3.5), we get dp jp = -"I dt div v. Comparing this result with the definition of the compressibility modulus K we see that K = 'YP and that thespeed of sound (compressional waves) in a gas at pressure p and density p is
c=
VYiiTP
= v'dpjdp
(2.3.22)
In flow of a compressible gas both pressure and density (and therefore the speed of sound) change from point to point in the fluid . The relation between them is obtained from Eqs. (2.3.15), where again we start by considering irrotational, steady-state flow (w = 0, av jat = 0 we also neglect the potential V). Since now p is not constant, the integration of grad U = 0 is a little less simple than before. Both pressure and density turn out to be functions of the air speed v at any point. The maximum pressure P» and related density Po and speed of sound Co are for those points where v = 0 (stagnation points). At any other point Eqs. (2.3.15) and (2.3.21) indicate that v2 = -2
l
v
dp POIl-r - = 2 -v=o P Po
l
p
pO
_
P
l/-y
2"1
v» [
dp = - - 1 "I - 1 Po
(p)(-r-1)/ -Y] po
This indicates that there is a maximum fluid velocity v... x = y2'YPojpO('Y - 1),
for which the pressure is zero. This would be the velocity of flow into a vacuum for instance. For air at 15PC at the stagnation points (To =
§2.3]
Motion of Fluids
165
288) this limiting velocity is 75,700 cm per sec. At this limiting velocity the speed of sound is zero (since the pressure is zero). At the stagnation point the air is at rest and the velocity of sound, vi "(Po/po, is maximum. Therefore as v goes from zero to Vmax , the dimensionless ratio M = vic goes from zero to infinity. This ratio is called the Mach number of the air flow at the point. If it is smaller than unity, the flow is subsonic; if it is larger than unity, the flow is supersonic. The equations giving pressure, density, temperature, sound velocity, and Mach number at a point in terms of the fluid velocity v at the point and the pressure and density P» and Po at a stagnation point are p = Po[l - (V/Vm ax)2]'Y/('Y-o = po(C/CO)2'Y/('Y- 1) P = po[l - (V/Vmax)2]1/ C'Y- 1) = po(c/co)2/('Y- 1) T = T o[l - (V/Vm ax)2] = T o(C/ CO)2 2 C = v·Hy - l)(v~ax - v )
M = Vm u Co
~ = ~ ('1 :
1)
(V~axV~
(2.3.23)
V 2)
= V2"(po/ poe '1 - 1) = 75,700 cm per sec = Vma x V('Y - 1)/2 = V"(po/po = 34,100 em per sec
The velocity v. at which the Mach number M is unity (fluid speed equals sound speed) turns out to be equal to V('Y - 1)/("( + 1) Vmax = 31,100 cm per sec for air at 15°0 (To = 288) at stagnation points. At this speed the pressure, density, etc ., are P. = 0.528po, P = 0.685po, T = 0.832To = 2400 K = -33°0, c. = v•. Subsonic and Supersonic Flow. Several examples will show the importance of the region where M = 1 and will indicate that the phe-
-
-
Fluid
Fluid
Velocity
Velocity
Fig. 2.14 Propagation of a disturbance through a fluid traveling past small obstruction at P with velocity smaller (left) and larger (right) than speed of sound.
nomen a of gas flow for speeds above this (supersonic flow) are quite different from the phenomena for speeds below this (subsonic flow). As a very simple example, we suppose air to be flowing past a small object at rest at point P in Fig. 2.14. The presence of the object continuously decelerates the air in front of it, which continuously produces a sound wave in front of it. If the air velocity is less than that of sound, these waves can travel upstream from the obstruction at P and warn the fluid of its impending encounter, so to speak. But if the fluid is moving faster than
166
Equations Governing Fields
[CR. 2
sound, then no warning can be sent upstream and the expanding wave fronts are carried downstream as shown in the right-hand sketch of Fig. 2.14. The envelop of these waves is a "bow wave" of disturbance, which is called a Mach line or Mach surface. The first intimation of the presence of the obstruction at P occurs when the air strikes this line or surface. Incidentally, it is not hard to see that the angle of inclination of this line , the Mach angle, is given by the equation a
= sirr'" (11M) = sin:' (cjv)
We shall come back to these Mach lines later. As another example, consider air to flow along a tube of varying cross section S(x), as in Fig. 2.15. In order that no air pile up anywhere (i.e., that there be steady flow), the same mass Q of air must pass through pressure p(x) Total Mass Flow Q density p (x)
l-----------~-----------x_
Fig. 2.16 Air flow in tube of varying cross section. Lower plot shows three curves of possible variation of M = vic along tube.
each cross section. If the tube does not change too rapidly with change of x, and if the inner surface of the tube is smooth, the density and velocity will be nearly uniform across each cross section, and with fairly good approximation, we can say that p and p and v are all functions of x alone. Then, to this approximation, for steady flow
Q = S(x)p(x)v(x) = or
In S =
~11n
~-
Vmax
(SPo jvmax2/t-y-ll)(v~.x
+ In (~) ~
- In v -
- v2)1 /(-y-ll v
_l-, In (v~ax ~-
- v2 )
Differentiating this last equation with respect to x and using the equation for c given in Eq. (2.3.23) , we obtain
1: dS = ! dv (M2 Sdx
vdx
_ 1) ' '
M = ~
(2.3.24)
c
Therefore if the flow is everywhere subsonic (M
< 1),
wherever S
decreases in size, the air speed v increases, and vice versa. On the other hand if the flow is everywhere supersonic (M > 1), wherever S decreases,
167
Motion of Fluids
§2.3]
the air speed v decreases, and vice versa. In each case wherever S has a maximum or minimum, there v has a maximum or minimum. These cases are shown in the lower plot of Fig. 2.15. If, however, the pressure Po, total flow Q, etc ., are adjusted properly, the Mach number M can be made to equal unity at a minimum of S . In this case dv/ dx need not be zero even though dS / dx is zero, and the velocity ' can increase from subsonic to supersonic as it passes through the constriction (or, of course , it could start supersonic and end up subsonic) . This case is shown by the center curve in the lower part of Fig . 2.15. Velocity Potential, Linear Approximation. We must now set up the equation which will enable us to compute the vector velocity field to satisfy any given boundary conditions. As with the irrotational flow of incompressible fluids, we assume that this field can be obtained from a scalar velocity potential field, v = grad 1/;. The equation for I/; comes from the equation of continuity, Eq. (2.3.1) for iJp/iJt = 0,
°°= =
or
Therefore, if v
div (pv) = div [(PO/Vm&X1/'Y-I)(v~&X - v2)1!'Y- I V] div [(v~&X - v2)1!'Y- I V] =
grad 1/1, (2.3.25)
where XI = X, X2 = y, Xa = Z, and c2 = ·H'Y- 1)(v~&X - [grad 1/11 2) . For two-dimensional flow the equation becomes
:~ [1 - ~ (:~YJ + ~~ [1 - ~
(:tYJ
=
~iJ:2~y:~:t
(2.3.26)
There is also a flow function which defines the lines of flow and which measures the mass flow of air between two flow lines. We obtain this from the equation of continuity for steady flow, div (pv) = 0, for we can set v
'"
iJl/; iJx
Po iJ - p iJy'
= - = - -'
v
11
iJl/; iJy
Po iJ p iJx
=-=--
and then div (pv) is automatically zero. Likewise , as we showed on page 156, the total mass flow between two flow lines (per unit of extent in z) is equal to Po times the difference between the values of for the two flow lines. The equation for is similar to that for 1/1:
168
Equations Governing Fields
[cH.2
Equation, (2.3.25) is, of course, -a nonlinear equation for if;, one quite difficult to solve -exactly. When M is small (subsonic flow), J may be neglected to the first approximation, and the equation reduces to the linear Laplace equation characteristic of incompressible fluids. When the solution of Laplace's equation, if;o, is found for the particular case of interest, J 0 may be calculated for each point from if;o by use of the equation for J. Then a second approximation to the correct if; may be obtained by solving the Poisson equation v 2if; = J o, and so on.If M is not small, however, such iterative methods cannot be used and other approximate methods must be used. One technique is useful when the flow is not greatly different from uniform flow, v = v", a constant. In this case the direction of the unperturbed flow can be taken along the x axis, and we can set v
= vJ
+ VI;
if;
= v.,x + if;1
where VI is small compared with V'" though v" is not necessarily small compared with c. To the first order in the small quantity vI/c, we have
+
iJ2if;1 (1 _ M2) iJ2if;1 iJx2 " i J y2
+ iJ2if;1 ~ 0 iJ z2
(2.3.28)
where M~ = [2/(1' - 1)][v~/(v~ax - v~)] = v~/c~ is the square of the Mach number of the unperturbed flow. This equation, being a linear one in if;, can be solved to determine the steady flow around irregularities in the boundary surfaces as long as the irregularities do not produce large changes in air velocity near them. Mach Lines and Shock Waves. Equation (2.3.28) again shows the essential difference in nature between subsonic and supersonic flow. The difference can be illustrated by a two-dimensional case, where the equation is iJ2if;1 (1 _ M2) + iJ2if;1 = 0 (2.3.29) iJx2 " i J y2 When M" is less than unity, this equation can be changed into a Laplace equation for if;1 by changing the scale of y to y' = y VI - M~ , x' = x. Therefore the flow lines and potential surfaces are similar to those for incompressible flow, except that the y axis is stretched by an amount I /V1 - M~ . However, if M" is larger than unity, we can no longer transform the equation into a Laplace equation, for the iJ2if;/ iJx2 term changes sign and the equation is more analogous to a wave equation (see page 124) with x analogous to time and the " wave velocity" c" = I/vM~ - 1. Solutions of the resulting equation are . - . if;1 =f(y - c"x)
+ F(y + c"x)
§2.3]
Motion of Fluids
169
As mentioned on page 166, any irregularity in the boundary shape (which is, of course, here a plane parallel to the z, z plane) produces a "bow wave" which spreads out at an angle a = tan- 1 (c,J = sin- 1 (l jM u ) to the x axis, the direction of unperturbed motion. This is the Mach angle mentioned on page 166. . In two dimensions we have also an approximate equation for the flow function cfl, discussed on page 167. We assume that cfl = (pj po)vuY + cfl l , and inserting in Eq. (2.3.27) and neglecting terms smaller than the first order in cfl l we have 2
aax2 cfl l (1
+ aay2 cfl 2
_ M2) u
l
= 0
(2.. 3 30)
which is similar to the approximate equation for the correction to the velocity potential y,..
p~ /b
A
- - - :----=-~/;
0 ,
Li~orized SOlut~-rI-,------
y:
i :
s-:': TO = no(p/po)lmQ.IMQm); if Qm/Q is independent of p (2.4.55) and the number of particles having momenta between p and p
+ dp is
§2.4]
Diffusion and Other Percolative Fluid Motion
199
for p < po. The last form d[ solution is valid only if Qm/Q does not vary with p. If there is no absorption, (Qa = 0) Vt is independent of T and p is inversely proportional to p4Qm. This must be, for if there is no absorption, the rate of degradation of momenta Vt must be the same for all speeds. This solution, of course, predicts an infinite number of particles having infinitesimal speeds. If our analysis of energy lost per collision holds to the limit of zero speed, we must have this infinite population in order to reach a steady state. In reality, of course, the atoms of the medium are not completely at rest, and therefore, at low enough particle velocities the analysis made above, assuming that the particles always lose energy, becomes invalid. Therefore Eq. (2.4 .55) does not hold for particles having kinetic energies of the same magnitude as the mean energy of the atoms or smaller. As another example, we can imagine one particle of momentum P» introduced (isotropically) per second at the point Xo , Yo , Zo in an unbounded space . Then Vt for T = TO would be a delta function o(x - xo) . . o(y - Yo)o(z - zo), and using the same methods which resulted in Eq. (2.4.43), we obtain Vt = G(x - Xo,Y - Yo,Z - zolr - TO), where G(X,y ,ZIT - To )
=
{[o;47r(T3niQ2 JI [ - TO) exp T=
r
ee
}p
o:
]
3niQ2r2 T < TO 4(T _ TO) - 7j (T - To) ; T > TO
(MQ) (d P); mQm p
p(r,p) =
(2.4.56)
MQVt 4 nt mp
Finally we can compute the result if q(xo,Yo,zoIPo) dpo particles are introduced isotropically per second per cubic centimeter at the point Xo , Yo, Zo in unbounded space, in the range of momentum between Po and Po dp«. The number of particles introduced per second between the" ages" TO and TO + dTo is, therefore, (pomQm / MQ)q(xo,Yo,zoh) dTo , where Po is related to TO in the same way that p and T are related in Eq. (2.4.56) . The resulting steady-state distribution of particles in space for different momenta p can be obtained from the solution for Vt :
+
Vt(X,y,ZIT) = mQmjT MQ _ '" Po dTo
f'"_ '" dxo f'"_ '" dyo
t
_ '" dzo•
• q(xo,Yo,zoh)G(x - Xo,Y - Yo,Z - ZolT - TO)
(2.4.57)
where the number of particles per cubic centimeter in the range of momentum between p and p + dp is p2p dp = (M1/I/ ntp2Qm) dp. For some simple forms of q the integration can be performed and a closed analytic solution can be found for 1/1. Recapitulation. Many other applications of the diffusion equation can be found . The only requirement for its occurrence is that some
200
Equations Governing Fields
[CH. 2
quantity (density, partial pressure, heat, etc.) satisfy two requirements : first, that it obey the equation of continuity, that the rate of change of the quantity with time ue equal to minus the divergence of the flow of the quantity and, second, that the flow of the quantity be proportional to the negative gradient of the quantity. Other solutions will be given in Chap. 12. Since the time derivative enters to the first order whereas the space derivatives are to the second order, the solutions ofthe diffusion equation are irreversible in time. Nearly all other equations we shaJI discuss represent reversible phenomena in the thermodynamic sense, whereas solutions of the diffusion equation represent degradation of entropy (this will be discussed again in the next chapter). All of this is roughly equivalent to saying that phenomena represented by the diffusion equation are inherently statistical events.
2.5 The Electromagnetic Field Another important branch of physics, where the concept of field turns out to be remunerative, is that of electricity. Some of the elementary particles of matter are electrically charged, and most, if not all, have magnetic moments. Electromagnetic theory has been elaborated to describe their interactions in bulk (the interactions between individual particles usually involve quantum phenomena, which will be described later in this chapter). As with the fields encountered earlier in this chapter, the electric charge can often be considered to be a continuous fluid rather than a swarm of charged particles. Classical electromagnetic theory deals with the fields produced by various configurations of such a fluid and with the interaction of these fields with other parts of the fluid. Of course it might be possible to discuss the force on one portion of the fluid (or on one particle) due to another portion (or particle) without talking about a field at all. But it seems considerably easier , and perhaps also better, to break the problem into two parts : first , the" creation" of an electromagnetic field by a distribution of charge and current, second, the effect of this field on the distribution of charge and current. The Electrostatic Field. The effect of one charged particle on another is quite similar to the interactive effect of gravity. The magnitude of the force on each particle is inversely proportional to the square of their distance of separation and directly proportional to the product of the "strength" of their charges; the direction of the force is along the line joining them (as long as the particles are relatively at rest) . In the case of gravitation the force is always attractive and the "strength " of the gravitational charge is proportional to its mass;
§2.5)
The Electromagnetic Field
201
between static electric charges the force is repulsive if the charges are the same sign and attractive if the charges are opposite in sign. The force on anyone particle or portion of charge is therefore proportional to the "strength" of its own charge . We can thus define a vector field, called the electrostatic (or the gravitational) field , E, at some point which is the ratio of the force on a test particle at the point to the strength of charge of the test particle. The vector E is called th e electric (or gravitational) intensity at the point. This field, being the sum of forces with magnitudes inversely proportional to the squares of the distances from the various charges present, is the one which was discussed on page 17, due to a number of "source points." We showed in Eq. (1.2.10) that the net outflow integral for this sort of field, over any closed surface, is equal. to 411" times the sum of the charges of all the particles inside the surface. As long as we are considering only large-scale effects, we need not consider the microscopic irregularities of the field due to the fact that the charge is concentrated on discrete particles instead of being spread out smoothly; we need consider only average fields .over elements of surface area large compared with interparticle distances but small compared with the total surface considered. When this can be done, the resulting averaged field is equivalent to one caused by a "smoothed-out," continuous charge distribution, and we do not need to bother ourselves about the exact positions of each individual particle. We can choose an element of volume dx dy dz "around" the point x, y , z, containing a fairly large number of particles. The total charge inside the element is the average density of charge "at" x, y, z, times dx dy dz. This is proportional to the net outflow integral over the surface of the element, which is, by Eq . (1.4.5), equal to dx dy dz times the divergence of the field E. Consequently, for large-scale effects, we can replace the swarm of charged particles by a smooth distribution of charge of density p(x ,y,z,t). When this density is independent of time , Eq. (1.4.5) shows that the resulting static field E is related to p by the equation div E = (471/e)p The factor of proportionality e is characteristic of the medium and is called the dielectric constant of the medium. Whenever E changes from point to point, it is better to compute a vector field related to E, called the displacement field D, where div D = 411"p ; D = EE
(2.5.1)
We solve for D in terms of o; then, knowing E, compute E and from E compute the force pE on a cubic centimeter of the electric fluid.
Equations Governing Fields
202
lCH.2
The vector E can always be expressed in terms of the curl of a vector and the gradient of a scalar potential, as was shown on page 53. But since the divergence of a curl is zero, the vector potential for E is not determined by Eq. (2.5.1) and does not enter into electrostatic calculations. The scalar potential", must satisfy the following equation : div [e grad ",J
= eV2",
+ (grad e) . (grad ",) =
-41rp; E = - grad",
(2.5.2)
When e is constant, this becomes a Poisson equation for", [see Eq. (2.1.2)]. In the case of gravity the quantity e is everywhere constant and the equation for the intensity has a reversed sign , div E = - (41rp je) , corresponding to the fact that the force is an attraction, not a repulsion. In this case also there is a scalar potential, which is everywhere a solution of Poisson's equation V2", = -(41rpje). The Magnetostatic Field. Ferromagnetic materials, having atoms with unneutralized magnetic moments, behave as though they were charged with a magnetic fluid 1 analogous to the electrical fluid we have just discussed. If there were a unit positive magnetic charge, it would be acted on by a force represented by a vector field H, analogous to the electric field E. Analogous to the dielectric constant e is the permeability p., and analogous to the displacement vector D = eE is the magnetic induction B = p.H. The important difference between electricity and magnetism, however, is that there is no magnetic charge. The equation for the induction field B, instead of Eq. (2.5.1), is
div B = 0
(2.5.3)
One could, of course, express B as the gradient of a scalar potential which would always be a solution of Laplace's equation [Eq. (2.3.6)]. But it is better to utilize the fact that the divergence of the curl of any vector is zero and express D in terms of a vector potential, B = curl A. This would be about as far as one could go with magnetostatics were it not for the fact that the magnetic field turns out to be related to the flow of electric charge, the electric current. For instance, if a long straight wire of negligible diameter carries current I, where the direction of the vector is the direction of the current along the wire, then the magnetic field around the wire is given by the equation H
=
(21 X r) jr 2
1 Since there are no magnetic charges, but only magnetic moments, it would be more logical to derive the magnetic equations by considering the torque on a magnetic dipole. This is not a text on electromagnetics, however, and we may be forgiven (we hope) for saving space by deriving the equat ions by analogy with electrostatics rather than using a few more pages to present the more logical derivation, whi ch is given,.in detail in such texts as Frank, "Introduction to Electricity and Optics," McGraw-Hill, or Stratton, "Electromagnetic Theory," McGraw-Hill.
The Electromagnetic Field
§2.5]
203
where r is the vector, perpendicular to the wire, from the wire. to the point at which B is measured. But this is just the field, discussed on page 19, due to a simple vortex. Reference to this and to the definition of the vorticity vector, given on page 41, leads to a general relation between B and a steady current. If the charge fluid is moving, the velocity v of the charge times the charge density p is expressible as a current density J, charge per square centimeter per second , having the direction of the velocity of the charge at each point. This current is related to the vorticity vector for H by the simple equation curl H
=
471"J
(2.5.4)
(Incidentally, it is possible to have a current and yet not have free charge if the current is caused by the opposing motion of equal amounts of positive and negative charge.) Reciprocally, there is a force on a current in a magnetic field. The force on a cubic centimeter of moving charge at point x, y , z is F
=
pv
X B
= J X B; B = }lH
(2.5.5)
For steady-state problems these equations suffice. The charge sets up the electrostatic field (E,D) ; the current sets up the magnetic field (H,B). The electric field in turn acts on the charge, and the magnetic field on the current. Electric field is caused by, and causes force on, static charges ; magnetic field is caused by , and causes force on, moving charges. A rather far-fetched analogy can be made between the scalar potential, determining the electric field, and the scalar wave potential for purely compressional waves in elastic media and between the vector potential, determining the magnetic field, and the vector wave potential for shear waves. Thus far, however, there can be no wave motion, for we have considered only the steady state. For unbounded space containing finite, steady-state charge and current distribution and with dielectric constant E everywhere the same, the solution of Eq. (2.5.2) is, according to Eq . (1.4.8), cp(x,y,z)
=
~
..
fff ~
p(x',y',z') dx' dy' dz'
(2.5.6)
where . R2 = (x - X' )2 + (y - y')2 + (z - z')2 and E = - grad cp, D = -E grad cpo The vector potential A can just as well be adjusted so that its divergence is zero (since we are interested only in that part of A which has a nonzero curl ). Since V 2A :;: grad divA - curl curl A, we have, from Eq. (2.5.4), (2.5.7)
Equations Governing Fields
204
[cH.2
and, from Eq. (1.5.16),
Iff (l /R)J(x',y',z') dx' dy' dz' 00
A(x,y,z) =
(2.5.8 )
where H = curl A, B = p, curl A. Dependence on Time. So far we have been discussing steady-state conditions, where ap/at and div J are both zero [div J must be zero if ap/at is zero because of the equation of continuity (2.3.1)]. But if p and J vary with time, Eqs. (2.5.1) and (2.5.4) must become related by reason of the equation of continuity relating charge density and charge flow. Here we must relate the units of charge, current, and field in the two equations. If we use the mks system, the equations stand as written, with E for vacuum being EO ~ (A-) X 10-9 and with p, for vacuum being P,o ~ 10-7 • We prefer, however, to use the mixed system of Gauss, measuring charge in statcoulombs and current in statamperes, magnetic field in electromagnetic units (p, for vacuum being 1) and electric field in electrostatic units (s for vacuum being 1). Then Eq. (2.5.4) for the steady state becomes
-c curl H = -471'J
(2.5.9)
and Eq. (2.5.5) becomesF = (l /c) J X B, wherec = Vl/JLoEo ~ 3 X 108 meters per sec = 3 X 1010 em per sec. Since the equation of continuity ap/at = - div J must hold , we should obtain an identity by taking the divergence of Eq. (2.5.9) and the time derivative of Eq. (2.5.1) and equating the two . The lefthand sides do not balance, however, for there is a term div(aD/at) left over . This is not surprising, for these two equations were set up for steady-state fields, and any term in the time derivative of D should have vanished. The equation for the time-dependent magnetic field evidently should be
c curl H - (aD /at) = 471'J
(2.5.10)
This is confirmed by experiment. The equation is called the equation for magnetic induction or the Ampere circuital law for H. Not only does electric current produces a magnetic field; a change in the electric field also produces it. Maxwell's Equations. We have nearly arrived at a symmetric pattern for the forms of the field equations; there are two equations dealing with the divergences of Band D [Eqs. (2.5.1) and (2.5.3)] and an equation dealing with the curl of H [Eq. (2.5.10)]. We need a fourth equation, dealing with the curl of E, in order to obtain a symmetric pattern. The fourth equation, however, cannot be completely symmetric to Eq. (2.5.10), for there is no magnetic current, any more than there is a
The Electromagnetic Field
§2.5]
205
magnetic charge. The nearest we can come to symmetry is an equation relating curl E and aB/at . This equation is also confirmed experimentally; it is called the Faraday law of electric induction and relates the change of magnetic field to the vorticity of the electric field. The experimental results show that a factor - (l /c) must be included on the right side of the equation. We can therefore write down the four symmetric equations relating the fields to the currents, 1 1 sn curIH=cat+c 411"J;
1 sa cUrlE=-cat
div B = 0; B = JLH ;
div D = 411"p D = EE
(2.5.11)
which are called Maxwell's equations. The force on a cubic centimeter of charge current is F
=
pE
+
(l/c)J X B
(2.5.12)
These are the fundamental equations defining the classical electromagnetic field, resulting from a "smoothed-out" charge and current density. The equations for the scalar and vector potentials are also modified by the variation with time. We still set B = curl A, for it automatically makes div B = O. Placing this into the equation for curl E results in curl E = -
c1curl (aA) at
or
. + curl [ E
c1aAJ at
= 0
A vector whose curl is zero can be derived from a scalar potential function, so that [E + (l /c )(aA/at)] is the gradient of some scalar. In the steadystate case we had that E = - grad /11.) instead of 5;>, to bring the equation in line with Eq. (2.6.11) relating p and q. In Chap. 1 we also showed that
5;> • e = ih(ae /at)
(2.6.33)
where we have here also included the additional factor - (1/11.). This equation is similar to Eq. (2.6.12) for the effect on p on e(q) . But there
§2.6]
249
Quantum Mechanics
is a fundamental difference, which enables us to write ae/at here but does not allow us to make this limiting transition in Eq. (2.6.12) . In the earlier equation, giving the effect of p, we are dealing with eigenvectors e(q) for the operator q, so that each e is perpendicular to the other and there is no possibility of taking a limit. In the present case t is only a parameter; e is not its eigenvector, for t is not an operator. All eigenvectors for the system having S) as an operator are continuous functions of the parameter t, rotating in abstract space as t increases. Consequently we can here talk about a derivative of e with respect to t. The operator (S)/ih) dt produces an infinitesimal rotation from its direction at time t to its direction at time t + dt, and this difference in direction becomes continuously smaller as dt approaches zero. Kinematics in classical mechanics is concerned with the time variation of variables such as position q with time. In order to be able to draw the necessary analogies in quantum mechanics and so to determine S), it is necessary for us to consider the variation of operators with time. The development of quantum mechanics considered up to now in this chapter assumes that the meaning of the operator is independent of t as far as its operation on its eigenvector, so that the equation \5 • f = jf gives the same eigenvalue j for all values of time (as long as \5 does not depend on t explicitly) . In many cases the state vector itself changes with time, in the manner we have just been discussing. However, we could just as well consider that the state vector is independent of time and blame all the variation with time on the operator. This formal change, of course, must not affect the measurable quantities, such as the eigenvalues j or the expansion coefficients defined in Eq. (2.6.20), for instance. In other words the operator ~(t), including the time parameter, can be obtained from the constant operator ~(O) by the use of Eq. (2.6 .33) and the requirement that the quantity f*(t) . ~(O) • f(t)
= f*(O) • ~(t) • f(O) = f*(O) • e(i/")~t~(O)e-(i/h)~t . f(O)
be independent of t.
Consequently ~(t)
=
e(i/h)~t~(O)e- (i/h)~t
(2.6 .34)
gives the dependence of ~(t) on t if we are to consider the operator as time dependent. Letting t become the infinitesimal dt, we can obtain an equation relating the time rate of change of the operator ~(t) to the as yet unknown operator S):
~(dt) or
S) •
~
-
= [1
~ • S)
=
+
i
S) dt ]
[S),~]
=
~(O) [1
-
i
S) dt]
~ ~(dt) ;;; ~(O)
=
~ :t ~
(2.6.35)
250
Equations Governing Fields
[cH.2
From the way we have derived this equation, the expression diJi. jdt can be considered to be the time rate of change of the operator iJi. if we have let the state vectors be constant and put the time variation on iJi., or it can be the operator corresponding to the classical rate of change of the dynamical variable A when we let the operators be constant and let the state vectors vary. For instance, the operator corresponding to the rate of change qm of a coordinate can be computed from Eq. (2.6.35) : (2.6.36) where we have used Eq. (2.6.8) to obtain the last expression. • But this last expression is just the one needed to determine the nature of the operator jp. In the limit of large energies and momenta this operator equation should reduce to the classical equations in ordinary variables. This will be true if the operator .p is the Hamiltonian for the system, with the p's and q's occurring in it changed to operators. In other words if .p is the Hamiltonian operator of Eq. (2.6.27), then Eq. (2.6.36) will correspond to the classical Eq. (2.6.25). This can be double-checked, for if we set iJi. = ~m in Eq. (2.6.35) and use Eq. (2.6.8), again we obtain (2.6.37) which corresponds to classical equation (2.6.26). Thus we may conclude that the equations of motion of the operators in quantum mechanics have precisely the same form as their classical counterparts, with the classic quantities p and q replaced by their corresponding operators ~ and q. For example, Newton's equation of motion becomes m(d 2 q/ dt2 ) = -(a5l3jaq). By taking average values of any of these equations we see immediately that the classical Newtonian orbit is just the average of the possible quantum mechanical orbits. Stated in another way, the effect of the uncertainty principle is to introduce fluctuations away from the classical orbit. These average out. Of course, the average of the square of the fluctuations is not zero and is therefore observable, but in the limit of large energies the uncertainties become negligible, and quantum mechanics fades imperceptibly into classical mechanics. This statement is known as the correspondence principle. Its counterpart in equations is the statement that in the limit the commutator (ijh) [iJi.,58] goes into the classical (A,B) . Because of the correspondence between commutator and the Poisson bracket, it follows that any classical constant of the motion is also a quantum mechanical constant of the motion. Of course we may wish to work with transformation functions instead of eigenvectors. These functions also change with time, and corresponding to Eq. (2.6.33) we have the time-dependent Schroedinger
§2.6]
251
Quantum Mechanics
equation (2.6.38) where H(p ,q) is the classical Hamiltonian and the time-dependent state vector is given by the integral e(t) =
J-.. .
,p(t,q) e(q) dq
Of course for stationary states
.p . e(E) = so that
ih(aejat) = Ee(E) e(E,t) = e(E,O)e-i(E lhlt
(2.6.39)
where E is an eigenvalue of the energy. Thus the time dependence for the stationary state is simple harmonic, with frequency equal to the value of the energy divided tJy h, so that the Planck equation E = hv, given in Eq. (2.6.1), is satisfied. We have thus shown that the transformation function ,p(E,q) is the "wave function" which we spoke about at the beginning of this section. The square of its magnitude gives the probability density of the various configurations of the system, and integrals of the form
f
y,B
[~ a~' qJ,p dq
give the average value of a sequence of measurements of the dynamical variable B(p,q) when the system is in the state corresponding to ,p. This probability density and these average values are all that can be obtained experimentally from the system. For large systems, having considerable energy, the results correspond fairly closely to the precise predictions of classical dynamics, but for atomic systems the uncertainties are proportionally large and the results may differ considerably from the classical results. We have also shown that these transformation functions have wave properties and exhibit interference effects which affect the probability density. The wave number of the wave in a given direction is equal to I lh times the component of the momentum in that direction, and the frequency of the wave is equal to I lh times the energy of the system, as pointed out in Eq. (2.6.1). Only by the use of the machinery of abstract vectors and operators and transformation functions is it possible to produce a theory of atomic dynamics which corresponds to the experimentally determined facts, such as the inherent uncertainties arising when atomic systems are observed. Time-dependent Hamiltonian. Having discussed the case where time does not enter explicitly into the energy expression H, where time turns out to be a parameter rather than an operator, let us now consider the case when H does depend explicitly on the time t. In this case the
252
[CR. 2
Equations Governing Fields
time used in describing the change of energy should be considered as an operator (just as are the coordinates) rather than as a convenient parameter for keeping track of the system's motion. The distinction is clearer in the quantum mechanical treatment than in the classical treatment, for we can distinguish between the operator corresponding to time and the continuous distribution of its eigenvalues. Classically we shall make the distinction by letting the explicit time be qt, so that the total energy is a function of qt . . . qn and of the momenta PI . . . pn, which we can indicate formally by H(qtiP ,q). This function gives the proper classical equations of motion (2.6.25) and (2.6.26) for IiI ... qn and 111 . .. Pn but does not give a corresponding set of equations for qt. As a matter of fact we have not yet considered the conjugate momentum Pt. It appears that we must modify the Hamiltonian function in the cases where H depends explicitly on time, so that the new Hamiltonian H(pt ,qtiP,q) satisfies the equation . dq, aH qt=-=dt apt
But before we decide on the form of H, we must decide on the meaning of qt. Since qt is the explicit time, we should expect that in classical dynamics it should be proportional to the time parameter t and in fact that dqtldt = 1. Consequently we must have that the new Hamiltonian e is related to the total energy H(qtiP,q) and the new momentum Pt by the equation (2.6.40) e(pt,qt;p,q) = H(qt;p,q) + Pt Then the equations of motion are
ae
aqm =
.
r
p«;
m = t, 1, . .. ,n
(2.6.41)
The total rate of change of e with time (due to change of all the p's and q's with time) can be shown to be zero, for, using Eqs. (2.6.41), (2.6.25), and (2.6.26)
de
Cit ·=
ae .
ae.
aqt qt + apt Pt +
\' [ae .
'-'
aqm qm
+
ae . ]
apm »-
=0
m
Therefore the new Hamiltonian is constant even though the total energy H changes explicitly with time. We may as well add enough to make the constant zero; e = H + p, = O. This means that the quantity Pt, the variable conjugate to the explicit time qt, is just the negative of the value of the total energy, Pt = -E (we write E because H is considered to be written out as an explicit function of qt and the other p's and q's, whereas E is just a numerical value which changes with time). Therefore the explicit time is conjugate to minus the energy value.
§2.6]
Quantum Mechanics
253
The classical Poisson bracket expressions can also be generalized to include the new pair of variables, (u ,v) =
~ [au av Lt iJpm iJqm -
J
aU aV aqm iJpm ; m = t, 1, . . . , n
m
The Poisson bracket including the Hamiltonian may be evaluated by the use of Eqs. (2.6.41) : (ev) =
'
~[ae ~ Lt iJpm iJqm
iJe ~J iJqm iJpm
m
n
=
qm ~ [~ dd + ~ ddPmJ + [!!!.. + ae !!!..J Lt aqm t iJpm t iJqt iJqt iJPt m=l
=
V dd (2.6.42)
t
since iJv/iJqt = av/at and iJe /iJqt = O. Introduction of the explicit time and its conjugate momentum into quantum mechanics is now fairly straightforward. We introduce the operator qt, having a nondenumerable, continuous sequence of eigenvalues t, which can be used to specify the particular state which is of interest. The conjugate operator Pt has eigenvalues equal to minus the allowed values of the energy. These operators are on a par with the operators for the different configuration coordinates and momenta. The commutator is [Pt ,qt] = h/i so that the corresponding uncertainty relation is AE At ~ h. The operators Pt,qt commute with all other p's and q's. Equations (2.6.8) and (2.6.11) also hold for this pair. The Hamiltonian operator 4) is now obtained by changing the p's and q's in the total energy function into the corresponding operators and the explicit time into the operator qt; then (2.6.43) The unitary operator transforming a state vector at time t to the state vector at time t' is exp [(i/h)4)(t' - t)] [see Eq. (2.6.11)]. The equation of motion of a state vector e is 4)' e(t) = ih lim [ e(t
+ d2t -
.e(t)
J
as with Eq. (2.6.12), and the equation of motion for an operator [4),~] = (h/i)(d~/dt)
In particular
CIt =
~ [4),qt]
=
~ [Pt,qtl = 3'
(2.6.44) ~1
is
Equations Governing Fields
254
[cH.2
where 3' is the constant operator which transforms every vector into itself (in Chap. 1 we called it the idemfactor) . We can now go to the properties of transformation functions for systems where the Hamiltonian varies explicitly with time. We still define the transformation function from q to E (often called the Schroedinger wave function) by the equation e(O) =
f .. . f 1f(OJq,t) dt dq,
. .. dq; e(q,t)
where t is the eigenvalue for qt, qm the eigenvalue for qm, and 0 is the eigenvalue for S), the operator given in Eq. (2.6.43) . Just as was shown earlier that the operator Pm operating on e corresponded to the differential operator (h/ i) (a/aqm) operating on the transformation function, so here the operator Pt corresponds to the differential operator (h/i)(a /at) acting on 1f. The differential equation for 1f corresponding to the vector equation S) . e = 0 is
h a ) H ( t j i aq' q 1f(lq,t)
+ ii ate 1f(lq,t)
(2.6.45)
=0
This is called the time-dependent Schroedinger equation and is to be compared with Eq. (2.6.38), where we considered time as simply a parameter. As we see, it is a consistent extension of the earlier procedure for changing the classical equations for time-dependent Hamiltonian to a quantum mechanical equation for the wave function 1f. The quantity 11f1 2 is the probability density for a given configuration at a given time. The mean value of the particle current density at any point would seem to be proportional to e* . P • e =
f . . . f y; ~ grad 1f dq, z
. . . dq;
except that this is not necessarily a real quantity. However, we can now calculate what the current is. Particle in Electromagnetic Field. For instance, for a particle of charge e (the electronic charge is - e) and mass m, moving through an electromagnetic field with potentials A and I{J, the force on the particle [by Eq. (2.5.12) is erE + (l /cm)p X H] and the total energy (nonrelativistic) of the particle is
as will be shown in the next chapter (page 296). We substitute (h/i) (a/aq) for each p in H to obtain the differential equation for 1f. There is no ambiguity in order in the terms in p . A; if div A = 0, then proper order is A . p. The resulting equation for 1f is 2 - -h
2m
V~
- -.eh A . grad 1f zmc
e2A2] ha1f + [ -2mc 2 + el{J 1f + -,z at
= 0
(2.6.46)
255
Quantum Mechanics
§2.6]
As with Eq. (2.6.45) the imaginary quantity i enters explicitly into the equation. This means that the equation for the complex conjugate I/; is
_ .!!::.- V21/; + ~ 2m
'/.mc
A . grad I/;
+ (e
2
A2
2me2
+ ecp) I/;
_
~ al/; = '/.
at
0
If el/;I/; is to be the charge density p for the electromagnetic equations, then the current density J must be such as to satisfy the equation of continuity (apjat) + div J = O. We use the equations for y,. and I/; to determine J. Multiplying the .equation for y,. by I/; and the one for I/; by y,. and subtracting, we obtain
h2
ieh
2m (I/;V2y,. - y,.V21/;) - me A . grad (I/;y,.)
+ '/.h. ata (1f;t/!)
=
0
But from the rules of vector operation we can show that
I/;V2tf; - y,.V21/; = div (I/; grad y,. - y,. grad 1/;) and if div A = 0, we have A . grad (1f;t/!)
;t (e1f;t/!) + div [2~~ (I/; grad y,. -
= div (A1f;t/!) . Therefore y,. grad 1/;) - :c A1f;t/!] = 0
and if p = e1f;t/!, the current density turns out to be
eh
e2
J = -2' (I/; grad y,. - y; grad 1/;) - -mc A1f;t/! '/.m .
(2.6.47)
This expression is real, and since p and J satisfy the equation of continuity, presumably they are the expressions to insert in Maxwell's equations for the charge current. We note that these expressions are only probability densities, not "true densities" in the classical sense. This result is, however, in accord with our new conception of what is observable; since the "actual" positions and momenta of individual electrons cannot be known accurately, the only available expressions for the densities must come from the wave function y,.. As indicated at the beginning of this section, they involve the square of the magnitude of y,., a characteristic of quantum densities and probabilities. Relativity and Spin. The relationship between the four momentum operators for a single particle and the corresponding differential operators for use on the transformation function y,.( Iq,t) (where the blank before the vertical bar indicates that y,. could be for any eigenvector and eigenvalue) , (2.6.48) pm ~ (hji) (ajaqm) ; qm = X, y, z, t is a four-vector relationship which can satisfy the requirements of special relativity. The time-dependent Schroedinger equation (2.6.46) is not Lorentz invariant, however, even for the case of free flight , where A and cp are zero. The space operators occur in the second derivatives, and the time operator occurs in the first derivative, and no .combination of p;, p;, p;, and PI = - E can be Lorentz invariant.
256
[CH. 2
Equations Governing Fields
The difficulty, ofcourse, lies in the fact that the expression we used for H(p,q) for a particle in an electromagnetic field was not relativistically invariant but was simply the first approximation to the correct relativistic Hamiltonian. This quantity can be obtained by combining the four-vector p", Pu, P., -(i/c)H (see page 97) with the four-vector A", Au, A., i
From this we can obtain the relativistic equations for the Hamiltonian : H(p,q) =
-ep
+ c vm 2c + p2 2
(2e/c)A· p
+ (e/c)2A2
(2.6.50)
It is this function which should be converted into a differential operator to obtain the correct time-dependent Schroedinger equation. This result, however, only poses a more serious problem : how do we interpret an operator involving the square root of a second derivative? Conceivably we could expand the radical in a series involving increasing powers of 1/m 2c 2 (the Hamiltonian of page 254 is the first two terms of this expansion, with the constant me" omitted) , but this series will involve all the higher derivatives of 1/1 and would be an extremely "untidy" equation even if it did happen to converge. A possible solution is to use Eq. (2.6.49) as it stands, remembering that - (l /c)H is the fourth component of the momentum vector and should be replaced by (h/ ic) (a/at). When the fields are zero, this results in the equation V 2./ . 'I'
.!.a2"JI _ c2 at2
(mc)2 h
.1, 'I'
=0
(2 6 51) •
•
which is the Kle in-Gordon equation [see Eq. (2.1.27)]. This equation for the transformation function is relativistically invariant but has the disadvantage that, if el1/l1 2 is the charge density, then the quantity given in Eq. (2.6.47) is not the current density. As a matter of fact the integral of 11/11 2 over all space is no longer always a constant, as it is for a solution of Eq. (2.6.46), so that it is not clear that el1/l1 2 is the charge density. We shall defer to the next chapter the determination of the correct expressions for p and J; it is only necessary to state here that the Klein-Gordon equation is not the correct one for electrons or for any particle with spin. The time-dependent Schroedinger equation (2.6.46) is fairly satisfactory for particles moving slowly compared with the speed of light, but it does neglect two items: relativity and spin. We know that the electron has a spin and have discussed in Secs. 1.6 and 1.7 the properties of spin operators. These spin operators correspond to an additional degree of freedom for the electron, with, presumably, a new coordinate
§2.6}
Quantum Mechanics
257
and momentum. We could, therefore, compute a transformation function including this new coordinate if we wished and obtain expressions for the spin operator in terms of differentiation with respect to this coordinate. Since the rules for operation of the spin operator are fairly simple, it is usually easier to keep the state vector. The function used is therefore a mongrel one, consisting of a transformation function for the space and time components and the state vector a for the spin part. The total state vector is thus expanded as follows:
where s is one or the other of the two eigenvalues ± hl2 for the spin operator el and a is one of the spin vectors defined in Eq. (1.6.44) . Therefore if we have a Hamiltonian (nonrelativistic) which includes the spin operator el as well as the p's and q's and time, the hybrid wavefunction spin vector is '1' = ¥'+(lq,t)a(h I2) + ¥'_Iq,t)a( - hI2), and the equation is
) ha i » H ( t i i aq' q iel '1' + i at '1' = 0 which corresponds to Eq. (2.6.45). The average value of the quantity B(p,q iel) the state denoted by ¥' is then
f ... f
'1'B
(~:q' q;el) '1' dql . . .
dqn
where the el part of the operator works on the spin vectors a and the differential operators work on the wave functions ¥'. But this solution still does not give us a particle wave function which includes spin and also is relativistic. To attain this we turn to the spinor operators discussed in Sec. 1.7. The unit vectors dl . . . d4 defined in Eq. (1.7.17) provide operators which behave like components of a four-vector. They operate on state vectors e which have only two different directions, one corresponding to the z component of spin equal to hl2 and the other corresponding to it being - hl2 (the direction of the z axis is arbitrary) . A suggestion for the Lorentz-invariant wave equation with spin is to form a scalar product of the four-vector pr, PlI' p., PIIc = - EI c with the four-vector spin operator. Since a scalar product of two four-vectors is Lorentz invariant, we shall thus obtain a wave equation which has a first derivative with respect to time [as with Eq . (2.6.45) but not (2.6.51)} and which is also relativistic [as in Eq. (2.6.51) but not in (2.6.45)}. We should expect to set up an equation of continuity with such an equation so we can determine a charge and current density [as we did in Eq . (2.6.47) for solutions of Eq. (2.6.45)].
258
Equations Governing Fields
[CR. 2
The simplest form for such an equation would be that the scalar product of the four-vectors d and p operating on a spin or e would equal a const ant times e, or for a wave function 'V, consisting of two functions of position multiplied by the two spin vectors, as before, we should have
~ [dl!"+ dz !..+ d3.iJ 'V = ~ ~ ~ I
[ constant -
~ ~J 'V
u~
since d4 = 3 and E becomes -(h/i)a /at) for the wave function. The only difficulty is that the vector d = (dl,d z,d 3) is an axial vector (see page 104) whereas the gradient is a true vector, so that the quantity in brackets on the right-hand side should be a pseudoscalar (see page 11) changing sign for a reversal of sign of the coordinates. It is extremely difficult to see what fundamental constant we could find which would be a pseudoscalar, so difficult, in fact, that we are forced to look for a less simple form which will circumvent our need for a pseudoscalar. Such a less simple form consists of a pair of equations (d ' p)e
=
[a + ~J f;
(d· p)f
=
[b + ~J e
where e and f are different state vectors. By eliminating f and using Prob. 1.33, we discover that b = -a; then a can be a true scalar, not a pseudoscalar, and e is different from f. As a matter of interest this pair of equations is analogous to the equations curl H = (l /c)(aE /at);
curl E = -(l /c)(aH /at)
for the electromagnetic field in free space. There again, we could not have described the electromagnetic field in terms of just one vector (say E) using a curl operator on one side and a time derivative on the other. For in the attempted equation curl E = a(aE /at) the curl operator changes sign when we change from right- to left-handed systems, whereas a/at does not. Therefore a would have to be a pseudoscalar, which would be just as distressing for electromagnetics as it is with the wave equation for e. We can consider e and f to be vectors in the same two-dimensional spin space , related by the equations above, just as we consider E and H to be vectors in the same three space . But since e and f are independent vectors (in the same sense that E and H are) and since spin space is not quite so "physical" as three space," it is usual to consider f to be in another spin space, perpendicular to that for e. In other words, we set up a four-dimensional spin space, with unit vectors el, e2, e3, e4 all mutually perpendicular, and we ensure that e and f be mutually independent by making e a combination of el and ez and f a combination of e3 and e4.
259
Quantum Mechanics
§2.6]
In this representation the change from e to f represents a rotation from one subspace to the other, which can be represented by an operator !' such that !' . e = f and !' . f = e. Likewise the change from a to - a in the earlier pair of equations may be expressed in operator form by !,o, such that !'oe = e and !'of = -f. In terms of this representation the two equations written down above can now be condensed into one equation : (2.6.52) !' . (d' p)e = [-!'oa + (E /c)]e where e stands for either e or f . We must now extend our operator definitions of the spin operator d to four-space, and these definitions, together with the detailed ones for 0, !,o, and a = !' . d, are]
= e2; d"e2 = = i e2; dye2 = d.e2 = =' el i !'e2 = = ea; = eli !'Oe2 = = es: a"e2 = aye 1 = ie4; aye2 = a.el = e a; a.e2 =
d"el dyel d.el !'el !'Oel a"el
d"ea = eli -iel ; dyea = -e2; d.ea = e, ; !'ea = !'oea = e2; ea; a"ea = -iea; ayea = -e4; a.ea =
= = = = = = aye, = a.e, = d"e, dye, d.e, !,e, !'oe, a"e,
e,; i e4; ea; el i -ea ; e2; ie2; eli
ea -iea -e, e2 -e, el -iel -e2
(2.6.53)
We notice that the operator !' commutes with the operators d", dy, d., a", ay, a. but that !'!'o !'o!' = O. The operator !'o therefore commutes with the d'S but does not commute with the a's. In tabular form these operators become
+
H~), (~ -~ ~ -D; -H ~);!'=(H ~ !); ! j ~) (H! ~), J ~ -~); (~ -D dy
o
=
1 0
o
0
0
-1
0
=
o
0
ao; a"
1
o
=
0-1
a.
o
0
0
o o o
=
0
0
1 000 1
0 0
-1 0
where we shall use the symbols ao and !'o interchangeably from now on.
260
[CR. 2
Equations Governing Fields
In terms of these operators we can set up the operator equation (2.6.52) to operate on some vector e, a combination of the four-unit vectors el, e2, ea, e4, as follows: [e(d. p)
+ eoa]· e
= (E jc)e or [a",p",
+ aypy + a.p. + eoa] . e
= (E jc)e (2.6.54)
We must now "square" this equation to obtain a form analogous to Eq. (2.6.49). The Dirac Equation. When the electromagnetic field is zero, Eq. (2.6.49) becomes [p; + p~ + p; + m 2c2] = (Ptlc)2 Taking Eq. (2.6.54) and squaring the operators on both sides (and remembering that a",ay must not equal aya"" etc.) , we obtain
{[a;p;
+ a;p; + a;p; + e~a21 + [a",ay + aya",]p",py + [a.eo + eoa.]a~.l : e
= (ptl c) 2e
To have this correspond to Eq. (2.6.49) as written above means that
a; = and
a~
= a; =
e~
a",ay + aya", = . . . = a.eo + eoa, = 0 a = mc (2.6.55)
= 1;
An examination of Eqs. (2.6 .53) indicates that the operators defined there do obey the requirements of Eqs. (2.6.55), so that we have finally arrived at a relativistic equation for a single particle of mass m which has a firstorder term in the operator for E (or PI). To obtain it we have been forced to extend our "spin space" from two to four dimensions. Two of these spin states (el ,e2) correspond to a term + mc 2 in the total energy, and the other two correspond to a term - me", a negative mass energy. . We know now that the negative energy states are related to the positron, a particle of opposite charge but the same mass as the electron. The wave equation for a single particle of charge e and mass m in an electromagnetic field operates on a wave-function, spin-vector combination W = !flel + !f2e2 + !f3e3 + !f4e4; (2.6.56) iF = ~lei + ~2e~ + Y,3e! + ~4et where the !f'S are functions of z, y, z, and t and the e's are orthogonal unit vectors in spin space. The equation, which is called the Dirac equation, is
[ aomc
+ a.
+ a'" (~~ t ax
- ~c A '")
+ a (~~ - ~c A ) t ay
G:z - ~ A.) + e",]
y
W
y
= {aomcw
+ a • [~grad
+ e"'w} ~
=
-
W -
~ ~at W
tC
~ AWJ (2 .6 .57)
261
Quantum Mechanics __
§2.6J
where the operator a is a vector of components a "" a y, a . and where ao = !l0. The a operators obey the rules set forth in Eqs. (2.6.53) and (2.6.55). The equation for \Y is obtained by reversing the signs of all the i's in Eq. (2.6.57). We must now see if all this work has resulted in an equation which will allow sensible expressions for charge-current density. It would seem reasonable that the charge density would be (2.6.58) carrying on as before, multiplying the equation for 'V by \Y and the equation for \Y by 'V and subtracting, we obtain - (ajat) (IF'V) = c['Va . grad 'V
+ 'Va' grad wJ
= c
div (\Ya'V)
Therefore the vector whose components are ce(Wll",'V) = ce(Wlly'V) = ce(Wa. 'V) =
ce[,yt¥'4 + ,y2t/!3 + ,y3t/!2 + ,y4t/!d = J", -ice[,ylt/!4 - ,y2t/!. + ,y3t/!2 - ,y4t/!tl = J y ce[,ylt/!3 - ,y2t/!4 + ,y3t/!1 - ,y4t/!2] = J.
(2.6.59)
is the current density vector J. It is interesting to note that, whereas the momentum density for the particle is WP'V = (h /i)'V grad 'V, the velocity density seems to be cWa'V. This can be shown in another way. We use the Hamiltonian equation aH/ iJp = Ii = u on Eq. (2.6.50) (we leave out the fields for simplicity and assume that the z axis is along p or u) . Then u =
cp
Vp2
+
m 2c2
or
p =
mu
vI -
(U /C)2
[see Eq. (1.7.5)J and H =
mc 2
VI -
(u /c)2
= u· p
+ vI
- (u /c)2 mc
2
Comparing this classical equation for the total energy of the free particle (relativistic) with Eq. (2.6.54), we see that the vector operator Ca takes the place of the particle velocity u and the operator ao takes the place of VI - (U /C)2 when we change to the Dirac equation. The transformations of the operators a, the spin vectors e, and the wave function '1' for a Lorentz rotation of space-time or for a space rotation can be worked out from the data given in Sec. 1.7. For instance, if the transformation corresponds to a relative velocity u = c tanh () along the x axis, the p's and A's transform as four-vectors: p", = p~ cosh () PI
+ (l /c)p: =
p~
sinh (); P» = p~ ; cosh () + cp~ sinh ()
P. = p~ ;
The spin vectors e are transformed according to the formula e' = g. e,
Equations Governing Fields
262
[cH.2
where
'"
2:
(8a x)n= cosh -28 + a
1 -n! 2
x
. h -(J sm 2
n=O
since a; = 1. The conjugate operator g*, such that e = g* • e', is equal to g in this case. Therefore the new wave function is
w' =
gW
+ lf2e2 + lf3e3 + lf4e4] + sinh(8 /2) [lf4el + lf3e2 + lf2e3 + lfle4]
= cosh(8/2) [lflel
(2.6.60)
The operators a are transformed according to the formula g* • a· g = a ' . For symmetry we set at = Sic, [see Eq . (1.7.17)] where S is the idemfactor. We then have a~
+ +
=
at cosh 8 (l /c)a x sinh 8 ax cosh 8 Cat sinh 8 e8az/2aye8a./2 = e8a./2e-8a./2ay
a~ = a~
a~
= =
a. ;
a~
=
=
ay
ao
so that the a's (al .. . a4) transform like a four-vector. Therefore the scalar product of the a's with the momentum four-vector is a Lorentz invariant, so that
l
axpx
=
[axpx
+ aypy + a .p. + aIPI]
=
xy.t
l
xy.1
g*axgp~ =
l
xy.t
a~p~
Therefore the equation in unprimed coordinates can be changed as follows:
[2:
g*axg
(p~ - ~ A~) + g*aomc g] W =
g*
[2:
ax
(p~ - ~ A~) + aomc] w'
which is the equation in the primed coordinates. For a rotation in space by an angle 8 about the x axis the rotation operator for the e's and a's is, appropriately enough, (2.6.61)
and the transformation equations are pe p.
=
=
p;;
-p~
P«
=
sin 8 +
p~ ; p~
py = p~ cos 8 + p~ sin 8 cos 8; e' = ge; a' = g*ag
More complex rotations can always be made up by a series of rotations of the types discussed here ; the corresponding rotation operators g are just
Quantum Mechanics
§2.6]
263
the 'p roduct s of the individual g's for the component simple rotations, taken in the right order. Total Angular Momentum. As an exercise in use of the operators a and d we shall show that, when there is no electromagnetic field, the total angular momentum of the particle is not just the mechanical momentum W1 [see Eq. (1.6.42)] but is a combination of W1 and the spin vector d. In other words we must include the spin of the particle in order to obtain the constant of the motion which we call the total angular momentum. Referring to Eq. (2.6.35) we see that for a constant of the motion represented by the operator ~ we must have
where.p is the Hamiltonian operator. In the present case, with the potentials A and 'P equal to zero, we have for the Hamiltonian, according to Eq. (2.6.57), .p
=
aomc2
+ c( a"p" +
aypy
+
azpz)
The operator for the z component of the mechanical angular momentum is W1z = 1;Py - tJP". We form the commutator .pW1z - W1 z.p to show that it is not zero. N ow the a'S commute with the p's and the coordinates, the n's commute With each other, so that the only term in .p not commuting with the first term of W1 z is ca"\l,, and the only term not commuting with the second is Caypy, giving .pW1z - W1 z.p = -cayp,,(py\) - \)py)
However, P,,1; - 1;P"
=
+ ca"py(p,,1;
- 1;P,,)
(h/i), etc ., so that this expression comes out to be
which is certainly not zero, so that W1 z is not a constant of the motion. By using the operator rules of Eqs. (2.6.53) we can show that [compare with Eqs. (1.6.43)] (d,,)2 = (dy)2 = (dz)2 = 1; d"dy = -dyd" = id z; d"dz = - dzd" = - id y; dyd z - dzd y = id" Also, since the operator ~ commutes with the d's and since a can obtain other equations, such as
(2.6.62)
=
~d,
we
Therefore we can show that .pdz - dz.p = (2c/i)(p"a y - pya,,) Comparing the commutators for W1 z and dz we see that the combination [W1 z + (h/2)d z] does commute With .p and hence is a constant of the motion. This is also true of the x and y components.
264
Equations Governing Fields
[CR. 2
Therefore the total angular momentum, which is a constant of the motion, is the mechanical angular momentum 9)( plus h/2 times the spin vector d. Field-free Wave Function. As another example we shall obtain the wave function when no field acts. For the field-free case the Dirac equation (2.6.57) takes on the form O:omc'F
+ -;-h~ 0:' grad
h a'F
'F = - - ic at
A solution of this equation is 'F = [AIel
+ A2e2 + A aea + A4e4]eWh)(p.r-Et)
(2.6.63)
where the A's are numerical coefficients, the radius vector r = xi + yj + zk, the vector p = p",i + pyj + pzk is a classical momentum vector with components p"" py, pz which are constants, not operators, and the number E is the magnitude of the electronic energy for the state designated by 'F. Inserting this in the Dirac equation, performing the differential operations and the spinor operations according to Eqs. (2.6.53), we finally obtain (we have set p", = py = 0, pz = P with no loss of generality) , [(mc2 - E)A l + cp Aa]el + [(mc2 - E)A 2 - cp A 4]e2 + [(-mc 2 - E)A 3 + cp Adea + [( -mc 2 - E)A 4 - cp Ade4 = 0 This requires the four coefficients of the vectors e to be zero, which results in four homogeneous equations in the four coefficients A, two involving Al and A a and two involving A 2 and A 4• Both can be solved if the equation E2 = m 2c4 + C2p2 is satisfied. Therefore there are four solutions : two corresponding to the energy value E =
r-
mc" VI
which are 'F l
=
'F 2
=
+
(p /mc)2
----+
p«mc
-[mc 2 + ~(p2/m)]
G[ -{3el + (1 + 'Y)ea]e(ilh)(pz+ mc2'Yl) G[{3e2 + (1 + 'Y)e4]eWh)(px+mc2'Yt)
(2.6.64)
and the other two corresponding to the energy value E = +mc 2'Y ----t [mc2
+ ~(p2/m)]
{j-O
which are 'F a = G[ (1 'F 4 = G[(I
+
'Y) el
+ (3ea]eWh)(p",-mc2'Yl);
+ 'Y)e2 -
(3e4]e(i/h)(p",-mC2'Yl)
where{3 = (p /mc) and v = ~ and G is a normalization constant. The functions 'F 1 and 'Fa correspond to a spin hd./2.equal to +(h/2)
§2.6]
Quantum Mechanics
265
(because d. \1\ = \11\, etc.), and , 'V 2 and 'V 4 correspond to a spin of - (hj2) .
Recapitulation. In this chapter we have endeavored to outline the basic connections between various phenomena in classical and quantum physics and the various sorts of fields discussed in Chap. 1. The connection usually can be represented, to a certain degree of approximation, by a differential equation specifying the point-to-point behavior of the field which is to describe a particular physical phenomenon. In the case of classical physics we found that we had to average over atomic discontinuities; in quantum physics we found that the uncertainty princi pie precluded our "going behind" the continuous "square-rootprobability" wave function; the discontinuous details of the elementary particle trajectories turn out to be impossible to measure or predict. In either case we are left with a continuous field-scalar, vector, or dyadic-subject to a partial differential equation and specified uniquely by some set of boundary conditions (or initial conditions or both) . We have seen the same fields and the same differential equations turn up in connection with many and various physical phenomena. We find, for instance, that a scalar-field solution of Laplace's equation can represent either the electric field around a collection of charges or the density of a diffusing fluid under steady-state conditions or the velocity potential of a steadily flowing incompressible fluid or the gravitational potential around a collection of matter, and so on. From the point of view of this book, this lack of mathematical originality on the part of nature represents a great saving in effort and space. When we come to discuss the solutions of one equation, we shall be simultaneously solving several dozen problems in different parts of physics . We have not gone in great detail into the physics of the various examples discussed in this chapter; this book is primarily concerned with working out solutions of the equations once they are derived. Other books, concentrating on various parts of physics, are available for detailed study of the physics involved. In the case of quantum mechanics, for instance, we have resisted the temptation to go beyond the bare outlines of the new point of view in dynamics. Only enough is given so that later, when solutions of Schroedinger's or Dirac's equations are studied, we shall be able to understand the physical implications of the solutions. More than this would make the section a text on quantum mechanics-a supererogative task at present. It is true that more time was spent discussing the equations of quantum mechanics than was spent on the equations for classical fields. These newer equations are less familiar, and thus they have provided a chance to demonstrate what procedures must be used when carving out new field equations to describe new phenomena. The classical field equations have withstood the scrutiny of several generations of workers,
266
Equations Governing Fields
[CR. 2
and the logical structure of the relation to "reality" has become" second nature" to physicists. In quantum mechanics we have not quite completed the process of rationalization transforming an unfamiliar equation which "works " into a logically justified theory which is "obvious to anyone." A new equation for the description of new phenomena is seldom first obtained by strictly logical reasoning from well-known physical facts ; a pleasingly rigorous derivation of the equation usually is evolved only about the time the theory becomes" obvious ." The first finding of the equation usually comes by the less deductive paths of analogy and "working backward" and by the constant use of a modern counterpart to Occam's razor. In the Dirac equation, for instance, we were held to a certain general form of equation because it was likely that the equation should be relativistically invariant, and we searched for the simplest equation which would result in "sensible" (i.e., not redundantly complex) expressions for charge, current, and other measurable quantities. The result may not appear very simple at first sight, but a few days of prospecting by the reader (or of reading back issues of journals for the days when the Dirac equation was being put together) should suffice to demonstrate that it is much easier to construct equations of greater complication than it is to find one more simple . Among the general principles which can be used to point the direction for search for a new equation, the requirement of invariance, particularly of Lorentz invariance, is one of the most valuable. But there are many others. One usually looks first for linear equations, for instance, and the Laplacian operator often turns up. When an equation has been devised, all the related quantities must be investigated to see whether they can satisfactorily " correspond " to various physical quantities. There should usually be an energy density, for instance, and the quantity chosen to correspond should not turn out to have the annoying property of becoming negative somewhere or sometime. We used the choice of charge and current density expressions and the requirement that they satisfy the equation of continuity to guide us in our choice of the Dirac equation. A formal machinery to obtain these subsidiary quantities is the variational method, which will be discussed in the next chapter. Once these quantities are ground out, it is then possible to decide whether they are too complicated or not. Another useful means of testing an equation which has just been devised is to find another aspect of physics where the same equation can be applied. The properties of the solutions of the Klein-Gordon equation could be tried out by devising a string-in-rubber system (see page 139) which satisfied the same equation and which could be pictured more easily than a wave function, because the motions of a string are familiar. Analogies of this sort occur throughout theoretical physics and represent
CR.
Problems
2]
267
a sort of cross-fertilization which is most useful. The early analysis of alternating electric currents was greatly aided by the analogy with more familiar mechanical oscillators. N ow that "everyone knows about" alternating currents, we are prone to discuss other sorts of vibrational and wave motion (even mechanical oscillators) in terms of an analogy with alternating-current behavior, speaking of impedances, capacitances, and so on. In the next chapter we discuss in detail an analogy between field behavior and the variational properties of classical dynamics, as developed by Hamilton. We shall find it a useful unifying agent for all the equations we have discussed in this chapter (as well as others).
Problems for Chapter 2 2.1 A membrane is stretched over one end of an airtight vessel so that both the tension T in the membrane and the excess pressure p of the air in the vessel act upon the membrane. If y; gives the displacement of the membrane from equilibrium, show that p
= -
(pc 2 IV)N dA
where p, V, and c are the equilibrium values of the density, volume, and velocity of sound of the air in the vessel. Show that the equation of motion for the membrane is therefore
where v2 = T l u, where T is the tension and a is the mass per unit area of the membrane. What assumptions have been made in obtaining this equation ? 2.2 An infinite, elastic medium has piezoelectric properties for compression in the x direction and electric field in the y direction (for displacement s in the x direction, the electric intensity E and displacement vector D are in the y direction) . The dielectric polarization P, also in the y direction, is related to D and E by the usual equation D = E + 47rP and is related to the stress component X = T zz and E by the coupling equation P = 0 X + xE, where X is the dielectric susceptibility and 0 the piezoelectric constant. Alternately the strain S",,,, = u is related to the stress and to the electric intensity by the equations u = uX + oE, where a is the reciprocal of the elastic modulus. By use of the elastic equations and Maxwell's equations, set up the two simultaneous equations for compressional wave motion in the z direction. Show that two electroelastic waves are possible, one with velocity a little smaller than that for pure compressional waves (value if 0 were zero) and the other a little larger than that for pure electromagnetic waves.
268
Equations Governing Fields
[CH: 2
2.3 During the passage of a sound wave through matter, the temperature of the regions under compression is greater than the average temperature, while in the regions under expansion the temperature will be less than the average. These temperature differences will give rise to a flow of heat from one part of the material to another. a. Show that the equations governing the heat flow and sound propagation are aT/at = (aTo /apo)s(ap /at) + (K/Cppo)V2T a2p /at 2 = (apo/apo)rv2 p + (apo/aT o)pV2T where the subscript zero is used to denote equilibrium values . b. Assume that T and p propagate as plane waves T = A exp [i(kx - wt)] ; p = B exp [i(kx - wt)]
Show that the relation between k and w is given by o = i (K/CppoW)[(apo/apoh k 2 - w2] - {w2 - k 2[(apo /apo)r + (aTo /apo)s(apo /aTo)p]} Determine the velocity of propagation for K/cppow « 1, for K/CppoW» 1. Discuss the propagation when K/CppoW ~ 1. 2.4 A conducting fluid (electrical conductivity a, permeability p.) in motion will generate a magnetic field, which in turn will affect the motion of the fluid. Show that the equations coupling the velocity v and the magnetic induction Bare
a~ p ::
= curl (v X B)
+ p(v' V)v =
- Vp -
+
(4;:u)
2
V B
(4~J [B X
(v X B)]
2.6 When dissolved in an appropriate solvent, many salts break up into positive and negative ions. Under the influence of an electric field these will diffuse. Show that the equations describing the motion of the positive ions, assuming that these move in a viscous medium with their terminal velocity, is acdat = AiV2c 1
+ B1Q div
(Cl
grad
I{J)
where Cl is the concentration, Ai the diffusion constant, B 1 the ratio between the terminal velocity and the applied force, Q the ionic charge, and I{J the electrostatic potential. Show that I{J satisfies the equation V21{J
= -41r(F / e) (cl - C2)
where F is Faraday's constant. 2.6 Particles of microscopic size are found to have a random motion, called Brownian motion, arising from molecular collisions. Let the
269
Problems
CH.2]
number of particles at a time to having a position between Xo and Xo + dxo, a velocity between Vo and ve + dvo be f(xo,vo,to) dxo dvo. Let the fraction of these particles which, in a time T, are found in the region between z and z + dx with a velocity v + dv be w(Ax,Av,Tlxo,vo,to) dx dv, where Ax = x - Xo, Av = v - Vo.
a. Show that !(x,v,t)
=
f-.. . f-"", w(Ax,Av,Tlxo,vo,to)!(xo,vo,to)
b. Show that, for small
T,
dx« dvo
Ax, and Av,
of(x,v,to) oto
f-.. . f-"",
where Ax = Ax(x,v,t,T) = Ax w(Ax,Av,Tlx,v,to) dxo dvo with corresponding definitions for the other average quantities. c. If the particles move in a viscous medium and if the molecular collisions are random, Av = - aVT and Av2 = AT, where a and A are constants. Show that in the limit of small T of _ 0 ot - - ox (vj)
+
0 a ov (vj)
1
+ "2"A
o2f ov2
d. Show that under steady-state conditions
J-"", f(x,v) dx =
F oe -
(a I A l v
2
Show that the average of the nth power of the velocity defined by V" =
f-"'. f-"'", vnf(x,v) dx dv
satisfies the differential equation dvn
dt =
2.7
-naV"
+ j-An(n -
l)v n -
2
a. Two operators a and a* obey the following commutation rule : aa* - a*a
= 1
Show that the eigenvalues of the operator a*a are 0, 1, 2, 3, . . . . If the corresponding states are en, show that
270
Equations Governing Fields
[CR. 2
b. Two operators a and a* obey the following commutation rule:
aa*
+ a*a
=
I
Also aa = 0, a*a* = O. Show that the eigenvalues of a*a are only 0 or 1. If eo and el are corresponding states, show that
2.8 An electron moves in the coulomb field of a nucleus of charge Z . a. Prove that if
= [x 2
r
+ y2 + z21~
then the appropriate conjugate momentum pr is p,
(I jr)(r· p - ih)
=
b. Show that the Hamiltonian for the electron may be written
where ~ is the angular momentum operator. c. Determine the energy values E n for an electron having a given angular moment l using the following method. Find an operator ~(r) such that
Hence show that En! is independent of l, that for a given E n! there is a maximum l which we shall call n - 1. Express En! in terms of n. 2.9 Show that under the transformation e = exp (-is)ot jh)f The Schroedinger equation (S)o
becomes where
S)l(t)f S)l(t)
= exp
Show that where
U = I
+ S)l)e
+ (I j ih) + (l jih)2
=
=
i h(iJejiJt )
ih(iJf jiJt)
(is)ot jh)S)l exp (-is)ot jh)
f", f",
f
= Ufo
S)l(t') dt'
where f o is time independent.
S)l(t') dt'
J~
'"
S)l(t") dt"
+
Relate £0 to the solutions of
.\)oeo = ih(iJeoj iJt)
Standard Forms of Equations
CH.2]
2.10
271
Decompose the solution e of the Dirac wave equation as follows: e
=f
+ g;
f = j(l
+ ao)e;
g = i(l - ao)e
Show that f* . g = 0 and that (E (E
+ eV + me
2)f
+ eV + mc )g 2
=
e[a • (p
+ eAje)]g; + eAje)]f
= -e[a· (p
Show , for positive energy states and eA and eV small compared with me", that g* . g « f* . f. 2.11 Define a set of four states ei which satisfy the Dirac equation for a particle at rest : (aOme 2)e i = EOei Show that the four solutions of the Dirac equation for a particle of momentum p are [C(a . p) + ao(me2 + IEi)]ei
Standard Forms for Some of the Partial Differential Equations of Theoretical Physics EQ. NO .
Laplace Equation : V2.J; = 0 Vector Form: curl (curl A) = 0 ; div A = 0 Poisson Equation : V2!J; = -411"p Vector Form : curl (curl A)
=
(1.1.4), (2.3.6) (1.1.5), (2.1.2) , (2.5.2) (2.5.7)
4'11"J; div A = 0
Helmholtz Equation : V2.J; + k 2.J; = 0 Vector Form : curl (curl A) - k 2A = 0 ; div A = 0 . 1 a2.J; Wave Equation : 02!J; = V2!J; - (;2at2 = 0
(2.1.10) (2.1.9) , (2.2.2)
a A = 0; + (;21 ai,2 2
Vector Form : curl (curl A)
div A, = 0
(2.2.3), (2.5.15)
D 1iffUSlOn . E quation . : V 2./Y', = a2 1 a.J;
at
Vector Form: cur l (curl A)
(2.4.4)
aA + a1 at = 2
0 ; div A = O·
(2.3.19)
272
Equations Governing Fields
=
Klein-Gordon Equation : 021Y
Vector Form : curl (curl A)
}.l21Y 1 iJ2A
Maxwell's Equations : div B
1 iJD B
= 0;
(Proca Equation)
(2.5.37)
OJ div D = 471"p
=
curl H = - c iJt
(2.1.27)
+ C2 7Ji2 + }.l2A
div A = 0
[cH.2
+ -c1 471"J'' =
}.lH;
D
1 iJB
curl E = - - c at
(2.5.11)
= fE
Electromagnetic Potential Equations : 0\0 = -4n-p/f j
02A = -471"}.lJ/c 1 iJA
B = curl A; E = - grad
«J -
cat;
(2.5.15)
div A = _ f}.l a«J c iJt
(For the forms of these equations for other gauges, see pages 207 and 332.) Elastic Wave Equation (isotropic media) : iJ 2s p iJt2 = V · [AS div s
+ }.l(Vs) + }.l(sV)] =
(A
+ 2}.l)
grad div s - }.l curl curl s
(2.2.1)
Viscous Fluid Equation : p
~; + pv '
(Vv)
=
V · [- (p
+ 'Y div v)S + 17(VV) + 17(VV)]
iJv
Pat = - grad where A =
h -
[p - (t17
+ A) div v + jr
(2.3.14)
2] - 17 curl curl v
+ pv
X curl v
'Y.
Schroedinger Equation for single particle of mass m in potential V : -
:~ V21Y + V1Y =
ih
~~ ;
11Y1 2 is probability density
(2.6.38)
Dirac Equation for electron in electromagnetic field :
aomc~ + a . (~ grad ~ + ~ A~) + (~ ~i - e«J~) 4
~ =
L en1Yn; n=l
~* . ~ is probability density
=
0
(2.6.57)
CH.2]
Bibliography
273
Bibliography General references for material in this chapter: Joos , G. : "Theoretical Physics ," Stechert, New York, 1944. Landau, L. D., and E. Lifschitz : "Classical Theory of Fields," Addison-Wesley, Cambridge, 1951. Lindsay, R. B., and H . Margenau: "Foundations of Physics, " Wiley, New York, 1936. Margenau, H., and G. M. Murphy : "Mathematics of Physics and Chemistry," Van Nostrand, New York, 1943. Rayleigh, J . W. S. : "The Theory of Sound," Macmillan & Co., Ltd., London, 1896, reprinted, Dover, New York , 1945. Riemann-Weber, "Differential- und Integralgleichungen der Mechanik und Physik," ed. by P. Frank and R. von Mises, Vieweg, Brunswick 1935, reprint Rosenberg, New York , 1943. Slater, J. C., and N. H. Frank: "Introduction to Theoretical Physics," McGrawHill, New York, 1933. Schaeffer, C. : "Einfuhrung in die Theoretische Physik," 3 vols., De Gruyter, Berlin , 1937. Sommerfeld, A.: "Partial Differential Equations in Physi cs," Academic Press , New York , 1949. Webster, A. G.: "Partial Differential Equations of Mathematical Physics ," Stechert, New York , 1933. Additional texts of particular interest in connection with the subject of vibration and sound: Coulson , C. A.: "Waves, a Mathematical Account of the Common Types of Wave Motion ," Oliver & Boyd, Edinburgh, 1941. Lamb, H .: "The Dynamical Theory of Sound ," E. Arnold & Co., London, 1925. Morse, P. M. : "Vibration and Sound," McGraw-Hill, New York, 1948. Books on elasticity and elastic vibrations: Brillouin, L. : "Les tenseurs en mechanique et en elastique," Masson et Cie, Paris, 1938. . Love, A. E. H .: " Mat hematical Theory of Elasticity," Cambridge, New York, 1927, reprint Dover, New York , 1945. Sokolnikoff, 1. S. : "Mathematical Theory of Elasticity," McGraw-Hill, New York,1946. Timoshenko, S. : "Theory of Elasticity," McGraw-Hill, New York, 1934. Additional material on hydrodynamics and compressional wave motion : Chapman, S., and T. G. Cowling: " Mathematical Theory of Non-uniform Gases," Cambridge, New York, 1939. Hadamard, .J. S.: " Lecons sur la propagation des ondes et les equations de l'hydrodynamique," Hermann & Cie, Paris, 1903. Lamb, H . : "Hydrodynamics," Cambridge, New York , 1932, reprint Dover , New York, 1945. . Milne-Thomson, L. M. : "Theoretical Hydrodynamics," Macmillan & Co., Ltd., London, 1938. Sauer, R. : "Introduction to Theoretical Gas Dynamics," Edwards Bros., Inc ., Ann Arbor , Mich., 1947.
274
Equations Governing Fields
[cH.2
Books on diffusion, heat flow, and transport theory : Chandrasekhar, S. : "Radiative Transfer," Oxford, New York, 1950. Chapman, S., and T . G. Cowling :" Mathematical Theory of Non-uniform Gases," Cambridge, New York, 1939. Fowler, R. H .: "Statistical Mechanics, " Cambridge, New York , 1936. Hopf, E.: " Mat hematical Problems of Radiative Equilibrium, " Cambridge, New York, 1934. Lorentz, H . A. : "Theory of Electrons," B. G. Teubner, Leipzig, 1909. Texts on electromagnetic theory, particularly on fundamental concepts : Abraham, M., and R. Becker : " Classical Theory of Electricity and Magnetism, " Blackie, Glasgow, 1932. Stratton, J. A. : " Elect romagnetic Theory," McGraw-Hill, New York, 1941. Van Vleck, J . H .: "Theory of Electric and Magnetic Susceptibilities ," Chaps. 1-4, Oxford, New York, 1932. Discussions of the fundamental principles of quantum mechanics from various points of view: Bohm, D. : "Quantum Theory," Prentice-Hall, New York, 1951. De Broglie, L. : "L'Electron magnetique," Hermann & Cie, Paris, 1945. Dirac, P. A. M. : "Principles of Quantum Mechanics," Oxford, New York, 1935. Jordan, P. :" Anschauliche Quantentheorie," Springer, Berlin, 1936. Kemble, E. C.: "Fundamental Principles of Quantum Mechanics," McGrawHill, New York, 1937. Kramers, H. A.: "Grundlagen der Quantentheorie," Akadem ische Verlagsgesellschaft m.b.H., Leipzig, 1938. Sommerfeld , A. : "Wellenmechanik," Vieweg, Brunswick, 1939, reprinted, Ungar, New York, 1946. Van der Waerden, B. L. : "Gruppentheoretische Methode in der Quantenmechan ik," Springer, Berlin , 1932. Von Neumann, J .: "Mathematische Grundlagen der Quantenmechanik," Springer, Berlin, 1932. '
CHAPTER
3
Fields and the Variational Principle
The use of superlatives enables one to express in concise form a general principle covering a wide variety of phenomena. The statements, for instance, that a straight line is the shortest distance between two points or that a circle is the shortest line which encloses a given area, are deceptively simple ways of defining geometrical entities. To say that electric current distributes itself in a network of resistors so that the least power is converted into heat is a description of direct-current flow which encompasses many individual cases without the use of mathematical complexities (though the complexities inevitably intrude when the general principle is applied to the individual case). The statement that a physical system so acts that some function of its behavior is least (or greatest) is often both the 'starting point for theoretical investigation and the ultimate distillation of all the relationships between facts in a large segment of physics. The mathematical formulation of the superlative is usually that the integral of some function, typical of the system, has a smaller (or else larger) value for the actual performance of the system than it would have for any other imagined performance subject to the same very general requirements which serve to particularize the system under study. We can call the integrand L; it is a function of a number of independent variables of the system (coordinates, field amplitudes, or other quantities ) and of the derivatives of these variables with respect to the parameters of integration (velocities or gradients of fields, etc.). If the variables are rpl, . . . , rpn, the parameters Xl, . . . ,Xm and the derivatives arp./ ox. = rprs, then the integral which is to be minimized is .£
=
lb. a.
. . .
Ibm L (rp,...J!, 0 X) oX
am
dXI'"
dXm
(3.1.1)
From the minimization of this function we can obtain the partial differential equations governing the rp's as functions of the x's and many other things. This pro cess of obtaining the rp's is called the variational method. In the present chapter we shall first indicate in a general way how 275
276
Fields and the Variational Principle
[CR. 3
the variational method can be used to obtain equations for the variables involved, then discuss in more detail the variational principles of classical dynamics, because they provide a well-explored example of this technique and its utility ; and then we shall proceed to apply the method to the various types of fields which are to be studied in this book.
3.1 The Variational Integral and the Euler Equations The integrand L of the integral to be minimized (or maximized) will be called the Lagrange density of the system. It is a function of functions of the basic parameters of the system. In the case of classical dynamics, for instance, the parameter is the time and the functions are the coordinates and velocities, at.various times, of the configuration of the system as it moves in conformity with the applied forces and the initial conditions; in the case of fields the basic parameters are the coordinates locating every point where the field is to be measured, and the functions are the various components of the field and their gradients, which are determined, as functions of these coordinates, by the distribution of the various "sources" (or charges) in space and by the boundary conditions. Thus when we require that the integral of L should be minimized (or maximized), we mean that the functions in terms of which L is expressed (the coordinates and velocities or fields and gradients) are to be adjusted, at every value of the parameters, so that the integral of L has the least possible value. We wish so to adjust the functions P that the integral of L, a function of the p's and iJp/iJx's, is as small as it can become, subject to the conditions imposed by the state of the system. In order to solve the problem we must first make the step from a variational requirement on an integral of L to a set of equations which determine the best values of the functions p. The Euler Equations. Before we can make even this step we must be more specific about what we mean by "minimizing the integral" and "best values of the functions." To do this, suppose we arbitrarily choose a functional form for each of the functions PI, • • • , Pn , as functions of the parameters Xl, • . • ,Xm • This arbitrary choice will, of course, fix the form of the functions pra = iJpr/iJx a and therefore will determine the value of the integral .c given in Eq. (3.1.1). Now let us change the p's a little bit ; for the function pr suppose the change to be expressed by the term Er1/r, where 1/r is an arbitrary function of the parameters and E is a small quantity, independent of the parameters. The shorthand notation Opr is often used instead of Er1/r, where op is considered to be an arbitrary small" variation" of the function p. This modification of the p's will also result in a change in the components pra of the gradients. These are related to the changes in the p's, for iJE r1/r/ aXa = ' Er1/r.. In the
277
The Variational I nlegral and the Euler Equations
§3.1]
shorthand variational notation, this relation is represented as 5.pr. = of!.pr/ AX•. By using a Taylor's series expansion of L we can show that the firstorder change in the integral £ due to the small changes of the .p's can be written
r n
5£ =
l
bl
. . .
lbm am
at
r m
et. 7Jr e; [ ;vtpr
+
r=1
st. a!}7J r ] dXl
!l" VtpT8
UX s
' . . da.;
.~1
We assume that the parameters are so chosen that the limits of integration can all be constants and that all the 71'S go to zero at these limits, which would be true, for instance, if the limits coincided with physical boundaries where certain boundary conditions are imposed on the .p's. This situation is usually the case, so we shall confine ourselves here to its study; the more general case, where limits are all varied, will be touched upon later. The term (oL/o.pr.)(07Jr/ax.) can be integrated by parts over x., giving
st. ]b. [ -a.pr. 7Jr a, -
l b'
- a (aL) 7Jr dx.. The first term is zero, since 7Jr = 0 ax. a.pr. Therefore the first-order variation of £ is a,
at a. and b; 5£ =
l
bt
at
•
i bmrn am
T=31
r m
E
r
[OL O.pr
-
a (aL)] -71 dXl . ax. a.pr. r
-
(3.1.2)
l ICIt
If 5£ is not zero, £ cannot be a maximum or minimum. When 5£ is zero, no matter what (small) values the Er'S have, then the functional forms of all the .p's have been chosen so that £, as a function of the Er'S, has either a minimum or a maximum or a point of inflection for Er = O. Usually we can tell from the physical situation which of these cases is true ; if we are not sure, it is always possible to calculate the secondorder terms in the E.r'S in the Taylor's series expansion of £ to see whether they are positive, negative, or zero. To save space from here on we shall use the .terms "minimize" and" minimum" when we mean "minimized or maximized or minimaxed" and "minimum or maximum or point of inflection." Therefore in order that £ have its extreme value (maximum or minimum), the functional form of the .p's must be chosen so ' that the coefficient of each of the Er'S in the integral for 5£ is zero. This results in a 'set of equations for the desired behavior of the .p's: r
= 1, . . . ,
n
(3.1.3)
278
Fields and the Variational Principle
[cH.3
where 'Pro = a'Pr/ax,. These equations, which serve to determine the optimum functional form of the 'P's, are called the Euler equations. We shall use these equations extensively later in this chapter. Several general comments should be made concerning these results. In the first place, if the variational principle is to be of general validity, then .£ should be an invariant, and the density L, or L divided by the scale factors coming into the element of integration, should be invariant to coordinate transformation of the parameters of integration. This will be of use later in finding new Lagrange densities. A still more general comment is that the variational principle is generally useful in unifying a subject and consolidating a theory rather than in breaking ground for a new advance. It usually happens that the differential equations for a given phenomenon are worked out first, and only later is the Lagrange function found, from which the differential equations can be obtained. This is not to minimize the importance of finding the Lagrange density L, for it is of considerable utility to find what physical quantity must be minimized to obtain the differential . equations for the phenomena, and the form of the variational equations often suggest fruitful analogies and generalizations. Auxiliary Conditions. In many cases the Lagrange integral is to be minimized (or maximized) subject to some one or more additional requirements. further restricting the independent variables and parameters. L---=---=.....:::..------...:==~x In this case we use the method of Lagrange multipliers to obtain Fig. 3.1 Maximum point (xo,Yo ) for function !(x,y, ) represented by contour lines the modified answer. Just how 0, 1, 2, . . .. Maximum point (X" y ,,) these multipliers work can best along line Y = Ya(X). be shown by an example. Suppose that the function f(x,y) is to be maximized. If there are no auxiliary conditions, we solve the two equations aj/ax = 0;
aj/ay = 0
(3.1.4)
simultaneously. The resulting pair (or pairs) of values of x and y, (xo,Yo), specify the point (or points) at which j has a maximum, minimum, or saddle point, and the value j(xo,yo) is the value of j at this maximum or minimum. Here the function is of two variables, and the two equations (3.1.4) are needed to obtain a pair of values (xo,Yo) for the maximum or minimum. A typical case is pictured in Fig. 3.1, where the functionj is depicted in terms of contour lines. But suppose that we wish to find the maximum of j(x,y) along the line
§3,l]
The Variational Integral and the Euler Equations
279
given by the auxiliary equation y = Ya(X). This line does not usually run through the point (xo,Yo), so the solution cannot be the same, There may be one or more points along the line, however, where f(x ,y) has a maximum (or minimum) value, such as the point (XI,YI) shown in Fig. 3.1. This may be computed by inserting the expression for Y in terms of z into the form for f, which gives the value of f along the lineas a function the single parameter z . We then differentiate with respect to x to find the maximum value, d af dxf(x ,Ya(x» = ax
+
af d ay dx [Ya(X)] = 0
(3.1.5)
The position of the maximum is then the solution Xl of this equation and the related value YI = Ya(XI) ' However, we can solve this same problem by a method which at first sight appears to be different from and more complicated than the one resulting in Eq. (3.1.5) . Suppose the auxiliary equation is g(x,y) = O. We first introduce a third unknown, A, and then maximize the new function f + Ag, subject to the relation g = O. In other words we are to solve the three equations (af /ax)
+ A(ag/aX)
= 0;
(af /ay)
+ A(agjay)
= 0;
g = 0
(3.1.6)
simultaneously to determine the proper values for X, y, and A. It is not immediately apparent that the solution of Eqs. (3.1.6Y is identical with the solution of Eq. (3.1.5), but the connection becomes clearer when we write the auxiliary equation g(x ,y) = 0 in the form used above, Ya(x) - Y = O. Then the first two of Eqs. (3.1.6) are (af /ax)
+ A(dYa/dx )
= 0 ; (af /ay) - A = 0
Substituting the second of these into the first gives us (af /ax)
+ (dYa /dx)(afjay) = 0
which is just Eq. (3.1.5) . Therefore in this simple case the method of Lagrange multipliers gives us the same result as the straightforward method. This is true in general. In this simple case it appears to be a more cumbersome method than the use of Eq. (3.1.5) , but in more complex cases it turns out to be an easier method. As applied to the variational integral of Eq. (3.1.1) the method of Lagrange multipliers can be stated as follows: Suppose that the Lagrange density is L('Pr,'Pr.,X.), (s = 1, 2, . . . , m), (r = 1, 2, . , . , n) , and the auxiliary equations are
l
b'
a,
. . .
lbm G ( am
t
where t = 1, 2, ' .. , k (k
'P,
a X) dXI . . . dXm = Ct --!£., ax
m) and where the
(3.1.7)
C's are constants.
280
Fields and the Variational Principle
[cH.3
Then the modified variational integral is £' =
lb! .. .ibmL' ('P' O'P, x) dXl . . . dXm ox a!
am
where k
L'
= L('Pr,'Prs,x.) +
LAIG'('Pr,'Prs
X. )
(3.1.8)
1=1
Then the m new Euler equations,
~ 0 (oL') L..t oXs 0'Pr8 8=1
=
eu
O'P8
(3.1.9)
plus the k equations (3.1.7) serve to determine the 'P's as well as the values of the A's. In this case the Lagrange multiplier method is definitely the easier.
3.2 Hamilton's Principle and Classical Dynamics In classical dynamics the parameter is time t and the functions 'P in the Lagrange function are the coordinates q which specify the configuration of the system at each time. If the system has n degrees of freedom, we can choose n independent q's (ql' . .. , qn) which will completely specify the configuration; the corresponding velocities will be qr = dqrjdt. No matter how the q's are chosen, the kinetic energy of an inertial system always turns out to be a quadratic function of the tis :
(3.2.1) where the a's may be functions of the q's. If the system is conservative (i .e., has a total mechanical energy which is constant in time), then the external force on the system can be represented in terms of the gradient of a scalar potential function V, so that the equivalent force along the qr coordinate is
(3.2.2) The potential energy may depend on time explicitly, but it is not a function of the q's. When the system is conservative, the variational principle determining the equations of motion, called Hamilton's principle, uses the kinetic potential (see page 229) T - V as the Lagrange function and is
a (" Jlo
(T - V) dt = 0
(3.2.3)
§3.2]
Hamilton's Principle and Classical Dynamics
281
This states that for any actual motion of the system, under the influence of the conservative forces, when it is started according to any reasonable initial conditions, the system will move so that the time average of the difference between kinetic and potential energies will be a minimum (or in a few cases, a maximum) . Lagrange's Equations. The Euler equations for the coordinates for this case, (3.2.4)
are called Lagrange) s equations of motion for the system. The left-hand terms represent the accelerations of the system, and the right-hand terms are the corresponding forces , derived from the potential energy V, plus the "kinetic forces " (such as centrifugal force) due to the motions. When the forces are not conservative, so that there is no potential energy, the variational principle is n
1:'[vr + L r. Oqr] dt
= 0
r=1
and Lagrange's equations are
!!.. (aT) = aT _ F dt aqr
aqr
(3.2.5)
'
Lagrange's equations are among the most generally useful equations of classical dynamics. The kinetic and potential energies are scalar invariants to coordinate transformations, so that they can be expressed in terms of any complete set of coordinates, and Lagrange's equations will have the same form in each case. In each case the quantity
a
-:-" (T - V) a qr
=
aT
a-:-" qr
=
is called the momentum for the rth coordinate. be written
pr Lagrange's equations can
dpr _ aT = { - (aV / aqr; if the system is conservative dt aq, F r; in general and if we use rectangular coordinates for the q's , so that T does not depend on the q's but only on the qs, the equations reduce to the familiar Newton's equations (d/dt)(momentum) = force Consequently Hamilton's principle represents, in simple invariant form, all the equations of classical dynamics.
282
Fields and the Variational Principle
[CH.3
Energy and the Hamiltonian. When the system is conservative, a function of the coordinates and momenta which remains constant throughout the motion of the system is the total energy E, the sum of kinetic and potential energy. When this is expressed in terms of the coordinates q and the momenta p, it is called the Hamiltonian function H for the system. Since the Lagrange function L is T - V and the energy E is T + V, the two can be related by the equation E = 2T - L. Therefore we can express the variational principle as of(2T - E) dt = O. From this we can obtain equations relating the Hamiltonian to the velocity and acceleration. Although these are just other forms of the same old equations of motion, their form turns out to be particularly well adapted for translation into quantum language. First we must translate (2T - E) into functions of q andn instead of q and q. As shown earlier the momentum PT is obtained by differentiating the kinetic energy T with respect to qTo Once we know Pr as a function of qT' it is not hard to eliminate the q's from Land T . From Eq. (3.2.1) we see that
n
2T =
LPTqT = L + H
(3.2.6)
T=l
This equation enables us to answer a question which we have so far ignored : whether H can be expressed only in terms of the q's and the p's (as defined in Eq. 3.2.6) with no dependence on q left over. To demonstrate, we use the equation H = ~pq - L, where L, in general, cannot be expressed in terms of just p's and q's but can be expressed in terms of just q's and q's. We now make a small change in the q's, p's, and q's.
dH
=
~p
dq +
~qdp
-
~(aLjaq)
dq -
~(aLjaq)
dq
But by definition, p = aL jaq, so that dH = ~q dp - ~(aLjaq) dq. Therefore the total change in H is given in terms of changes of the q's and p's ; so we have proved that H can be expressed as a function of the q's and p's alone (unlessL depends on t explicitly, when aH jat = -aLjat and H is a function of the q's, of the p's, and of t) . When the energy is expressed in terms of the p's and q's (and t, if necessary), we call it the Hamiltonian H. The variation of fL dt becomes
Hamilton's Principle and Classical Dynamics
§3.2]
283
where the oq's and op's represent the variations of the quantities q and p from the values taken on along the actual path (corresponding to the quantities E'T/ used earlier) . Integrating the terms pr oqr = Pr(d oqrldt) by parts, we can finally separate the variation of the integrand into a part due to the variation of.the q's and another part due to the variation of the p's;
o
f
L dt =
n
1 L[(11
p, -
~~) oqr + (qr - :~) opr] dt =
0
r = 1
Since we are assuming that we can vary the p's independently of the q's, each of the parentheses must be zero, and we arrive at the alternate form of the equations of motion (3.2.7) which are called Hamilton's canonical equations. They have been used a number of times in the previous chapter (see pages 229 and 242). We shall return again to use them later in this chapter. It is not difficult to see that the Hamiltonian function of the p's and q's is independent of time . For the time rate of change of H is
which is zero according to Eqs. (3.2.7) unless H depends explicitly on the time. Any Hamiltonian for which the canonical equations hold is thus a constant of the motion for the system. Note that this equation states that for conservative systems the total change of H with time is zero. In some cases H may depend explicitly on t; in these cases the variation of H with t due to changes of the q's and p's is still zero, and the total change dHIdt is equal to the explicit change aHI at which is, incidentally, equal to - (aLI at), as we have already seen. Impedance. In Chap. 2 (page 128) we introduced the concept of mechanical impedance, the ratio between a simple harmonic driving force Foe-WI and the corresponding velocity. If the system is a linear one, this ratio is independent of the amplitude of motion and is a function of wand of the constants of the system. As we shall see in the next chapter, the use of the impedance function enables us to transform a problem from one of studying response as a function of time to one of studying impedance as a function of frequency. Often the latter problem is a simpler one. At any rate the canonical equations (3.2.7) enable us to approach the concept of impedance from a new direction. We first note that, if an external force F, is applied to the coordinate q-, then the canonical
284
Fields and the Variational Principle
[CR. 3
equations become
qr = aH/apr; Pr
+
(aH/aqr) = F;
We now see that the canonical equations have broken the second-order Lagrange equations into two simultaneous first-order equations of precisely the form required to compute the impedance (if the system is of such a form that an impedance function has meaning) . For the ratio of applied force F, to velocity qr, for the rth coordinate is just
z
= Pr
+ (aH/aqr) (aH/apr)
r
From this new point of view we can imagine ourselves learning the fundamental properties of a system by "probing" it with oscillating forces. To each coordinate in turn we apply such a force, and we measure the ratio between force and coordinate velocity. If this ratio is independent of amplitude, then it can be used to reconstruct the system. Alternately, if we know the nature of the Lagrange function for the system, we can compute the impedance for each of its coordinates. We assume that qr = Areiwl (in Chap. 2 and many times later in the book, we use the exponent e- iwl to represent simple harmonic motion; here and in Chap. 4, we shall be considering the behavior of Z for all values of w, both positive and negative and imaginary, so we might as well compute it first for the positive exponential) . The equation Pr = aL/aqr enables us to compute pr as a function of the q. and therefore as a function of the exponential eiw l and of the amplitudes A r • Therefore the derivatives aH/apr and aH/aqr can be expressed in this form, and eventually the ratio Zr. When this ratio turns out to be independent of the A's and of time, it is an impedance. When the potential energy of the system has a minimum value for a certain set of values of the coordinates q, we can set the origin of coordinates at this minimum, and for sufficiently small values of displacement away from this origin, the potential energy can be expressed as a quadratic function of the q's: V =
t
Lb.. qrq. + V
min
(3.2.8)
m ,n
analogous to Eq. (3.2.1). Sometimes this minimum is not an absolute one but depends on a "dynamical equilibrium." For instance we may find that one of the p's (say Pn) must be a constant; that is, aH/aqn = O. It is then possible to eliminate qn from the equations and consider the constant pn as a property of a system with one fewer coordinates, with possible additional terms in the potential energy which depend on pn and are "caused" by the motion of constant momentum (they can be called dynamic potential energies). This new system may have equilib-
§3.2]
Hamilton's Principle and Classical Dynamics
285
rium points where the II dynamic" forces are just balanced by the II true" forces, and then again the new potential energy takes on the form of Eq. (3.2.8) near the equilibrium, where some of the b's depend on the constant p« (which is no longer considered as a momentum) . For displacements of the system from equilibrium (dynamical or otherwise) which are sufficiently small, therefore, the Hamiltonian function H is a quadratic function of the p's and q's. The expression equal to the applied force F T is
This can be written in simpler form in abstract vector space. We say that the component of the displacement vector q along the rth coordinate is qT and the corresponding component of the force vector F is FT' The equation relating the force, displacement, and acceleration vectors is then
where the components of the dyadics & and mare amn and b mn , respectively. If, now, the vector F is simple harmonic with frequency w/27r, it can be represented by the vector FOe i", ! and the steady-state velocities q can be represented by the vector Ue;,.,t, where the components UT of U are complex numbers with magnitude equal to the amplitude of the velocity of the rth coordinate. In this case we can write the previous equation in the form
FO = where
.8(w)
=
.8. U iw~(
(3.2.9)
- (i/w)m
is called the im pedance dyadic for the system near the equilibrium point under study. Therefore the impedance concept always has meaning sufficiently near to points of equilibrium of the system (if there are such) . The diagonal element Zmm is called the input impedance for the mth coordinate, and the nondiagonal element Zmn is called the transfer impedance coupling the mth and nth coordinates. It is always possible (see page 59) to make a transformation to the principal axes of .8, to normal coordinates q~, so that all the transfer impedances are zero and the diagonal elements Z~(w) are the principal values of the impedance. This transformation may be different for different values of w. One can also express the displacements q = Ae i"' ! in terms of .8 and FO: F '= iw.8.A = (-w 2&
+ m)·A
where IAml is the amplitude of motion of qm. This transformation to normal coordinates is a special case of the rotation of axes in abstract vector space. The new coordinates q' are
Fields and the Variational Principle
286
[CR. 3
related to the old ones q by the equation n
n
q~ =
I
I
where
'Yrmqm ;
m=1
'Yrm'Y.m
=
Ora
m=l
In other words the abstract vector operator, with elements 'Ymn, is a unitary dyadic. The quantities 'Yrm are then the generalizations of the direction cosines of page 22. For a rotational transformation they are independent of the q's. As we showed on page 61, the sum of the diagonal terms of the dyadic 8 is invariant under rotation: n
181 =
I
n
Zmm
I
=
z;
m=1
m=l
as is also the determinant of the components of 8. Az = IZmnl = Z~(W)Z2(W) . . . Z~(w)
It is also useful to compute the dyadic
ID reciprocal to 8, such that
where 3 is the idemfactor. Reference to page 57 indicates that the relationship between the components Y mn and Zmn is given by the equation Y mr = Z'mr/ Az
where Z'mr is the first minor of Zmr in the determinant Az• It should also be obvious that the principal axes for ID are the same as the principal axes for 8 and that the principal values of ID are Y~
=
1 /Z~
The dyadic ID = 8- is called the admittance dyadic for the system. When the determinant Az is zero, we naturally cannot compute ID. This occurs whenever the angular velocity W has a value which makes one of the principal impedances Z~ zero. A reference to the functional dependence of 8 on w indicates that the determinant 1
( - iw)n Az
= Iw2amr - bmrl =
(-iw)nZ~(w)Z~(w)
. ..
Z~(w,
is an nth order polynomial in which goes to zero for n different values of w2 (though two or more of these roots may be equal). Corresponding to the root of lowest value (which we shall call wi) one of the Z~'s becomes zero, and so on, for W2 . . . wn. Since we have been arbitrary about the numbering of the principal axes, we may as well choose it so that Z~(w) goes to zero when t» = ±wr. In other words Z~ goes to zero for the lowest root WI , Z~ goes to zero for the next root W2, and so on. Furthermore, an application of the theory of equations indicates that, with w 2,
§3.2]
Hamilton's Principle and Classical Dynamics
287
these definitions, we can write the principal impedances in the form Z;(w) = iwM r - (i /w)K r = (i /w)M r(w2
-
w~)
=
l /Y~(w)
(3.2.10)
where M r and K, = w~Mr are constants determined by the values of a mr and bmr. We therefore see that the principal impedances for a conservative system are pure imaginary quantities which are odd functions of w, that is, Z( -w) = -Z(w) . When w = ±wr , the amplitude of the rt h normal coordinate becomes infinite unless F~ = 0, so a steady-state solution cannot be obtained. These frequencies Wr/21r, roots of the determinant A., are called the resonance frequencies for the system. Incidentally, we should note that the constants M; and K, and w~ are all positive, for otherwise the potential energy would not have an absolute minimum at q = O. Canonical Transformations. The general form of Eqs. (3.2.7) is of suggestive simplicity. In the first place the second-order Lagrange equations (3.2.4) are replaced with pairs of first-order equations, which are to be solved simultaneously to determine p and q. This separation of the description of the motion into two sets of independent variables, the p's and the q's, corresponds to the fundamental peculiarity of classical dynamics : that the acceleration, the second derivative, is proportional to the force and therefore that both the initial position and initial velocity can be chosen arbitrarily. The q's are the generalized components of position, and the p's are related to the corresponding velocities in such a way as to make the interrelations come out in symmetric form. The fundamental equations, which at the same time relate the p's and the q's for a given system and also determine the behavior of the system, are the canonical equations (3.2.7). Choice of the Hamiltonian H , a _ function of the p's and q's, determines the system. A set of pairs of variables q and p, related according to Eqs. (3.2.7) by a given Hamiltonian H, are called canonically conjugate variables for the Hamiltonian H (or, more, simply., conjugate variables). The same system can , of course , be described in terms of different coordinates (and conjugate momenta) . Just as the study of fields was clarified by investigating the effects of coordinate transformations on field components, so here it will be illuminating to discuss the effect of transformations on conjugate variables p and q. We can take the long way around by transforming the q's into new coordinates Q, which also are capable of describing the configuration of the system under study, by expressing the Lagrange function L in terms of the Q's and the Q's, by finding the conjugate momenta P, by the equations P, = - (aL/aQr) , and finally by obtaining a new Hamiltonian K = !,PQ - L in terms of the new conjugate variables P and Q. Or else we can develop new techniques of simultaneously transforming from the conjugate pairs p, q
Fields and the Variational Principle
288
[CR. 3
to the new pairs P, Q by a transformation which keeps the canonical equations (3.2.7) invariant in form . Such transformations are called canonical transformations. Canonical transformations are related to a family of transformations, called by mathematicians contact transformations, which transform line elements (i.e ., position and direction) rather than points. Since we wish to transform both position (the q's) and momentum (the p's, related to the direction of the motion of the system), the connection is obvious. The basis of a contact transformation is a function S of both old and new coordinates.
(x, y ') Plane
(x,y) Pla~e
Fig. 3.2 Contact transformation in two dimensions.
As an example we consider a two-dimensional case shown in Fig. 3.2 where S is a function of x, y and of x', y'. Corresponding to each point P in (x,y) space (i .e., to fixed values of x and y), the equation S(X,YiX',y')
= constant
defines a curve C in (x' ,y') space ; and, vice versa, to every point on the (x',y') plane there is a related curve on the (x,y) plane. If we move the point in the (x ,y) plane so it traces out a curve K, the related sequence of curves in the (x' ,y') plane may have an envelope curve E, which can be said to be the related curve traced out on the (x' ,y' ) plane. To each point in (x ,y) there is a related curve in (x',y'), but to each curve in (x ,y) there is a corresponding envelope curve in (x ',y') . Therefore to each line element (i.e., position plus direction) in (x,y) there is a corresponding line element in (x',y') . The correspondence between line elements can be shown by taking the two points (x,y) and (x + dx,y + dy) defining a line element in (x ,y) and working out the related line element in (x',y') . The curves in (x' ,y') are given by the two equations S(x,y-,x',y')
= Cj
S(x
+ dx, y + dy j x',y')
.
... S(x,y;x',Y')
as + as ax dx + ay dy
= C
Hamilton's Principle and Classical Dynamics
§3.2]
289
If we set dx = X de and dy = iJ ds , where X/iJ is the slope of the line element, then we arrive at two simultaneous equations: S(x,y;x',y') = C;
. as x -
ax
+ y. as -= ay
0
which we can solve to obtain the point (x ',y') corresponding to the point (x,y). The direction of the envelope curve in (x',y') is obtained by differentiating the first equation with respect to the primed coordinates, dx'
(:~,) + dy' (:~,) = 0
or x'
(:~,) + iJ' (:~,)
=0
if dx' = x' de and dy' = iJ' ds . The symmetry of the equations in the derivatives ensures that the transformation is symmetrical with respect to the two planes. A simple example might make this clearer. Suppose the transformation function is S = (x - x'F + (y - y')2 and the constant C is R2. The point x = a , y = b corresponds to the circle in the (x' ,y') plane of radius R and center (a,b). If the line element from (a,b) to (a + dx, b) is used, the two equations to solve are (in this case x = 1, iJ = 0) (x' - aF + (y' - b)! = R2; 2(x' - a) = 0 The envelope of circles of radius R drawn with centers on the horizontal line y = b is the horizontal line y = b + R (or else the line y = b - R). Therefore the transformed line element goes from (a, b ± R) to (a + dx, b ± R) . Incidentally, this example indicates the close relationship between contact transformations and Huygen's principle. For a dynamical system with a Hamiltonian function H which does not depend specifically on time, it turns out , as we shall show shortly, that the quantity !,p dq - !'P dQ, when expressed as a function of the p's and q's (or of the q's and Q's, etc.), is a complete differential whenever the transformation from the p's and q's to the P's and Q's is a canonical transformation. For then the transformation function S can be obtained by integrating the quantity !,pr dq; -
!'Pr dQr = dS
The function S can be expressed in terms of the q's and the Q's and is then the function defining the contact transformation. Such a transformation leaves Hamilton's variational principle unchanged and therefore leaves the canonical equations (3.2.7) unchanged in form; for adding the quantity (K - H) dt (which equals zero for K is the new Hamiltonian) to the equation for dS and integrating with
290
[cH.3
Fields and the Variational Principle
respect to time, we have
If we do not change the end points to and t l , the integral for S is unchanged when the intermediate path is varied, and so if of[2":pq - H] dt is zero, then 0f[ 2":PQ - K] dt is likewise zero. Therefore the transformation defined by the function S, so obtained, is a canonical transformation and the P's, Q's, and K are related by the equations Qr = aK /aPr; ?T = - (aK /aQr)
which are the canonical equations (3.2.7). Even when H (and therefore K) depends explicitly on the time, the transformation function S can be obtained by integrating the equation 2":prQT - H - 2":P rQr
+K
= dS/dt
or the equation in its differential form
+ (K
2":prdqr - 2":P rdQr
where dS is a complete differential.
- H) dt = dS
Since in any case
n
~S =
2: [(::)
T=1
dq;
+ (:g) dQr + ~~
-l
we have, by equating coefficients of the differentials, PT = as/aqr ; P, = - (as /aQr);
K - H = as/at
(3.2.11)
which gives us the expressions for the momenta, in terms of the coordinates for the transformation. Poisson Brackets. The study of the invariants for canonical transforms covers most of the basic aspects of classical dynamics. The energy function H is one such invariant (unless H depends explicitly on t). One whole class of invariants can be expressed most conveniently in terms of Poisson bracket expressions. These brackets, for two arbitrary functions u and v of the p's and q's, are defined as follows :
as was given in Eq. (2.6.4). The Poisson brackets have several interesting algebraic properties which are formally related to the properties of ordinary derivatives: (u,c) = 0
§3.2]
291
Hamilton's Principle and Classical Dynamics
where c is a constant independent of the p's and q's. (u,v
+ w)
= (u,v) (uv,w)
+ (u,w) ; =
u(v,w)
(u + v,w) + v(u,w);
=
(u,w) etc.
+
(v,w);
The brackets are , of course, antisymmetric, so that (u,v) = - (v,u). The reason the expressions are useful is that they are invariant with respect to a canonical transformation. If the set q, p are related to the set Q, P by the relation that !,p dq - !'P dQ is a complete differential, then, for any pair of functions u, v of p, q or of P, Q,
Therefore dynamical equations set up in terms of Poisson brackets are invariant under canonical transformation. The canonical equations of motion (3.2.7), for instance, become tir = (H,qr);
Pr = (H,Pr)
In fact, by using the canonical equations plus the definition of the Poisson brackets, we can show that for any function u of the q's and p's (or of the Q's and P's) du /dt = (H,u);
au/aqr = (Pr,U);
au /apr = (u,qr)
We can also use the Poisson brackets to test for a contact transformation. Starting with the original set of n coordinates q and the conjugate momenta, another set of n coordinates Q and conjugate momenta are related to the original set by a contact transformation if, and only if, the following relations are satisfied: (Qr,Q.) = 0;
(Pr,p.) = 0;
(Pr,Q.) = 5'8
where 5,. = 0 if r ~ s, = 1 if r = s. The Action Integral. We note that the transformation function S has the dimensions of action and that, when the Q's are held constant and S is considered as a function of the q's, then the vector p, being the gradient of S, is normal to the surface S = constant. In other words, a choice of values for the Q's and K picks out a family of action surfaces S = constant and a family of trajectories for the system which are orthogonal to the action surfaces. If we pick new values of the Q's and K, we get new families of surfaces and trajectories. From one point of view the Q's can be considered as initial conditions, and the subsequent behavior of the system can be considered to be the unfolding of a contact transformation as time marches on. A differential equation for the action function S, as function of initial and final coordinates, can be obtained by setting up the equation
292
[CH. 3
Fields and the Variational Principle
stating that the Hamiltonian is constant, H(p,q) = E, and substituting in for each PT the expression aSjaqT, from Eqs. (3.2.11). This results in a differential equation H (::'
q) = E
(3.2.12)
which is called the H amilton-Jacobi equation. Its solution is a function of the n q's, of E, and of n constants of integration, which we can call Q2, QI, . . . ,Qn, a (we can always juggle the constants around so that a is simply an additive constant) . If we let E = QI, then we can consider the other Q's to be a new set of coordinates for the system. According to Eqs. (3.2.11) the conjugate momenta are P T = -(aSjaQT), and the transformed coordinates and momenta satisfy the canonical equations PI = -aHfaQI = -iJHjiJE = -1; r
= 2,3, . . .
PT =
-iJH jiJQT
= 0;
,n
for in this case K = H, and H is independent of the initial condition constants QT' Therefore the equations of motion can be written iJS jiJE
=t
+ CI;
iJSjiJQT
= cT; r = 2,3, .. . ,n
where the c's are another set of constants (corresponding, along with the Q's, to the initial conditions). It should be noted that PI is related to the quantity qt of page 252. The relations between the Hamilton-Jacobi equation and Schroedinger's equation (2.6.28), H[(hji)(iJ jiJq),q]1/; = EI/;, for the quantum mechanical wave function 1/;, are important and interesting but cannot be expanded on here . We need point out only that, if we set Yt = e(ilh)S, the Schroedinger equation reduces to the Hamilton-Jacobi, if S is so much larger than h that we can neglect (i jh)(iJ 2S jiJq2) compared with (ijh)2(iJSjiJq) 2. In the limit of large values of action and energy, the surfaces of constant phase for the wave function I/; become the surfaces of constant action S for the corresponding classical system. Wave mechanics goes over to geometrical mechanics, just as wave optics goes over to geometrical optics for vanishingly small wavelengths. We have displayed here a large part of classical dynamical theory, undiluted with any examples. Before we go on to the real business of this chapter, the application of the variational principle to fields, we had better consolidate our position thus far by discussing a few examples. The Two-dimensional Oscillator. A useful example is that of the motion of a mass on the end of a stiff rod which can bend in both directions. For small oscillations the motion is in a plane, and the coordinates ql, q2 for the mass can be taken to be the principal axes for the spring, so that the force in the ql direction is proportional to ql only , and similarly with q2. The kinetic energy of the mass is then }m(qi + ~m. If
§3.2]
Hamilton's Principle and Classical Dynamics
293
the spring is equally stiff in both directions, the potential energy can be written jmw 2(qr + qi). The Lagrange equations (3.2.4) are just the N ewton's equations ql = - W 2ql ; q2 = - W 2q2 The solution is, therefore, that q) and q2 vary sinusoidally with time, with a frequency w/27r. The related momenta are, of course, PI = mql, P2 = mq2, so that the Hamiltonian is (3.2.13) The solutions are already determined from the Lagrange equations, but we are now seeking to illustrate some of the concepts and quantities discussed above, so we shall try solving the problem by the use of a contact transformation. It would be handy if the new momenta were constant, and this could be accomplished by letting the new Hamiltonian K be independent of QI and Q2. The easiest way is to let K be proportional to (PI P 2) , for then P = -(iJK/iJQ) = O. Since Q = iJK/iJP, the Q's will be proportional to time t. This suggests the following transformation :
+
or
q = A sin(wt); P = mwA cos(wt) ; Q ex: wi and P ex: A 2 QI = tan- l (mwql/PI); Q2 = tan- l (mwq2/P2) PI = (1/2mw)(pr + m 2w2qi); P 2 = (1/2mw)(pi + m 2w2qi)
The quantity PI dql + P2 dq2 - PI dQl - P 2 dQ2, expressed in terms of the p's and q's, turns out to be ·H P I dql + ql dp, + P2 dq2 + q2 dq2), which is a complete differential of the quantity
S
=
j-(Plql
+ P2q2) =
j-mw[qr cOt(QI)
which is the transformation function. formation . The inverse equations
q; = v 2/mw PI = ~
vlPI sin
+ qi cot(Q2»)
(3.2.14)
Therefore this is a contact trans-
QI; q2 = v2/mw ...;P;, sin Q2 (3.2.15) P2 = ~ ...;p;, cos Q,
vp;, cos QI;
enable us to see that the new Hamiltonian is
K = w(P I
+P
2)
Since this is a contact transformation, Hamilton's equations (3.2.7) still hold, and since Q = iJK/ap, we have
QI = wt + 'PI ;
Q2
=
wt + 'P2
Also, since P = - (iJK/aQ) = 0, we have that PI and P 2 are constant. This completes the solution, for we can substitute these simple expressions back in Eqs. (3.2.15) to obtain formulas for the coordinates and momenta in terms of time and the energy of motion K .
294
Fields and the Variational Principle
[cH.3
We notice that the P's havethe dimensions of action and the Q''ci are angles. In fact, if we integrate p dq over a whole cycle of the oscillation and express this in terms of P and Q, the result is
Jp dq
= 2P
102.. cos" Q dQ
= 27rP
proportional to P itself . These canonically conjugate variables Q and P are called angle and action variables. All vibrational problems with sinusoidal solutions can be simplified and solved by performing a contact transformation to the appropriate action and angle variables. But to return to the harmonic oscillator in two dimensions, we can also express the motion in terms of the polar coordinates r, lp instead of the rectangular coordinates ql and q2. The contact transformation is given by the equations
Vq¥ + q~ ; lp pr = (l /r) (Plql + P2q2); r =
=
tan- 1 (qdql) Pp
= (P2ql - Plq2)
(3.2.16)
and the Hamilton ian is (3.2.17) Since iJK liJlp is zero, pp, the angular momentum of the system, is a constant. It is not difficult to see that this is true no matter what the potential energy is as long as it depends on r alone and not on lp . The rest of the solution, if we wish it , can be obtained from the solution in rectangular coordinates. Charged Particle in Electromagnetic Field. There are times when it is not obvious what form the Lagrange function L = T - V should have, for substitution in the variational integral (3.2.3). This is particularly true when forces due to fields enter. In many such cases we must combine our knowledge of scalar invariants of the system with investigations of simple, limiting cases to arrive at the correct answer. For instance, in the case of a charged particle in an electromagnetic field, should we call the energy of interaction between the magnetic field and the motion of the particle part of the kinetic energy T (since it depends on the velocity of the particle) or part of the potential energy V (since it is due to an externally produced field)? We start by putting down all the scalar invariants (in three space) of the particle and of the field. The Lagrange function L = T - V must be an invariant, for Hamilton's principle must hold in any space coordinate system. The kinetic energy of the particle alone, Vnv 2 , is such an invariant, for it is proportional to the dot product of the vector v with itself. The electric potential lp is also an invariant (in three space). So are the squares of the magnitudes of the fields, E2 and H2, and the square of the vector potential A 2. However, the fields are obtained from the potentials by
§3.2]
Hamilton's Principle and Classical Dynamics
295
differentiation and the forces on the particle are obtained from the Lagrange function by differentiation [see Eqs. (3.2.2) and (3.2.4)], so it would be natural to expect only the potentials A and lp to appear in the L for the particle. Another invariant which might enter is the dot product v . A. The forces on the particle are eE = -e grad lp - (e/c)(aA /at) [see Eq. (2.5.13)] and (e/c)v X B = (e/c)v X curl A [see (Eq. 2.5.5)], and these must come from the Lagrange equations (3.2.4) by differentiation of L . Since there is to be a time derivative of A, we must have a term involving the product of v and A, presumably the v A term. In considering the equation for the particle only (we are not yet considering the equations for the fields), we have no term to the second order in A. Therefore L for the particle must be a combination of v2 , v A, and lp . The first term must obviously be the kinetic energy of the particle, lmv 2• The term in lp must be a potential energy, and if the charge on the particle is e, this term must be - elp. The third term must give the term - (e/c) (aA /at) , for the rest of eE , and also the term (e/c)v X curl A. Since grad (v - A) = v X curl A + v (VA) (see page 115), it appears that the term should be (e/c)v· A. Therefore the Lagrange function for the charged particle in an electromagnetic field is »
»
»
L
= jmv2
+ (e/c)v· A
(3.2.18)
- elp
Remembering that the coordinates in A and lp are the coordinates x, y, z of the particle at time t, we can express the three equations (3.2.4) for the three coordinates in terms of a single vector equation. Since (a/avz)L = mv",
+ (e/c)A", =
p",
the vector equation is
!i (mv + ~A) dt c
= grad L
=
-e grad
lp
+ ~v X curl A + ~v. (VA) c c
The term entering into the expression for E [Eq. (2.5.13)] has the partial time derivative aA /at , representing the change of A at a fixed point in space, whereas the quantity on the left-hand side is the total derivative dA /dt, representing the total rate of change of A at the particle, which is moving . From Eq. (2.3.2) we have that the change in a field at a point moving with velocity v is dA = aA dt at
+ v . (VA)
Therefore, the vector equation of motion of the particle reduces to d dt (mv)
. e aA · -e grade -
e
cat + C V X
curl A = eE
e
+ CV X H
(3.2.19)
296
[CR. 3
Fields and the Variational Principle
which corresponds to Eq. (2.5.12) for the effective force on the charged particle. Weare now in a position to set up the Hamiltonian for the particle. The "momentum" of the particle is the vector with x component fJL/fJvz;: p = mv
+ (e/c)A
In this case the action of the field is to change continuously the particle velocity, so that mv is no longer a quantity which is "conserved." If we are to have conservation of momentum, p cannot equal my. According to Eq. (3.2.6) , the Hamiltonian is H
=
p •v
- L = (mv + ~
A) .v -
jmv 2 =
-
~ v . A+ etp
2~ (p - ~AY + etp · (3.2.20)
This is the result quoted on pages 256 and 254. This gives us a foretaste of the consequences of the introduction of fields and field interactions into the formalism of classi cal dynamics. The momenta are no longer simply proportional to the velocities, it is no longer quite so obvious what is kinetic energy or H or L, and we must depend more on the formal equations, such as Eqs. (3.2.4), (3.2.6), and (3.2.7), instead of an "intuition," to reach the correct results. As a specific example we might consider the case of a particle of . mass m and charge e in a constant magnetic field of magnitude B = mcw/e in the z direction. The scalar potential is tp = 0, and the vector potential is A = (mcw/2e) ( -yi + xj) The Lagrange function is L = jm(x 2 + y2)
+ jmw( -xy + yx);
X = dx /dt;
etc.
and the two momenta are pz;
= m(x -
twy);
p" = m(iJ
+ twx)
The Lagrange equations are d 2x dy dt2 = w dt ; and the solutions turn out to be
x = R sin (wt + a)
+ Xo ;
y = R cos (wt + a)
+ Yo
representing a circular orbit of radius R with center at Xo, yo. Hamiltonian is, of course, 1 H = -2m (p z;
1 + imwy) 2 + -2m (py -
imwx)2 = jmw 2R2 = jmv 2
The
Hamilton's Principle and Classical Dynamics
§3.2)
297
We note that the radius of the orbit is equal to v/w, where w = ef mc times the magnitude of the magnetic field B. A contact transformation which simplifies the Hamiltonian considerably is
x = vl/mw [~sin QI + P 2); P,. = y = Vl /mw [~ cos QI + Q2)i PII =
l l
vnu;; [~cos QI - Qd vnu;; [- ~ sin QI + P 2)
'Use of the Poisson bracket equations (2.6.4) in reverse (using the Q's and P's as independent variables) will confirm that this is a contact transformation. Substituting in for H, we find that the transformed Hamiltonian is just
K = wPI Therefore using Eqs. (3.2.8), we see that PI, P 2, and Q2 are constants and QI is linearly dependent on time, with the proportionality constant equal to w, the angular velocity of the particle in its circular orbit. Relativistic Particle. Another example of the use of the classical dynamical equations, which will be needed later in the chapter, involves the behavior of a particle moving so fast that we can no longer neglect v2 compared with c2• We have pointed out earlier that L is not a Lorentz invariant. It should not be, for the variational integral fL dt, integrated along the world line of the particle, should be the invariant. If the particle is moving with speed u with respect to the observer, the differential dt of the observer's time is related to the proper time dr for the world line of the particle by the relation dr = VI - (U/C)2 dt. Consequently if the integral fL dt = f[L/VI - (U/C)2) dr is invariant and dr is invariant, then the integrand [L/Vl - (U/C)2) is invariant and L must be some Lorentz invariant multiplied by VI - (u/c )2. For instance, for a force-free particle the relativistic Lagrange function is L = -moc 2 VI - (U/C)2 ~ -moc 2 lmou 2 ..• ; u« c (3.2.21)
+
The term corresponds to the kinetic energy term minus the rest energy moc 2 ; if there were a potential energy V, it would be subtracted from the term shown here . The momentum is then obtained by differentiating L with respect to the components of u : mou P = -Y/'I;'=-=;(;=u=/;=c=:=: )2 which corresponds to Eq. (1.7.5). The Hamiltonian function is H
= pu
- L
=V
moc2 2 = moc 1 - (U/C)2
VI +
(p/moc )2
~ moc + C~) p + . . . ; 2
2
p-«; moc (3.2.22)
Fields and the Yariaiional Principle
298
[CR. 3
which is the expression used on pages 256 and 260, in connection with the Dirac equation for the electron. This expression is, of course, the time component of a four-vector, the space components of which are the components of cp. Any potential energy term which is to be added should thus also be the time component of a four-vector. Dissipative Systems. Finally, before we turn to the application of Hamilton's principle to fields, we shall introduce a formalism which will enable us to carryon calculations for dissipative systems (i.e., ones with friction nonnegligible) as though they were conservative systems (i .e., ones with negligible friction) . The dodge is to consider, simultaneously with the system having the usual friction, a "mirror-image" system with negative friction, into which the energy goes which is drained from the dissipative system. In this way the total energy is conserved, and we can have an invariant Lagrange function, at the sacrifice of acertain amount of "reality" in some of the incidental results. For an example of what we mean, let us take the one-dimensional oscillator with friction , having the equation of motion mx
+ Rx + Kx
(3.2.23)
= 0
We wish to obtain this equation from some Lagrange function by the usual variational technique. In order to do this we set up the purely formal expression L = m(xx*) - tR(x*i; - xi;*) - Kxx*
(3.2.24)
This is to be considered as the Lagrange function for two coordinates, x and x* . The coordinate x* represents the " mirror-image " oscillator with negative friction. Applying our formalism, we obtain for the two "momenta" p = mx" - tRx*; p* = mi; + tRx which have little to do with the actual momentum of the oscillator. Nevertheless a continuation of the formal machinery results in two Lagrange equations for the two systems: mx* - Ri;*
+ Kx*
= 0; mx
+ Ri; + Kx
=
°
The equation for x is just Eq. (3.2.23), which we started out to get. The equation for x* involves a negative frictional term, as we mentioned above. The Hamiltonian is H = pi;
+ p*i;* -
+
L = mi;i;* Kxx* = (l /m)(p tRx*)(p* - tRx)
+
+ Kxx*
(3.2.25)
Since x* increases in amplitude as fast as x decreases, then H will stay constant.
Hamilton's Principle and Classical Dynamics
§3.2]
299
By this arbitrary trick we are able to handle dissipative systems as though they were conservative. This is not very satisfactory if an alternate method of solution is known, but it will be necessary, in order to make any progress when we come to study dissipative fields, as in the diffusion equation. As an indication of the fact that such cases are far from typical we note that we had previously assumed that L was a quadratic function of the q's, whereas in the present case terms in x*x occur. Impedance and Admittance for Dissipative Systems. At this point it is worth while to take up again the discussion of mechanical impedance from the point where we left it on page 286, to discuss the effect of resistive forces. For a system with dynamic or static equilibrium, as we have seen, we can transform coordinates to the set Xl , X2, • • • , X n which go to zero at the point of equilibrium. For displacements sufficiently near to this equilibrium the potential energy will be
v
L
= j
bmTxmX T
+ Vo =
j-x . 58 . x
+ Vo
m.T
where 58 is the dyadic with elements bmT and x is the n-dimensional vector with components x The kinetic energy may, as always, be given by the expression T•
T =
1 '\'
~ ~
.
.
amrXmXT
=
l'
~x
.
If
•
u • X
m,T
where ~ is the dyadic with elements a mT• In case there is friction the retarding force on the mth coordinate can be written in the form
L
r mTx T
= (m· i)m
T
where m is the resistance dyadic, with components r mT• Here, as with the potential and kinetic energies, we have included terms corresponding to the coupling between different displacements X m • represented by the nondiagonal terms of m. Following the pattern of Eq. (3.2.24) we write the Lagrange function
m,T
=
i* .
~
. i - jx* . m . i
+ ji* . m . x -
where x* is the vector conjugate to x, conjugate are then p = i* ·
~
x* . 58 . x
(3.2.26)
The momentum vector and its
- jx*. m ; p* =
~ .
i
+ jm· x
300
Fields and the Variational Principle
[cH.3
and the Hamiltonian is
+
+
H = P•i i* . p* - L = i* . ~ • i x* . 58 . x = (p jx* . ffi) . (~-1) • (p* - jffi . x) x* . 58 . x
+
where the dyadic
=
~-1. ~
(~-l)
+
is reciprocal to the dyadic
~,
(3.2.27)
such that
~
.
~-l
= 3.
The generalized driving force vector, acting on the displacement X m but not on the mirror-image displacements, is the one causing changes in the p*'s . The Hamilton canonical equation for the component acting on the mth coordinate is or, in abstract vector space , F = p*
n
l
+
(aH/a x:')em
= p*
+ jffi . (~-l) . (p* -
jffi· x)
+ 58 . X
m=l
= ~ .
i
+ ffi . i + 58 . x
(3.2.28)
If now the driving force is oscillatory, F = FOei"'t, each displacement (for steady-state motion) will be oscillatory with the same frequency with x = Aei",t (where a component of A is Am, the amplitude of oscillation of the mth coordinate) and the velocity is i = Ue iwt (where the mth component of U is Um) . The relation between F O and A or U is
iwamr + rmr + (l /iw)b mr (3.2.29) The impedance dyadic is now complex instead of pure imaginary. The real part of each term is called the resistance, and the imaginary part is called the reactance. The reciprocal dyadic ID = B-1 can be computed from the determinant A. = IZmrl : F = B •U
= iwB . A; where Zmr
=
(3.2.30)
where Z:"r is the first minor of Zmr in the determinant A.. This reciprocal is called the admittance dyadic, the real part of each term is called the conductance, and the imaginary part is called the susceptance. Since B is a symmetric dyadic, it is possible to find the principal axes for it. In terms of these principal axes, or normal coordinates, the dyadic is diagonal, having diagonal elements Zr, which are the principal values of the impedance. The determinant transforms into the product of these principal values, so that (-iw)nA. = Iw 2a mr - iwrmr - bmrl = (- iWZl)( - iWZ2) . .. (-iwZ n) (3.2.31) Since all the diagonal elements of ~, ffi, and 58 are positive, the determinant can be factored into n factors of the form Mw 2 - iRw - K, where M, R, and K are all positive. The roots of these factors are i(R/2M) ± (1/2M) v'4KM - R2, and the factors can be ordered in
Scalar Fields
§3.3]
301
order of increasing real part of the roots. The factor iwZ1 has the smallest value of y(K/M) - (R /2M)2, and so on (unless one or more of the factors has roots which are pure imaginary, as when R2 4KM, in which case these factors will be labeled in order of decreasing magnitude of the root). Therefore the rth principal value of the impedance has the form
Zr =
i.W [M,.w2 -
iRrw - Krl = iMr [w 2 - 2ik,.w W
w~
-
= iMr (w - ik; - wr)(w - ik; W
k~l
+ wr)
(3.2.32)
where k; = (R r/2M r); W r = Y(Kr/M r) - k~ as long as Kr/M r is larger than k;. The constants M r, R r, K r, k; and Wr all are determined by the values of a mr, r mr, and bmr. . The dyadic ID therefore becomes infinite for 2n different complex values of w, n of them, (wr + ik r ) , having both real and imaginary values positive and being ordered so that Wm-l ::; W m; the other n of them, (-w n + ik r), have the same imaginary parts but a real part having a reversed sign. In other words, if we plot the roots on the complex plane, they will all be above the real axis and the set for negative W will be images of the set for positive W with respect to the imaginary axis. (Unless k; is larger than K r / M r , in which case W r is imaginary and all such roots are on the imaginary axis, above the real axis.) These roots correspond to the free vibrations of the system; for the mth normal mode of oscillation the time dependence will be given by the term e-krt+i"'r t and the relative amplitudes of motion will be proportional to the components of the unit vector along the rth normal coordinate in abstract vector space . The time term shows that the free oscillations are damped out . There are , of course, impedance and admittance dyadics for the conjugate coordinates, giving the relation between the forces F:' = pm + (iJH/iJxm ) and the velocities x:,. The elements of these dyadics tum out to differ from the elements of .8 and ID simply by having negative resistance terms instead of positive. Put another way, the components of [-iw.8*] are the complex conjugates of the components of [-iw.8], corresponding to the mirror-image relationship.
3.3 Scalar Fields In the case of classical dynamics the problem is solved when we have obtained expressions for the coordinates of the system as functions of time; the variational integral from which the solution is obtained contains the Lagrange function, in terms of these coordinates and their
Fields and the Variational Principle
302
[cH.3
time derivatives, integrated over time. "The material fields discussed in the previous chapter (elastic displacement, diffusion density, fluid velocity potential, etc.) all represent "smoothed-out" averages of the behavior of some complex system containing many atoms. To solve such problems we can obtain the equations of motion of the"particles and then average to obtain the equation for the field, as we did, in principle at least, in Chap. 2. Or we can average the Lagrange function for the whole system, before minimization, to obtain a variational integral for the field, which will be the approach used in this chapter. In many cases the resulting field is a scalar function of time and of the coordinates, which are in this case parameters of integration only. Here the field is the quantity which is varied to find the minimum value of the integrated Lagrange function, and the Euler equations (3.1.3) turn out to be partial differential equations giving the dependence of the field on the coordinates and on time. The Flexible String. The simple example of the flexible string under tension will illustrate several of these points and can serve as a guide in the analysis of more complex cases. We can start out with the Lagrange function for everyone of the atoms in the string (of course we should start with the Schroedinger equation, but here we deal with gross motions of many millions of atoms, so that quantum effects are negligible and classical dynamics may be used ). The total kinetic energy is N
T
=
l im.tJ; . =1
where we have assumed that there are N atoms present. The motion of each atom in an element of the string between x and x + dx can be considered as the vector sum of the average motion of the element j(d", jdt) (we here assume for simplicity that the average motion is transverse, in one plane, and that the average displacement of point x on the string from the equilibrium line is "') and the fluctuating motion w. of the individual atom away from this average. The total kinetic energy for the element dx is therefore
T =
i
L
m.[(,j,)2
+ 2,j,(j . w.) + w;] ;
,j, = iJ'" iJt
dx
where the time average of the terms in j . w. is zero. The sum is taken over all of the particles in the length dx of the string. Weare not interested here in the fluctuating motions of the individual atoms, so we shall not bother about setting up the individual Lagrange equations for the coordinates corresponding to the velocities w.. Consequently the last term in the bracket will be dropped out , not because it is negli-
§3.3]
Scalar Fields
303
gible (actually it represents the internal heat energy of the string and so is not small in the aggregate) but because such motion is not of interest to us here . The second term in the bracket can be neglected because the derivative of this with respect to t/t (which comes into the Lagrange equation for y;) has a zero time average. The total kinetic energy which is of interest to us is therefore .
(3.3.1) where p dx is equal to the sum of all the masses m. of all the particles in the element of length of the string between x and x + dx. The potential energy of the string is a complex function of the coordinates of all the atoms in the string. It also can be broken up into a term representing the average increase in potential energy of the string when it is displaced by an amount y;(x) from equilibrium, plus some terms involving the individual displacements of the particles away from their average position, which may be omitted from consideration here. The average term can be obtained by measuring the amount of work required to stretch the string when it is displaced from equilibrium. If the string is under tension T, this work is T times the increase in length of the string as long as this increase is a small fraction of the length of the string. Expressed mathematically, the part of the potential energy which is of interest to us here is
when (oy; /ox)2« 1 and when the string is stretched between supports at x = 0 and x = l. Consequently the Lagrange function for the average transverse motion of the string in a given plane is
(3.3.3) This over-all function is an integral of a Lagrange density L = ip[ (oy;/ ot)2 - c2(01/;/ax) 2] over the whole length of the string. As before, the time integral of the function £ is to be minimized. The Euler equation (3.1.3) corresponding to this minimization is a (_~) at a(oy;/ at)
~(
+ ax
ilL ) _ oL _ a(oy;/ ax) - oy; - 0
or
(02'iJt2!!.) _ c (02Y;) ox
which is the wave equation for the string [Eq. (2.1.9)].
2
2
=
0
304
Fields and the Variational Principle
[CH. 3
The Wave Equation. Consequently the wave equation for the string corresponds to the requirement that the difference between the total kinetic energy of the string and its potential energy be as small as possible, on the average and subject to the initial and boundary conditions. This is a very suggestive and useful result, from which many other relations can be derived. If, for instance, a transverse force F(x) per unit length of string is applied, an additional term - Fy, should be added to the potential energy; or if the string is embedded in an elastic medium (as discussed on page 139), the added term is iKy,2. The resulting Lagrange density is
L = ip
(~~y - iT (~~y - iKy,2 + Fy,
and the equation of motion is a~
a2Y,
p at2 - T ax2 = F - K y, which corresponds to Eq. (2.1.27). The derivative of £ with respect to J/; should correspond to the momentum in particle dynamics. The corresponding density
p = aLja~ = p(ay,jat) (3.3.4) will be called the canonical momentum density of the field at x. In the case of the string, as we see, it is the momentum of a unit length of string when moving with a velocity [ay,(x) jat]. The Hamiltonian density is, according to Eq. (3.2.6), H
=
pJ/; - L = ip
[(~~y + c e~Y] + iKy,2 - Fy, 1 = 2 p p2 + iT (~~Y + iKy,2 2
Fy,
(3.3.5)
which is to be compared with Eq. (2.1.11) . The integral of this density over the string is the total energy, but we notice, in contrast to the case of classical dynamics, that H is dependent not only on p and y, but also on ay,jax. Consequently we should expect that the equations corresponding to Hamilton's canonical equations (3.2.7) will be of more complex form . We obtain, by use of Hamilton's principle and by integration of two terms by parts,
o=
oJdtJdx[p~
- H(p,y"y,')]
= JdtJdx[J/; op + p o~ - (aHjap) op - (aHjay,) oy, - (aH jay,') oy,'] = JdtJdx{[J/; - (aH jap)] op - [p + (aH jay,) - (ajax)(aHjay,')] oy,} where J/; = ay, jat; p = apjat; y,' = ay,jax so that
aY,
en
at = ap '
ap
at =
a (aH) aH ax aY,' - aY,
(3.3.6)
§3.3]
Scalar Fields
305
which differs from the canonical equations by the term in oH/ 01/1'. These equations, when combined, give the equation of motion for the string
021/1 021/1 P fit2 = T ox2 - K1/I
+F
The situation here is more complicated than for dynamics, for the variable 1/1, which now corresponds to the dynamical coordinate q, in addition to depending on the parameter t, also depends on a parameter x. This means that the interrelations between momenta, field, and field gradient must be more complicated than those given by the canonical equations (3.2.7). This increased complexity is also apparent from the point of view of relativity. As we have seen on page 97, the energy of a particle is the time component of a four-vector, with space components proportional to the momentum. In the present case, however, the energy density
H
=
oL 0.J; if! - L
01/1 01/1
= P fit
at -
L
appears to be the (4,4) component of a tensor, 5ffi, with components
W ll W 41
01/1 oL - L = _.jp(.J;) 2 - .jTW)2 + .jK1/I2 - F1/I ax 01/1' _ 01/1 oL ,• 01/1 oL r - fit 01/1' = - T.J;1/I, W 14 = ax 0.J; = p# (3.3.7)
=
W 44 = a1/l oL - L = H at 0.J; These components satisfy the divergence equations
2 21/1 oif; [ 0 1/1 _ T 0 + K.', - F] ox p ot2 ox2 'I' 2 21/1 OW44 = 01/1 [ 0 1/12 _ T 0 + K·', - F] at at P ot ox2 'I'
oWn oX
+ oWat
oW 41 ox
+
14
=
=
O' ,
=0
(3.3.8)
which have interesting physical interpretations. Taking the second equation first and integrating it over x from a to b, we see that
- fb (OWu) & dx
= -
i. & fb
H dx = [W 41 ]b 4
But W41 is, by Eq. (2.1.12), the rate of energy flow along the string, it should be naturally related to the change of energy H in the manner given in the equation. The second divergence equation is therefore the equation of continuity for energy flow along the string. The first divergence equation relates the change in energy flow with time to the distribution of stress along the string, for W 14 = -(W 41 /c2)
306
Fields and the Variational Principle
[cH.3
has the dimensions of momentum density, the momentum related to the energy flow of wave motion. The integral of the first divergence equation,
a (b
- at l, W 14 dx
=
[Wll]~
indicates that, if W14 is the wave momentum, then W u is a force, which might be called the wave stress. The equation states that the rate of change of wave momentum in a certain portion of the string is equal to the net wave stress across the ends of the portion. The wave momentum density P = W 14 is more closely related to the wave motion than is the canonical momentum density p = p(aiftlat). For p is the transverse momentum of the various parts of the string, whereas P is related to the flow of energy along the string due to wave motion. Incidentally, Eqs. (3.3.6) can be used to show that the integral of the Hamiltonian density is a constant of the motion, independent of time, for
d dt
en , aH a ] Jor H(p,ift,ift), dx = Jor [aH. ap p + aift y; + aift' ax at dx 2ift
aH a J Jor ray; at axa ay;' + aift' ax at dx = 2y;
2H
=
[aHJ t 1/; aift' 0 = 0
since 1/; or aHI aift' is zero at both ends of the string. Helmholtz Equation. When the string vibrates with simple harmonic motion, the dependence on time can be expressed as ift = Y(x)e-u.t, where the function Y must satisfy the Helmholtz equation
(d2 y Idx 2 )
+k
2
y = 0;
k = wlc
where the value of the constant k must be determined by boundary conditions. This equation, too, can be derived from a variational principle. The Lagrange density in this case is simply the potential energy term - T(dY Idx)2. In order to obtain a nonzero result we must insist that the rms amplitude of Y be larger than zero. We can ensure this by requiring that fol P(x) dx = 1 This is, of course, a subsidiary requirement, which, together with the variational equation 5
Jor (dY)2 dx dx = 0
constitutes the requirement to be met by Y .
§3.3]
Scalar Fields
307
Reference to page 279 shows that the solution of such a variational problem can be obtained by the use of Lagrange multipliers. We include the auxiliary requirement by requiring that
be a minimum or maximum, where X is the multiplier to be determined. The Euler equation for this is (d2 y / dx 2 )
+ XY
= 0
which is the Helmholtz equation again. The best values for X must equal the required values for k 2• Velocity Potential. Going next to a three-dimensional case, we can consider the motion of it fluid , discussed earlier in Sec. 2.3. When the motion is irrotational, the fluid velocity can be expressed as the gradient of a velocity potential 1/;. The kinetic energy density of the fluid is then (3.3.9) If the fluid is incompressible, the potential energy is constant and the
Lagrange function is just L = T. is just the Laplace equation
In this case the Euler equation (3.1.3)
V 21/; = 0
Therefore the Laplace equation for the steady, irrotational motion of an incompressible fluid is equivalent to the requirement that the total kinetic energy of all the fluid is as small as it can be, subject to initial and boundary conditions. If the fluid is compressible but has negligible viscosity, then it will have a potential energy density which can be expressed in terms of the velocity potential 1/;. This energy is the work p dV required to compress a unit portion of the fluid from standard conditions of density, etc ., to the conditions of the problem. We shall assume here that the fractional change from standard conditions is small; if the density at standard conditions is p, that at the actual conditions can be written as p(l + s), where s is quite small compared with unity. At standard conditions 1 cc of fluid will occupy (1 - s) cc when the density is p(l + s) (to the first order in the small quantity s). To determine the potential energy we must know the relation between the pressure and the density as the state of the fluid is changed. For instance, we can consider the case of a gas, as discussed on page 163, where the ratio between pressure and density is given in Eq . (2.3.21). In the present case, we call pothe pressure at standard conditions and the pressure at actual conditions to be po + p, so that p is the difference in
308
Fields and the Variational Principle
[cH.3
pressure between actual and standard conditions. This definition is generally used in acoustics (note that in deference to custom we use p for the pressure in this section whereas everywhere else in this chapter, p stands for the canonical momentum). Rewriting Eq. (2.3.21) in our new notation, we have 1
+ (p jpo)
(1
=
+ s)'Y ~ 1 + vs
or
p '" pc2s
(3.3.10)
where c2 = -YPoj p for a gas. For other fluids the excess pressure is also proportional to the fractional increase in density [so that Eq . (3.3.10) still holds] but the constant c2 depends on different properties of the material. In any case c is the velocity of sound in the fluid, as we shall shortly see. As the fluid is being compressed from standard to actual conditions, an element of fluid will change volume from dV to dV (1 - s) = dV [1 (p jpc 2)]. The work done in making this change ,
d~ I" p dp Jo{s p dV ds = pC Jo
=
(212) p2 dV pc
is the potential energy of compression in an element of volume dV. The potential energy density is therefore p2 j 2pc2. This still is not in form to set in the variational equation, for we must relate p to the velocity potential 1J; (though it is perfectly possible to set up the variational equations in terms of the scalar p rather than 1J;) . The needed relationship can most easily be obtained from Eq . (2.3.14). When the viscosity coefficients 1/ and A are negligible , and when the external force F is zero, then p(avjal) = - grad p, If a velocity potential exists, v = grad 1J;, and equating gradients, we see that p
= -p(ay;;at)
+ Co
(3.3.11)
where the constant of integration Co is usually set zero. Therefore the pressure is the time component of the four-vector which has, as space components, the velocity components of the fluid. Compressional Waves. We are now ready to set up the Lagrange density for small vibrations of a nonviscous, compressible fluid :
L = T - V =
ip
{Igrad 1J;12 -
~ (:~y}
(3.3.12)
This is an invariant density, the space-time integral of which is to be minimized. The inversion here represented, where the velocity is given by space derivatives and the force by the time derivative, is due to the fact that the velocity must be a vector (a gradient) whereas the force (pressure) is a scalar. The Euler equation (3.1.3) for this Lagrange density is just the wave
Scalar Fields
§3.3]
309
equation in three space dimensions for a wave velocity c. It is of interest that the role of space and time derivatives is the reverse of that for the string. Here the space derivatives of 1/1 correspond to the kinetic energy and the time derivative corresponds to the potential energy. Here the so-called" canonical momentum density" [see Eq. (3.3.4)] aL/atit is proportional to the pressure, not to the fluid velocity at all. It shows that the simple pattern of canonically conjugate variables q and p, suitable for particle dynamics, must be replaced by a more complex pattern. The useful quantity here is the four-dyadic ~, having components (3.3.13)
The component W 44 is the energy density W 44
=
{p
[~(~~y + [grad 1/I\2J
=
{pv 2
+ 2~C2 p2
=
H
(3.3.14)
the integral of which is independent of time (as one can show by a procedure analogous to that on page 306 for the string). The time-space components are proportional to a three-vector S, ip a1/l a1/l 1 1 Wk 4 = W 4k = - -;-t -;- = 7 ' PVk = 7 ' Ski C u UXk zc sc
k = 1, 2,3
(3.3.15)
which gives the direction and magnitude of the flow of energy due to wave motion. The quantity P = S/c 2 is a vector having the dimensions of momentum density, which may be called the field momentum density . On the other hand the vector pS /p = (pv) is the momentum density of the moving fluid . We note also that the four-divergences of the vectors formed from ~ are all zero. This can be shown as follows :
where 1/1; = af/ax;. If L depends on and its derivatives, we have next,
Xi
only through the function 1/1
Fields and the Variational Principle
310
[cH. 3
Finally, using the Euler equations (3.1.3), we have
'\' aW;; _ aif; [aL '\' a aLJ Lt ax; - aXi aif; - Lt ax; aif;; i
0
i
which proves the statement regarding the divergences if L does not depend explicitly on the x's. From it we can obtain the equation of continuity for Sand H , div S
+
(aH /at) = 0
showing that, if H is the energy density, then S is the energy flow vector (which was pointed out earlier). Although the integral of H over all space is constant, H at any point can vary with time , for the energy can flow about within the boundaries. Wave Impedance. Returning to the Lagrange-Euler equation for if;, we see that, if a "force density" f is applied to if;, the equation relatingf to the changes of if; will be
f
=
~ (:~) + ~ a~i (a(a:~aXi)) - ~~ i
= -
(~) (~t~) + p div (grad if;)
In the case of classical dynamical systems we can usually apply a force to one coordinate of the system and observe the relationship between the applied force and the displacement or velocity of the system, as was shown on page 284. In the case of fields, however, a force concent rated at a point usually produces an infinite displacement at the point of contact; the force must usually be applied over an area or a volume to produce a physically realizable reaction. In fact it is usually the case that an externally applied force is applied over some part of the surface bounding the field. In the present case, for instance, a vibrating loudspeaker diaphragm produces sound waves in the fluid in front of it and these waves react back on the diaphragm. A measure of this reaction is the acoustic impedance of the fluid in front of the diaphragm, which is characteristic of the kind of wave produced. If the driving force is applied to the boundary surface, we must integrate the Lagrange-Euler equation over the volume enclosed by the boundary to find the total reaction. The result is
- ~ fff ~t~
dv
+p
ff
(grad if;) • dA
where the second term has been changed to a surface integral by the use of Gauss' theorem, Eq. (1.4.7). Any force acting on if; along the
§3.3]
Scalar Fields
311
boundary surface is balanced by this surface term, so that, if F is the driving "force" on if; per unit of surface area, then F = p grad if;. If F is a simple harmonic driving force, F = Foe-wI, then the corresponding rate of change of if; (analogous to the velocity) is -iwif;, so that the ratio of surface force density to rate of change of if; at the surface is -F / i wif; = - (p/iwif;) grad if;
Because of the reversal of the role between gradient and time derivative in this case, the quantity (iwif;) is proportional to the pressure and grad if; is proportional to the fluid velocity. In acoustics we usually consider the pressure to be analogous to the driving force, instead of F (which is the "force" causing change of if;), so that the ratio given above is more analogous to an admittance rather than an impedance. The usual definition of acoustic admittance is given by the equation y = v / p = (l /iwpif;) grad if;
(3.3.16)
where p is the pressure at some point on the boundary surface where a driving force is applied and v is the fluid velocity at the same point. To compute this admittance we must first solve the wave equation to find the field caused by the vibration of the given part of the boundary surface, after which we can compute the ratio (l /iwpif;) grad if; for the various parts of the driving surface and, if required, can integrate over this surface to find the acoustic admittance for the whole driving surface. The admittance Y is a vector because v is a vector and p a scalar. It is usually sufficient to compute the normal acoustic admittance, which is the component of Y normal to the surface. Y n = (l /iwpif;)(aif; /an)
The reciprocal of this, Zn = l /Yn = iwpif;(ay;;an)-l
is called the normal acoustic impedance . It is the ratio of pressure at the driving surface to normal velocity at the same point. Since the normal velocity of the fluid at the surface is equal to the velocity of the surface itself, this normal impedance is the most useful part of the impedance. Plane-wave Solution. As an example of the various quantities we have spoken about so glibly in this section, we shall consider a particularly simple sort of wave motion, a plane wave. Such a wave is represented by the formula if; = Cei k . r-i",1 where C = IClei 4 is a constant giving the amplitude ICI and phase angle cP of the velocity potential and where k is a constant vector of magnitude
312
Fields and the Variational Principle
[cH.3
wlc, pointing in the direction of the wave motion. The wave surfaces for this wave are planes perpendicular to k, traveling in the direction of k with a velocity c. The expressions for the pressure and fluid velocity for the plane wave can be obtained from the velocity potential by means of the relations given earlier. They are the real parts of the following expressions : p = -p(a/;Iat) = iwpCeik.r-i.t; v = grad /; = ikCeik·r-i.t In other words the actual value of the pressure at point x, y, z at time t is -wplCI sin [(wlc)(ax + (3y + 'YZ - ct) q,] where a, {3, 'Yare the direction cosines for k and Ikl = wlc. The fluid motion is in the direction of k , perpendicular to the wave fronts, and the velocity is in phase with the pressure for a plane wave. In computing the stress-energy tensor we must use the real parts of the quantities given above, for the components are quadratic expressions involving /;. Letting the symbol Q stand for [(wl c)(ax + (3y + 'YZ - ct) + q,] = (k· r - wt + q,), we have
+
= -kiCI sin Q; p2 pw2 W u = H = ipv 2 + 2~ = -2 ICI2 sin ! Q; oc c S = pv = pwklCI 2 sin- Q = P c2 ; pw 2 pw 2 W ll = - - 2 a 21CI2 sin" Q; W 12 = - - 2 a{3ICI2 sin Q c c
/;
= ICI cos Q; p = -wplCI sin Q; v
where a, {3, 'Yare the direction cosines for the propagation vector k. In matrix form the stress-energy tensor for the plane wave is, therefore, 2
5ill = - pw2 c
ICI2 sin"
;~ ';~Y =~;) i'Y
Q (;:
'Y{3 i{3
'Ya
ia
2
i'Y
-
-1
It is not difficult to confirm the divergence equations for 5ill,
L(a:~n) 4
= 0
n=l
or that the space components transform like a dyadic. The principal axes of the space part of 5ill are in the direction of k and in any two mutually perpendicular directions which are orthogonal to k. In terms of these coordinates, ~l, ~2, ~3, the matrix for 5ill becomes
5ill =
p~21C12 [~(b - (~~ ~ ~ ~) sin?
ct)]
-i
0 0
1
Scalar Fields
§3.3]
313
All the terms are proportional to the square of the frequency and to the square of the wave amplitude. We can imagine the plane wave to be produced by a vibrating plane surface perpendicular to k, oscillating in the direction of k with velocity equal to grad 1/; = zkCeik'r-i",'. The acoustic admittance for this case, measuring the reaction of the wave back on the vibrating surface, is Y
= v ip = k!pw = (l /p c)ak
where a, is the unit vector in the direction of k. The acoustic impedance normal to the plane wave fronts is therefore (pc), a real quantity. In other words a plane wave produces a resistive impedance, independent of frequency, on the driving surface. Impedances for other wave configurations will be worked out later in this book. Diffusion Equation. When we come to dissipative equations, such as the fluid-flow case with viscosity or the case of diffusion , we must use the technique discussed on page 298 to bring the system within the formal framework. For instance, for the diffusion equation the Lagrange density is L = - (grad 1/;) . (grad 1/;*) -
~a2 (1/;* ~~ -
1/;
a~*)
(3.3.17)
where 1/; is the density of the diffusing fluid, a 2 is the diffusion constant, and 1/;* refers to the mirror-image system where the fluid is "undiffusing" (or whatever it is that one calls the reverse of diffusing). The canonical momentum densities are p
= aL ia1/; =
-~a21/;* ;
p*
=
+~a21/;
which has little to do with a physical momentum. The Euler equations for this Lagrange density are V 21/;
= a2(aif;/at);
V 21/;*
= -a 2(a1/;* lat)
(3.3 .18)
The equation for 1/; is the usual diffusion equation; that for 1/;* is for the mirror-image system, which gains as much energy as the first loses. The Hamiltonian density is (grad 1/;) • (grad 1/;*), the integral of which over the volume is independent of time . It is the 4,4 component of the tensor 5ffi defined by the equations a1/; ei.
a1/;* st.
Wi; = LfJij - -a. - -a. a.I.* x, a.I.. 'Y) X, 'Y;
(3.3.19)
where 1/;; = a1/; I ax;. Tne components W k4 contain the components of the vector, grad 1/;, which gives the magnitude and direction of the diffusive flow. The introduction of the mirror-image field 1/;*, in order to set up a Lagrange function from which to obtain the diffusion equation, is prob-
Fields and the Var iational Principle
314
[cH.3
ably too artificial a procedure to expect to obtain much of physical significance from it . It is discussed here to show that the variational technique can also be applied to dissipative systems and also because similar introductions of 1/;* fields are necessary in some quantum equations and the diffusion case is a useful introduction to these more complicated cases. A similar procedure can be used to obtain a Lagrange function for the dissipative case of fluid motion when viscosity is taken into account. Schroedinger Equation. Somewhat the same procedure can be used to obtain the Schroedinger equation (2.6.38), though it is not a dissipative equation. The wave function 1/; is a complex quantity, so that its real and imaginary parts can be considered as independent variables or, what is the same thing, 1/; and its complex conjugate 1/;* can be considered as separate field variables, which can be varied independently. The product 1/;*1/; is to be real and, for the best values of the variables, will equal the probability density for the presence of the particle, i .e., for the configuration of the system specified by the coordinates x. As an example, we consider a single particle of mass m, acting under the influence of a potential field V(x,Y,z). The Lagrange density turns out to be 2 h (grad 1/;*) • (grad 1/;) - 2i h ( 1/;* a1/; L = - 2m
a1/;*) at - at 1/;
- 1/;*V1/;
(3.3.20)
and 1/;* and 1/; are to be varied until £ = If IfL dv dt is a minimum. The Lagrange-Euler equations are then
or - ~ .* v .1 2m ...2 P
+ V.I.* P
= -ih a1/;* ~
(3321) . .
It can easily be seen that these equations correspond to Eq . (2.6.38), when we write down the classical Hamiltonian for the particle, H(p ,q) = (1j2m)(p; + p~ + p;) + V . Substituting (h ji)(a jax) for pz, etc ., operating on 1/; turns Eq . (2.6.38) into the first of Eqs. (3.3.21). The equation for the complex conjugate 1/;* is obtained by changing the sign of the i in the time derivative term. The two canonical momenta are p
• = aL jat! = -(h1/;* j2i) ; p* = M j2i
(3.3.22)'
They will have use when we wish to perform the" second quantization" often resorted to in modern quantum theory, but not treated in this book. The" stress-energy " tensor 5lli has components defined by the equations
§3.3]
315
Scalar Fields
W mn
* st. = Y,m ay,:
st. + 1/Im aY,n
- Omn L
(3.3.23)
aY, {O; m;;c n Y,m = ax",; x", = (x,Y,z,t); Omn = 1; m = n
where
The energy density is the (4,4) component of 5ffi H = W u = (h2 /2m)(grad y,*) . (grad y,) + y,*Vy, It should be pointed out that in the present case, as with the diffusion equation, the time derivative terms (equivalent to the q's) enter as a linear function in L , rather than as a quadratic function of the form ~ar8qrq8 ' as had been assumed in classical dynamics. Whenever the q's occur as a linear function in L , the canonical momenta aLjaq will not be a function of q's but will be a function of just the q's, so that p and q are not independent variables. In this case the definition H = ~pq - L will ensure that H is a function of the q's alone , without either p's or q's, and of course, the canonical equations will not have the same form as heretofore [Eq. (3.3.6)]. In both the diffusion equation and the Schroedinger equation p is a function of y,* and p* a function of y" so that we cannot have one canonical equation for p and the other for q but must have one for y, and the other for ",*. As always, we have (where Y,2 = ay,jay, y,: = ay,*/az, etc .)
se =
f f f f
0 = 0
=
dt
+ if;*p*
dv [ op if;
dt
-
dv [pif;
(~~) oy, -
- H(y"y,*,y,n,y,:)]
+ p oif; + if;* op* + 00/;* p*
(:;) N* -
L(:~) n
OY,n -
2: (~~) N:] n
But now op = (dp/dy,*) oy,* and, integrating by parts, fdt p 00/; = - fdt P N = - fdt o/;*(dp/dy,*) N, etc . As before we also have
-f 2: (~~) dv
71
OY,n =
f 2: dv
a: n
n
(:~) N, etc.
Setting all these into the integral we find that oJ!, divides into an integral of a quantity times oy, plus another quantity times N*. Since oJ!, must be zero, no matter what values oy, and oy,* have, the two quantities must be zero, giving us the two equations which are the new canonical equations: dp d P*] su \' a (aH) if; [ dy,* - ay, = ay,* aXn ay,: ; n (3.3.24) * [d P* dP] _ aH \ ' a (aH) 0/; ay, - ay,* - aY, aXn aY,n
Lt
Lt n
316
Fields and the Variational Principle
[CH.3
These equations, when applied to the Hamiltonian for the diffusion equation or for the Schroedinger equation, will again give the equations of motion (3.3.18) Dr (3.3.21). It is not certain, however, how useful Eqs. (3.3.24) will turn out to be, since they do not appear to tell us more than the Lagrange-Euler equations do. The energy flow vector for the Schroedinger case is S = iW4 1
+
jW 42
+ kW43
2
= -
h 2m
N *) grad y; + (ay;) [(i7if at grad if;* ] (3.3.25)
This satisfies the equation of continuity with the energy density W H , div S + (aH/at) = O. The field momentum density vector is P
=
iW 14
+ jW + kW 24
34
= -(hj2i)[y;* grad if; - if;grady;*] . (3.3.26
Referring to page 255, we see that (when the magnetic field is zero) the current density corresponding to the wave function if; is J = - (ejm)P, so that the field momentum vector P is related to the probable density of flow of the particle associated with the wave function if;. Klein -Gordon Equation. Another equation which can be dealt with in a manner similar to the preceding is the Klein-Gordon equation (2.6.51), a possible wave equation for a relativistic particle (though it is not the correct equation for electron or proton). Here again we use two independent field variables, if; and 1/1*. The quantities (h/i) (ay; jax, aif;jaY,aif;/az,aif;/aict) are the components of a four-vector, as are the similar derivatives of if;*. Combining these and the four-vector (Ax,Ay,A"ip) for the electromagnetic potential in a manner suggested by Eq. (2.6.49),· we obtain a Lagrange density for a "particle " of charge e and mass m in an electromagnetic field: L
= -
;~ [ (grad y;* + ~~ Aif;*) . (grad if; - ~~ Aif;)
- GY (att* - ~ pif;*) (:t + ~ pif;) + (~cY if;*y;]
(3.3.27)
From this we can obtain the Lagrange-Euler equation for y;, 4
'\' (aL) 4 aranay;:
st.
- aif;* = 0;
rl
= x; r2 = Y;
arn' r3 = z; r4 = t; y;: = ar
n=l
resulting in the Klein-Gordon equation for the presence of an electromagnetic field: 3
2: (a~n - ~c AnY
n-l
y; -
~(~ + ~ pY y; = (~cY y;
(3.3.28)
Scalar Fields
§3.3]
317
The equation for 1/;* is of the same form. Here we have used the equation div A + (l jc)(acpjat) = 0 several times to obtain the above result. This equation reduces to the simple form given in Eq. (2.6.51) when A and cp are zero. To simplify the rest of the discussion we shall treat the case when A and cp are zero. The Lagrange function is then 2 h (grad 1/;*) . (grad 1/;) L = - 2m
v (a1/;*) + 2mc2 ----at (a1/;) at -! mc 21/;*1/;
and the canonical momenta are
(3.3.29)
/
The 4,4 component of the 'stress-energy tensor (3.3.31) is, of course, the energy density H . This can be expressed in terms of the canonical momenta p, p*, the 1/;'s and their gradients: 2 2 h (grad 1/;*) . (grad 1/;) + 2mc2 h (a1/;*) Wu = 2m at (a1/;) at +! mc21f;*1/; 2mc2 h2 = ----riT (p*p) + 2m (grad 1/;*) . (grad 1/;) + ! mc 21/;*1/; = H (3.3.32) From this, by using the canonical equations (3.3.6), a1/;
aH
at =ap
2mc2 *
= fi2" p
plus two others for a1/;* j at and ap jat. These again give the KleinGordon equations for1/; and 1/;*. The field momentum density vector is P = iW 14
+ jW + kW 24
34
=
2~C2 [att* grad v + ~~ grad 1/;* J
(3.3.33)
and the energy flow vector S is equal to -c2 p . The expressions for charge and current density for this equation may be obtained in several ways. One way, which will be useful later, is to refer ahead to Eq. (3.4.11) to note that the part of the Lagrange function which involves the interaction between electromagnetic potentials and charge current is the expression (l jc)A· J - ipp: Therefore the
318
Fields and the Variational Principle
[CR. 3
part of the Lagrange density (3.3.27) which involves the vis and the potentials;
he A . [* 2imc Vt grad should have this form. should be
J
=
.r.
'I' -
.r. 'I'
gra d 1/1 *]
+ 2imc2 he '" (* Vt aVt at -
aVt*) Vt---at
This indicates that the current density vector
(eh/2im)[Vt* grad Vt - Vt grad Vt*]
(3.3.34)
which is the same as the expression (2.6.47) for the Schroedinger equation, when A and '" are zero. The corresponding expression for the charge density where the potentials are zero is p = -
eh [ Vt * [j[ aVt 2imc2
-
aVt*J Vt ---at
(3.3.35)
which is not the same as for the Schroedinger equation. In fact this expression for charge density is not necessarily everywhere positive (or not everywhere negative, depending on the sign of e), which is not particularly satisfactory for a wave function (unless we are willing to consider the possibility of the change of sign of the charge!). Incidentally, these expressions for J and p can be obtained from the Klein-Gordon equation itself, by using the same methods as those given on page 255 to obtain p and J for the Schroedinger equation. ~.4
Vector Fields
When the field needed to describe some physical phenomenon has several components, the analysis becomes somewhat more complicated but the general principles are the same as those already discussed. The independent variables, which are to be varied until the integrated Lagrange function is a minimum, are the components Vt1, . . . , Vtn, functions of the parameters x, y, z, t (or another set of four-dimensional coordinates) . The Lagrange density L is an invariant function of the Vt'S and their derivatives Vtij = aVti/a~j, (h = X, ~2 = y, ~3 = z, ~4 = t), and the integral is to be minimized. field, are
The Euler equations, the equations of motion of the
4
~ a (aL) Lt a~8 iJVti8
=
et. . aVti; '" = 1, 2, . . . ,n
8=1
or
a (iJL) at aVti4
=
et. aVti -
~ aa~8 (aL) Lt aVtis 8=1
(3.4.1 )
Vector Fields
§3.4]
319
We note that the Lagrange integral £ and the corresponding LagrangeEuler equations have a sort of "gauge invariance" (see page 211). Addition to the density L of the four-divergence of some four-vector function of the field variables or their derivatives, which goes to zero at the boundaries of the volume, will not change the value of £ . For the fourfold integral of a four-divergence is equal to the four-dimensional equivalent of the net outflow integral of the vector function over the boundary surface, and this is zero if the vector function is zero at the boundary. Since £ is not changed by changing L to L + V • F = L', the new Lagrange density L' will also satisfy the Lagrange-Euler equations (3.4.1). Therefore £ and the Lagrange-Euler equations are invariant under such a change of L. General Field Properties. ,The quantity Pi = aL/ay"i4 is the canonical momentum density for the ith component ¥ti, though we have seen that its relation to what is usually considered momentum is sometimes quite tenuous. Nevertheless, the quantity api/at entering into the Euler equations is analogous to the mass times acceleration in a simpler system. The quantity 3
Fi =
st. ~ a (aL) ay"i - L.t a~8 ay"i8 8=1
which is equal to the time rate of change of Pi, is therefore analogous to a force component corresponding to the field component ¥ti. The first term aL/ay"i usually has to do with the presence of external forces acting on the field. The second term often represents the effect of the rest of the field on the ith component at x, y, Z, t. The tensor 5ffi, having components (3.4.2) is the stress-energy tensor. Its time component W 44 is the energy density H of the field, the integral of which is independent of time. We can show, as we did earlier, that H can be expressed in terms of the y,,;s, the canonical momenta pr, and the gradients ¥trio We can proceed as we did on page 304 to obtain the Hamilton canonical equations from the variational principle. These equations turn out to be 3
apr
at
=
~ a (aH)
L.t aXi
en
ay"ri - ay"r; r
= 1, 2, ..
j=1
From them also we can obtain the equations of motion (3.4.1).
. ,n
320
[CR. 3
Fields and the Variational Principle
The tensor 5ffi is often not symmetric, which can be a serious matter inasmuch as we have come to expect stress dyadics to be symmetric. If it is desirable to use a symmetric tensor, we can usually do so by utilizing the" gauge invariance" of the function .c and of the Lagrange equations. We add to the density function L the divergence of some particular vector function of the ",'s and their derivatives and at the same time adjust the scales of the coordinates in a manner which will make the tensor 5ffi symmetric and still have W 44 the energy density. This uncertainty in the exact form of the stress-energy tensor is analogous to the uncertainty in form of the energy density of the string, as discussed on page 127. It is only a formal indeterminacy, however, for physi cally measurable quantities are not affected. As we showed on page 309, the four-vector obtained by differentiation, having components (3.4.3) has zero magnitude. We note, however, that the proof that these divergences are zero depends on the assumption that Land 5ffi depend on the parameters ~i only through the junctions If L (and therefore 5ffi) contains other terms (such as potentials or current densities) which are explicit functions of the es, then Eqs. (3.4.3) will differ from zero by terms involving the derivatives of these extra terms with respect to the es. Explicit dependence of L on the coordinates occurs only when the field is coupled to a set of particles or to a material medium causing the field (such as electric charge current) . The Lorentz force on an electron, for instance, is given in terms of the field at a particular point in space, namely, the position of the electron. Interactions between the various parts of the field are expressed in terms of integrals over all space, and the dependence on the coordinates only enters through the ""s. At any rate, when Land 5ffi depend on the es only through the field variables "', Eqs. (3.4.3) hold, and in that case the three-vector
"'r .
n
S = iW 41 + jW 42 + kW43 =
Laftr [i :~, + :~. + j
k
:~.J
(3.4.4)
r=1
satisfies the equation of continuity, div S + (aH/at) = 0, for the energy. Therefore it must represent the density of energy flow in the field. It can be called the field intensity. The complementary vector n
P = iW 14
+ jW24 + kW 34
=
\' et.
L.t a",,, grad Y,r r=1
(3.4.5)
Vector Fields
§3.4}
321
has the dimensions of momentum per unit volume and can be called the field momentum density . If L has been modified so that m3 is a symmetric tensor, then P = S ; in any case P is closely related to S. The space part of the tensor m3 is a three dyadic where
U = iW 1 + jW 2 + kW a WI = Wlli W12j W13k
+
+
etc .
(3.4.6)
The other three divergence equations (3.4.3) are then given by the vector equation U· V = i div Wi
+ j div W + k div W a = 2
- (aPjat)
indicating that, if P is a momentum, the derivative of U is a force tensor, so that U is related to the potential energy due to the field. In tabular form, the stress energy tensor is
m3 =
W ll W 21
( W 31 SI
W12 W 13 ' PI) W 22 W 2a P 2 W 32 W a 3 r, S2 S3 H
with Wi; given by Eq. (3.4.2) and with W n4 = P nand W 4n = Sn. An angular momentum density vector may also be generated (if it is needed) by taking the cross product of the radius vector r from some origin to the point (x,y,z) with the vector Pat (x,y,z): n
M = r X P =
~
st. Lt a1/l" [r X
grad 1/Ir]
r=1
In quantum mechanics this property of the wave-function field turns out to be related to the probable angular momentum of the particles associated with the wave function. In the case of the nonviscous, compressible fluid, for instance, the angular momentum density would be M
= (p jc 2 ) (r
X v) = ps(r X v)
according to Eqs. (3.3.15) and (3.3.10). This is the angular momentum of the excess density due to the motion of the fluid. Therefore the change in energy density H with time requires an energy flow vector S, and a change in momentum density P with time requires an internal stress dyadic U. For these reasons the tensor m3 is called the stress-energy tensor, though it would be more accurate to call it the stress-momentum-energy tensor.
Fields and the Variational Principle
322
[oa . 3
Isotropic Elastic Media. Weare now in a position to apply these general formulas to a few interesting cases to see what some of this formalism means in physical terms. The first example is that of the motion of an elastic solid, discussed earlier in Sees. 1.6 and 2.2. From the latter section [Eqs. (2.2.17) and (2.2.18)] we see that the Lagrange density for an isotropic elastic medium is
L = T - V =
jp(as/at)2 -
[~~ + ~~ + ~:'r 2~ [(~:~Y + (~~Y + (~~Y]
= i {p [(aaStY -
-
jl~' ~I
+
eaStY + (aaS;Y] -
X
Y
~ [(~;~ + :; + (~:~ + ~:'Y + (~; + :;.)2]}. (3.4.7)
where the vector s is the displacement of the point x, y, z from undistorted equilibrium, p is the density of the medium, X and ~ its elastic constants, ~ is the strain dyadic i(Vs + sV), and ~ the stress dyadic ~
= X31~1 + 2~~ = X(div s)3 + ~(Vs + sV)
The field variables v; can be the three components of the displacement, s., which are to be varied until the total Lagrange function £ = ff ffL dx dy dz dt is a minimum. The Lagrange-Euler equation (3.4.1) for s~ turns out to be S~, Sy,
paa2;~ = X a~ (div
s)
+ ~V2S~ + ~ a~ (div s)
This is equivalent to the x component of the vector equation
p(a 2s/ W ) =
(X
+ ~) grad (div s) + ~V2S
which is the equation of motion given already in Eq. (2.2.1) . The time part of the tensor m3, defined in Eq. (3.4.2), is the energy density, WH =
jp(as/at) 2+ il~' ~I
=
H
and the field intensity vector, defined by Eq. (3.4.4), is S =
-(as/at).~
which is the energy flow vector given in Eq. (2.2.20). It satisfies the equation of continuity for energy, div S + (aH/at) = 0, as shown in Eq. (3.4.3), and proved on page 309 (for in this case L depends on the coordinates only through the field variables s). The tensor m3 is not a symmetric one. The space part, corresponding to the force dyadic defined in Eqs. (3.4.6), is
U = - (Vs) . ~ - L3
Vector Fields
§3.4J
323
The field momentum density, defined in Eq. (3.4.5), is P = p(Vs ) • (as/at) These two quantities satisfy the divergence equation U· V + (ap/at) = o. If P is a momentum.density, the dyadic U is related to the st ress density, as is verified by its definition in terms of ~, the stress dyadic. To illustrate the convenience and compactness of the dyadic and vector notation, we shall write out in full a few of the components of the st ress-energy tensor 5ID: W ll = i(X
+ (::zy +e~y +(~:zY] - [(a;:y +(a;;y,+ (~as;YJ + (~~) (~~) + (:s;Y e~y+(::zy - (~~Y+(~; +:~zy] 2JL) [ -
ip
X
iJL [
z+as W12 = _asz[ ax JL (as ay ax
l l ) ] _ a sl l [ ( ' + 2
ax
1\
)as ay
ll+,
JL
1\
z+, asz]. as ax az ' 1\
t
e c.
There are not many additional comments which are appropriate here. Certainly the variational principle has collected most of the equations and formulas we so laboriously derived in Chap. 2 all in one compact package. Whether or not we can squeeze further physical meaning from the synthesis or can discover a use for the by-product quantities such as field momentum and force dyadic, we have at least developed a straightforward, as well as a suggestive, method for obtaining such important quantities as intensity, energy density, and equations of motion from the expression for a Lagrange density. Plane-wave Solutions. To make more specific the formulas we have derived here, let us apply them to the plane wave, simple harmonic solutions of the equation .of motion (2.2.1). Following Eq. (2.2.2), one solution is s = grad .Jt, where
1/1 = Ceik.r-iot i k = w/cci
c~ =
(X
+ 2JL) /p
The actual displacement is then obtained by taking the gradient of 1/1, So = 2'kCeik·r-i..t = akAeik.r-iot i A = ikC = IAleifl k = ke; i a, = ai + ~j + ')'k where a, ~, ')' are the direction cosines for the propagation vector k . Therefore, the displacement in this compressional wave is in the direction of propagation (as we mentioned before) with an amplitude IAI· The strain tensor is then @5 = i(Vs + sV) = -kkCeik'r-iot = 2'kakAeik.r-fwti ~ = _[X(W2/C~)3' + 2JLkkJCe,l·r-i..t
where the dyadic kk is symmetric and its expansion factor Ikk l = (W/c o) 2.
324
Fields and the Variational Principle
[cH.3
In order to compute the stress-energy tensor we must take the real parts of these expressions. The energy density, for instance, is
W 44 = pw 21AI2 sin" n where n = k . r - wt energy flow vector is
+ 'P =
(w/ce)(ax
+ (3y + 'YZ
cet)
-
+ 'P.
The
S = akPCeW21A!2 sin n and the wave momentum vector is P dyadic turns out to be
= S/c~.
The space part of the
U = akakpw21AI 2 sin? n All of this, of course, is very nearly the same as the results given on
page 312 for compressional waves in a fluid . In the case of transverse or shear waves s
=
apBeik'r-i.Jt;
k
= w/c,; c; = p./ p; B =
where a p is a unit vector perpendicular to k. dyadics are given by the equations
st
= 2p.@5 =
ipc,wB(akap
IBl ei~
The stress and strain
+ apak)eik.r-i"'t
The dyadic (asa, + apak) is symmetric but has zero expansion factor, so that Istl and !@5! are both equal to zero. The various parts of the stress-energy tensor are
W 44 = pw 21BI2 sin? n; where n = (w /c,)(ax + (3y + 'YZ - c,t) S = akPc,w 21BI2 sin? n; P = Sic;; U = akakpw21BI2 sin" n
+ 'P
which have the same form as the expressions for a compressional plane wave. In other words the energy flow vector and the wave momentum vector are pointed along the propagation vector k, even though the displacement of the medium is perpendicular to k . Impedance. In the case of nonisotropic media the Lagrange density is [see Eq. (1.6.29)]
L = jplas/at!2 - j(Vs)
:~:(Vs)
where ~ (gimel) is a tetradic with elements gmnra determined by the nature of the medium. Due to the symmetry of the dyadics @5 and st, there are certain symmetries of the tetradic which always hold: gmnr. = gr.mn = gmn,r. In the case of an isotropic medium the elements are gmnra
or
~ =
Ay
=
+ p.' + p.'*;
[AOmnO ra
+ P.OmrOna + p.lim.onr]
y :~ = 1~13;
' :~ =~ ;
For the nonisotropic case, the equation of motion is
'* :~ = ~*
§3.4]
Vector Fields
325
which is a complicated second-order partial differential equation for the components of s. Here it is not always possible to separate out pure compressional and pure transverse waves ; also waves in different directions travel at different velocities. The elements of the stress-energy tensor are = -!p(as/at)2 + -!(Vs) :~:(Vs) S = - (as/at) • ~ :(Vs) ; P = p(Vs) . (as/at) U = - (vs) . [~:(Vs)] - £3
W 44
This same symbolism can be used to discuss the impedance for waves in an elastic medium. As we stated on page 310, the usual driving force is applied to the boundary surface of the medium and is equal to the volume integral of the inertial reaction p(as/at). But this is equal to a divergencelike expression, V . ~ :(Vs), and the volume integral becomes equal to a surface integral of the surface force density dyadic
\5 = ~ :(Vs) This expression is a dyadic (as are all stresses in an elastic medium) because the force is a vector which changes as the surface is changed in orientation. The force density on an element of boundary having the inward normal pointed along the unit vector an is an • ~ :(Vs) , which is a vector. When the driving force is simple harmonic, the steady-state displacement vector also has a factor ei"" (or e- i",t, in which case the impedance and admittance will be complex conjugates of the expressions for ei"'t) . The force density across the part of the boundary surface which is vibrating with velocity v = Ve i",t = i ws is given by the equation
where a, is a unit vector normal to the surface at the point where F is measured. The dyadic .8, which can be expressed in terms of the components g and the properties of the solution for s, is the impedance dyadic which measures the reaction of the medium to a driving force. For instance, for the isotropic case, ~ = X'Y + p.' + p.'*, so that an • 1:(Vs)
= (X div sja,
+ p.a
n •
(Vs
+ sV)
For a plane compressional wave, with the driving surface perpendicular to the propagation vector (that is, an = as), then (Vs) = iakkAeik.r-i",t = (sV) a, . ~ :(Vs) = akiwpceAeik.r-iwt ; pc~ = X + 2p. In this case the driving force is in the same direction as the velocity of the medium, akiwAeik.r-i",', so that the impedance dyadic is equal to the characteristic compressional impedance of the medium pCe times the idemfactor.
326
Fields and the Variational Principle
[cH.3
For a plane shear wave the velocity of the surface iwapAe- iwe is is perpendicular to the propagation vector, and using the formulas on page 323, a, .) :(Vs) = apiwpc.Beik.r-i.JI; pc; = !.L so that also in this case the driving force is parallel to the velocity and the impedance dyadic is the characteristic shear impedance PC. times the idem factor, although here the driving force and velocity are perpendicular to the propagation vector. The Electromagnetic Field. Next we come to a field which is expressed in terms of four-ve ctors, which, in fact, is the field for which the Lorentz transformation was devised, the electromagnetic field. A study of Sec. 2.5 suggests that the fundamental field quantities 1/Ii should be the components of the potential four-vector given on page 208, VI = Ax;
V 2 = All;
Va
= A . ; V 4 = icp
where A is the vector potential and cp the scalar potential. In this case we may as well discard the coordinates ~ used in the previous example in favor of the Lorentz coordinates Xl = X, X2 = y, X3 = Z, X4 = ict, as we did in Sec. 2.5. This choice will ensure Lorentz invariance but will require the factor ic to be added at times to retain the proper dimensions. The potential derivatives 1/Iij are therefore V 12 = (aA x/ay), etc . ; Vu = (l /ic)(oA x/ot) V41 = i(ocp/ax), etc.; V 44 = (l /c)(ocp/ot)
In the present notation, then, the field vectors become Ex = i(V 41 - Vu) = i!u ; Ell = i(V42 - V 24) = i!24 H x = (Va2 - V 23) = !23; H II = (V13 - Val) = fal
(3.4.8)
if we assume that!.L and E are both unity. We have as a new complication, not encountered in the case of the elastic solid, the fact that the components of the potential are interrelated by means of an auxiliary divergence condition 4
2:
,,=1
V"" = div A
+ (~) (~~)
(3.4.9)
=0
which is equivalent to Eq. (2.5.14), relating A and cpo This zero-value divergence can be added to or subtracted from various expressions to help simplify their form. What we must now do is to set up a Lagrange density which will generate the equations of motion (see Eq. 2.5.20)
2: n
0 !m" -0
x"
=
2: n
0 (V"m - Vm,,) = -a a -0
x"
Xm
2: -2: V",, --
n
n
02V m 4?r 0 2 = -1 m X"
C
(3.4.10)
§3.4)
Vector Fields
327
equivalent to Maxwell's equations or to the wave equations (2.5.15) and which will produce, as the (4,4) component of the stress energy tensor, the energy density [see Eq. (2.5.28))
when the four-vector I is zero. This vector I was defined on page 208, as the charge-current-density vector
It is a little difficult to use the definition that L should be the difference between the kinetic energy density and the potential energy density, for it is not obvious which is kinetic and which potential energy. Examination of possible invariants suggests that part of the expression be (1/811")(E2 - H2); presumably the rest includes a scalar product of vector I with the potential vector V. Computing the Lagrange-Euler equation for such an L and comparing with Eq. (3.4.10) show that the proper expression for the Lagrange density is
(3.4.11) Therefore Maxwell's equations for free space (J and p, zero) correspond to the requirement that E2 be as nearly equal to H2 as the boundary conditions allow. The Lagrange-Euler equations (3.4.1) are just the Maxwell equations (3.4.10). The canonical momentum density vector p, having components pn = (l /ic)(aL/aV n 4) (n = 1,2,3), which turn out to be (1/411"ic) (V 4n - V n 4) , is the vector -(1 /411"c)E. The "force vector" corresponding to the rate of change of this momentum with respect to time is then G/c) - (1/411") curl H. The time component of the canonical momentum density (aL /aV 41 ) , is zero, and the time component of the LagrangeEuler equations (3.4.10), div E = 411"p is a sort of equation of continuity for the canonical momentum density vector p = - (1/411"c)E. . Stress-energy Tensor. The time component of the momentumenergy tensor jffi should be the Hamiltonian density :
328
Fields and the Varialional Principle 4
WH
=
l:
[cH.3
3
V i4
aa~4 -
8~
L = -
i=1
l:
(V 4m - V m4)2
m=1
3
+ 17T'
l:
V 4m(V4m - V m4)
+ 8~ [(V
V 2l F
12 -
m-I
4
+ (V 23 = -
1
81r
4~
V 32)2
+
(V 31 - V13)2j .
l:nr + 1~7T' l:nr- ~ l: 8,r
r
(E2
+ H2) -
1
_!c Lt \' m=1
I"V"
n
- J .A c
I"V"
+ ~l:
V 4m!4m
m
+ PP + -41r1 E . grad P
. (3.4.12)
which differs from the results of Eq. (2.5.28) by the terms [PP + (l /47T')E . grad p]. However the expression E· grad p is equal to div (pE) p div E; and by remembering that div E = 41rp, we see that the extra terms are just equal to (1/41r) div (pE). Since the integral of a divergence over all space equals the net outflow integral at infinity, which is zero, we see that the average value of W 44 is equal to the average value of the Hamiltonian density
1 U = 87r (E2
+ H2) -
c1 J. A = T 44 - c1 J. A
(3.4.13)
where the tensor X is defined in Eq. (2.5.30) . This is an example of the gauge-invariant properties of the field mentioned on page 211 and also of the fact, noted on page 126, that energy density and energy flow are not uniquely determined, except in terms of the integral over all space. On the other hand, to obtain the correct results for the Hamilton canonical equations given on page 319, we must use the full expression for W 44, with the canonical momentum p inserted for -(1 /41rc)E. The Hamiltonian then takes on the form JC = 21rC 2p2
+ 81r1 H2 -
cp • grad p -
c1 J . A + PP
3
=
l:
(21rc
2p;
+ ic V 4"P,,)
,,-I
+ 87T'1 [(V 12 -
-
~
4
l:
V mIm
m=l
V 21)2
+ (V I 3 -
V 31) 2
+ (V 23 -
The equation ay,.,,/at = iCV"4 = aH /ap" becomes iC(V"4 - V 4,,) = 41rc 2p" or p" = (1/41ric)(V4n - V,,4)
V 32)2]
§3.4)
Vector Fields
329
which is the original definition of pn. The equations 3
apn
7it
'\' a (aH) aH = ~ aXr aV nr - aVr r=1
become the Maxwell equations 1 1 aE - 4'lTC at = - 4'lT curl H
1
+ CJ;
P4
=
0;
- J.. div E + p = 0 4'lT
Therefore one can use the component W 44 to calculate the Hamiltonian, but one should use U to compute the conventional energy density. A similar sort of adjustment must be made to obtain the familiar forms for intensity vector' and field momentum from the nondiagonal terms of the stress-energy tensor. We have 4
W mn
=
l: v., (a~..) 4~ l: rr., -
r= 1
-
= -
4~
l:
V rm(Vrn - V nr)
r
Vmr)(Vrn - V nr) -
4~
l:
(VmrVrn - VmrVnr)
r
(3.4.14) The second sum in the last expression can be modified by using the auxiliary condition (3.4.9) and also the wave equation for the potentials [Eqs. (2.5.15)]2: (aVnr/ax r) = -(4'lTI n/ c) :
The first sum is a four-divergence, which is zero on the average. The second sum is zero because of Eq. (3.4.9) and the third is equal to V mIn/ C. Therefore the average value of W mn (m ~ n) is equal to the average value of the terms (3.4.15)
In fact the average value of any of the terms W mn of the tensor Q:B is equal
Fields and the Variational Principle
330
[cH.3
to the average value of the tensor with terms
Ill:
T mn + - VmIn - - Omn c c
V rIr,
r
(3.4.16)
where
The tensor ~ has been discussed earlier, on page 216. In those parts of space where the charge-current vector is zero the tensor ~ is the stress-energy tensor. Expressed in terms of fields, the components are 1
Tll
= 811"
TH
= 811"
1
1
[E; - E~ - E; [E;
+ H; -
H'; - H ;];
+ E'; + E; + H; + H'; + H;] + HxH,,] =
etc. = U
(3.4.17)
T l2 =
411"
T 14
1 1 4---; [E"H. - E.H,,] = -4. (E X H), = T 41 ;
=
[E",E"
11"2
T 21 ;
etc. etc .
11"2
Field Momentum. The fact that we have discarded the tensor 5ffi for the tensor ~ need not trouble us unduly, for 5ffi does not satisfy the divergence conditions (3.4.3) unless J and p are zero, so 5ffi would not be very useful anyway. The divergence relations for ~ are not simple either because of the separation off of the four-d ivergence terms. We have \ ' aTmr = ~ \' f
'-' s»,
41r '-'
afro = ~ \' f I = k mr
ax.
r. s
c '-'
mr
r
m
(3.4.18)
T
where k m is the mth component of the force-density vector defined in Eq . (2.5.27). The space part
+ (l /c)J
XH gives the magnitude and direction of the force on the charge-current distribution. It should equal the time rate of change of the momentum of the charge , which , together with the rate of change of the momentum of the field, should equal the net force on the field plus charge . Taking the integral of k 1, for instance, over a given portion of space, and calling Ih the x component of the momentum of the charge current in this region, we have pE
Vector Fields
§3.4]
+
331
+
where F 1 = Tui Td T 13k is the net force acting on the x components of field and charge momenta, and where the last integral is a surface integral over the boundary of the given portion of space. If now 1 p = -4 (E X H)
7rC
= i.C [T 14i
+ T j + T3~]
(3.4.19)
24
is called the field momentum (see page 321), then the previous equation states that the net stress T acting over the surface of a portion of space equals the rate of change of the momentum IT of the charge current inside the surface plus the rate of change of the field momentum P inside the same surface. The time component k« of Eq. (3.4.18) is the time rate of change of the kinetic energy T of the charge current. This equation also has physical significance, which becomes clearer if we define the rate of flow of energy by the usual vector (called here the Poynting vector) S
= ic[T 41i
+ T j + T4~] 42
= (c/47r)(E X H)
(3.4.20)
The component T 44 is, of course, just the energy density U of the field. The m = 4 part of Eq. (3.4.18) is therefore div S
+ (au/at)
=
-(aT/at)
so that the equation of continuity for energy flow is that the net outflow integral of S over a closed boundary is equal to the negative rate of change of energy of charge current T and of the field U for the volume inside the boundary. Thus all the components of tensor ~ have physical significance. The density of angular momentum in a field with no charge current present is M
= r XP =
4;c r X (E X H) = 4;C [(r· H)E -
(r · E)H]
The total angular momentum of the field about the origin is obtained by integrating M over all the volume occupied by the field. When an electromagnetic field, divorced from charge current, is confined inside a finite volume of space (a wave packet) which moves about as time is changed, we can show that the integral of the four quantities (P""P lI,P z , U) = (iT 14/c, iT 24/ C, iT 34/ C, T 44 ) over the space occupied by the field (i .e., integrated over the three space perpendicular to the time axis at any given instant) is a four-vector satisfying 't he Lorentz requirements for transformation of a four-vector. For in this case
L(aTmr/axr) = 0 so that, if Cm are the components of a constant r
four-vector, the four-divergence of the four-vector with components
e. =
LCmTmr, L(aBr/ax r) m
is zero, and the integral of the normal
332
Fields and the Variational Principle
[cH.3
component of this vector over the surface of an arbitrary volume in four space is zero. We choose for the volume in four space the" four prism" with axis along the time dimension parallel to the motion of the wave packet and with space part perpendicular to this axis and large enough to contain the packet completely. The surface integral over the space part (along the sides of the four prism) is zero, for the field is zero outside the packet.
Hence the integral of the time component of B,
l CmTm4, m
over the packet at one end of the four prism must be equal to the same integral over the other end of the prism at an earlier time. Therefore, for this case, the integral of 'J:.CmTm4 over the packet is a Lorentz invariant and the components given by the integration of T m4 over the packet (over volumes perpendicular to the time axis) are components of a true four-vector . This is what we set out to prove. ' This result indicates that, if we have such a thing as a wave packet of electromagnetic field, the vector, having as components the integrated field momentum P and the integrated field energy U, is a true momentumenergy vector behaving just as if the packet were a material particle. Its angular momentum can be obtained by integrating the M just obtained. There are many other interesting properties of the electromagnetic field which can be obtained by means of the variational machinery we have set up. Gauge Transformation. Many of the difficulties we have encountered in going from Lagrange density to energy density can be simplified by choosing the right gauge. If, instead of the gauge defined by the equation div A + (l /c)(a'P/at) = 0, we use the gauge defined by the equation 'P = 0, the Maxwell equations reduce to curl A = B = J.lH; E = -(l /c)(aA /at) = D/E div (aA/at) = -(411"PC/E) curl (curl A) (eJ.l/c 2)(a2A /at 2) = (411"J.l /c)J
+
(3.4.21)
In other words, we use both longitudinal and transverse parts of A, the longitudinal part being determined by the charge density and the transverse part being largely determined by the current density. This gauge is particularly useful for cases where there is no free charge p, though it is also useful at other times. In this gauge, the Lagrange density is E \aA12 L = 811"c 2 7ft
1 - 811"J.llcurl AI2
+ c1 J . A
1 = 811" (E· D - H . B)
+ c1 J. A (3.4.22)
The canonical momentum density is, therefore, P = EA/411"c = - (D/ 411"c). The Lagrange-Euler equations give us the last of Eqs. (3.4.21) ; the first 2
Vector Fields
§3.4]
333
two equations define the relation between the fields and the potential, and the third equation fixes the gauge. The Hamiltonian density is then
W 44 = p '
A- L
=
8~ (E . D + H · B) 21rc 2
= -e- p2
-
~J.A
1 1 + 81rJ,l [curl AI2 - cJ . A =:JC
(3.4.23)
The second of the modified canonical equations (3.4.2) again corresponds to the last of Eqs. (3.4.21). To find the rest of the stress energy tensor lID in this gauge we note that the dyadic with (x,y) component iJL/iJ(iJAz/iJy) is (H X 3 /41r) = - (3 X H /41r). Utilizing this expression, we see that the energy flow vector is (for J,l = e = 1) . S
= -A..
(3 X H /41r)
=
(c/41r)(E X H)
which is the same as the expression given in Eq. (3.4.20). Therefore this particular choice of gauge gives the standard form for the energy density and the Poynting vector without all the fussing with divergences which the usual choice of gauge requires and which we displayed in earlier pages . The field momentum vector, on the other hand, has a modified form, 1 . p = - 41r (VA) • A
=
c
41r [(D X B)
.
+ D • (VA)]
and, correspondingly, the space part of the stress-energy tensor is modified, becoming . U = (1/41r)(VA) X H - 3L These quantities are not quite so familiar to us as are the energy density and Poynting vector, so they may, perhaps, be allowed to take on these modified forms (or else the divergence argument may be applied to arrive at the more familiar form) . Impedance Dyadic. To determine the field impedance for the electromagnetic field it is most convenient to use this latest choice of gauge, which gives the" correct" form for energy density and energy flow density. We return to the Lagrange-Euler equations (or the canonical equations) .
p = - ~ V • (3 41r
X H)
+ !C J
The quantity on the right is the" force" which causes a rate of change of the momentum p = eA/41rc 2 = - (D /41rc) . The part which can be applied at the boundary surface, according to the arguments of page 310, is the dyadic (-1 /41r)(3 X H), the divergence of which enters into the above expression for force density. If an electromagnetic wave is
334
Fields and the Variational Principle
[CR. 3
started at some part of the boundary surface, the" reaction" of the wave back on the element of area dA is, accordingly,
(1/41r)dA· (3 X H)
= (1/41r)(dA
X H)
a vector perpendicular to H and to dA (i .e., tangential to the boundary surface) . Relating this to the circuital rule (see page 220) cj'H • ds = 47rI we see that if the wave is "caused" by a surface current in the boundary surface, then the vector - (c/41r)(dA X H) is just equal, in amount and direction, to the part of this surface current which is contained in the element dA. The integral of this vector over all the driving surface gives us just the total current sheet. The "velocity" vector is A = -cE, so that the quantity corresponding to the impedance for the potential A, in the direction given by the unit vector a, is the dyadic which changes the vector -cE into the vector (a/41r) X H. However, the choice we have made as to which expression is to be "force" and which "velocity" is just the inverse of the usual definition of impedance, which is that Z is the ratio of voltage to current (H is proportional to current and E to voltage) . Consequently we define the impedance dyadic B of the electromagnetic field as the" ratio" between the electric field and c/ 41r times the magnetic field, and the admittance dyadic ID of the field as its inverse :
41rE = -cB· H;
cH =
-41rID· E ; ID = B-1
(3.4.24)
The admittance of the field in the direction of the unit vector a is then a X ID, as can be seen by vector multiplication of the second equation bya. Incidentally we notice that, if E is analogous to a voltage and (c/41r) (a X H) to a current, so that the" ratio" is an impedance, the" product" (c/47r)(E X H) is the ratio of energy consumption, i.e., the energy flow density [which Eq. (3.4.20) shows it to be]. Therefore our analogy is complete. Plane-wave Solution. If there is no charge current and if E = JI. = 1, a simple solution of the equations of motion (3.4.21) is
A = apAeik.r-i"'l;
k = (w /c)ak;
A =
IAlei~
where a, and a, are two mutually perpendicular unit vectors. are therefore E = i(w/c)apAe''k·r-ic.t;
The fields
H = i(w/c)(ak X ap)Ae''k·r-ic.t = a, X E
so that the vectors k, E, and H form a right-handed, orthogonal trio of vectors. As usual with plane-wave solutions, the value of the Lagrange function is zero. The energy density and the Poynting vector are E2 w21AI2 w21AI2 U = - = -sin? n· S = - - ak sin n 2 41r 41rc 41rC , n = k(ax + {3y + 'YZ - ct + 1')
Vector Fields
§3.4]
335
where a, {3, and -yare the direction cosines of k on the x, y, z axes. Dyadic (VA) is i(W/c)akapAe ik'r-iwt so that the field momentum density is 2
w 1A I2 =- a, sin 2 n = s
p
4?rc
and the space part of the stress-energy tensor is the symmetric dyadic w 21A I2 2 U = -4 2 aia, sin 71'C
n
In matrix form the whole stress-energy tensor has the following symmetric form:
~
=
w21At sin- (k . r - wt 471'c
+ 'P)
(;:
;~ -y2;~ c-y~;)
-ya -y{3 Ca c{3
c-y
1
Finally the impedance of the plane wave is the ratio between the vector E and the vector - (c/471')H, which is the dyadic
B=
(471'/c)3' X a,
and the admittance is The impedance of the wave in the direction of propagation is thus (471' /c)ak X
3' X a,
= -(471'/c)(3 - akak)
In the Gaussian units, which we are using here, the" magnitude" of the impedance of an electromagnetic plane wave in vacuum is thus 471'/ c. There are only a few other fields meriting attention in this chapter. Dirac Equation. For instance, we should be able to set up a Lagrange density for the Dirac equation for the electron, Eq. (2.6.57). Here we have eight independent field functions, the four components of 1/;, 1/;1, 1/;2, 'h 1/;4 [given in Eq. (2.6.56)] along the four directions in spin space and the corresponding components of 1/;*. A little juggling of expressions will show that the Lagrange density is
L = hc 2i [(grad 1/;*) • a1/; - 1/;*a • (grad 1/;)] -
+ 2ih [O1/;* tit 1/; - 1/;* O1/;] at e1/;*a ' A1/; + ec1/;*'P1/; - mc 21/;*ao1/;
(3.4.25)
where A and 'P are the electromagnetic potentials at the position of the ' electron, m and e the electronic mass and charge, where 1/; and 1/;* represent all four components of each vector and where the operators a = aJ + allj + azk and ao are those defined in Eqs. (2.6.55). The Lagrange-Euler equations may be obtained in the usual manner after substituting for 1/;*, 1/; in terms of 1/;i, 1/;~ , 1/;:, 1/;:, 1/;1, 1/;2, 1/;3, 1/;4 in
336
Fields and the Variational Principle
[cH.3
Eq. (3.4.25) and performing the- necessary operations required by the operators a . For instance, the equation
~ (~) + ~ (~) + s. ( aL) + ~at (aL) ax aif;:% ay aif;:. az aif;:. aif;:,
_ aL _ 0 aif;~ -
results in
(3.4.26) which is one term of the Dirac equations (2.6.57) . However, · we can obtain the same result more easily by considering that only tivo field variables, if; and if;* , are involved and performing the necessary partial derivatives formally as though they were simple functions instead of vectors in spin space. For instance, the Lagrange-Euler equation
a (aL) ax aif;=
a (aL) · a (aL) + aya (aL) aif;: + az aif;= + at aif;;
ot. - aif;*
= 0
corresponds to the whole of Eq. (2.6.57),
c [ aomcif;
+a.
o
grad if;
+ ~ Aif;) + (ic ~~ - e~if;) ]
one part of which is Eq. (3.4.26). spin vector if;* is
c[ if;*mcao
= 0 (3.4.27)
The corresponding equation for the
+ ( - ~ grad if;* + ~ Aif;*) . a + ( - ~ atl* - e~if;*) ]
= 0
The energy is again the (4,4) component of the tensor lID ;
* aL aL W44 = if;t aif;'i + if;t aif;t - L = H = mc 2(if;*aoif;) + eA· (if;*aif;) -
+
ec(if;*~if;)
he . 2i [if;*a . grad if; - grad if;* . aif;]
(3.4.28)
and the" field intensity" vector is S = iW4 1
+ jW + kW 42
43
if;*a ~~]
(3.4.29)
[(grad if;*)if; - if;*(grad if;)]
(3.4 .30)
=
~ [ate* aif; -
whereas the" field momentum" vector is P = iW 14
+ jW + kW 24
34
=
;i
Problems
CH.3]
337
Neither of these vectors is proportional to the current density vector
J = cey;*al/; given in Eq. (2.6.59). As a matter of fact, since L is only a linear function of the time derivatives of the fields, the canonical moments are proportional to the fields themselves and the whole formalism of the Hamilton canonical equations must be modified in the manner described on page 315. What is more important is the expression for the Lagrangian, which is minimized, and the expression for the energy and momentum densities.
Problems for Chapter 3 . 3.1 a. Show that a generating function S'(q,P,t) may be defined as follows : S' = S(q,Q,t) + PQ p = as' jaq; Q = aS jap ; K = H + (as' jat) and that
b. Show that S' = qP is the identity transformation. c. Show under an infinitesimal transformation S' = qP
+ eT(q,P)
= qp
+ eT(q,p ) ; e« 1
that P - p = -e(aTjaq) ; Q - q = e(aT jap) d. Show that !J.f = f(P,Q) - f(p,q) is given by !J.f = e[f,T] (where (f,T] is the Poisson bracket), and therefore show that the quantity T is a constant of the motion if the corresponding transformation leaves the Hamiltonian invariant. e. Show that the proper T for an infinitesimal rotation about the z axis is (r X p). = M z3.2 Show that the Lagrange equations are not changed when a total time derivative is added to the Lagrangian. Hence show that the Lagrangian for a nonrelativistic particle moving in an electromagnetic field may be written £
= imv 2
-
etp - (ejc)[(aAjat)
+v
»
(VA)]· r
where vA is a dyadic (the gradient operating on A only). the corresponding Hamiltonian is x = (im)lp
+ (ejc)(VA) . rj2 + (ejc)r· (aA jat) + etp
Show that
(Richards)
338
Fields and the Variational Principle
[CR. 3
3.3 Show that the Lagrange-Euler equation for a generalized orthogonal coordinate system ~1, ~2 , ~3 is
Employing (Vy;)2 as the Lagrangian density, derive the result
3.4
Show that the tensor of the third rank (Tp.p = W uv ; see page 319)
satisfies the continuity equation
only if TJ'v is symmetric. Show that M 4j k is just the angular momentum density and that the continuity equation yields, upon integration, the principle of the conservation of angular momentum. 3.5 a. Show in an infinitesimal Lorentz transformation
b. From the fact that the Lagrangian density is an invariant against Lorentz transformation, show, in the electromagnetic case, where
L =
-i
2: [(~~:) - (~~~)r 2:
that
va
where Show also that
r.,
=
-n: +
wvur vu = 0
2:J' a~J' [a(a:~axJ') Au] r vu = r uv
3.6 When TJ'v is not symmetric, it is always possible to find a symmetric tensor SJ'v which is symmetric and which has all the physical properties of T,.•.
CR.
. Problems
3] 0..
339
Show that S". must satisfy the conditions S". = S.,,;
L(iJ /iJx")S,,. = 0 ; JS4. dV = JT 4• dV "
b. Show that S". must have the form
L(iJ~;:.)
S". = T". -
A
where
GA". = -G"A'
and
T". - T." =
LiJ~A
[GA". - GA''']
A
c. Using the results of Prob. 3.5, part b, show that GA". - GA'" Hence show that
iJL
iJL
~ H A". = iJ(iJA,, /iJx A) A. - iJ(iJA./iJx A) A" G'''A = i(H'''A
+ H"A' + H A".)
d. Evaluate S". for the electromagnetic case. 3.7 Show that the homogeneous integral equation if;(x) = A
f
K(xlxo)if;(xo) dxo
follows from the variational requirement 5
f
if;(x) [if;(x) - A
lab K(xlxo)if;(xo) dxo1dx
= 0
K(xlxo) = K(xolx)
if
Show that, if K(xlxo) ~ K(xolx), 5
lab t/t(x) [if;(X)
- A
lab K(xlxo)if;(xo) dXo] dx = 0
where -Ji satisfies the integral equation -Ji(x)
=
A
lab K(xolx)-Ji(xo) dx«
3.8 The equation of motion of a membrane stretched over one end of an airtight vessel is given in Prob. 2.1 as (1/c 2)(iJ 21/;/iJt2) = V 2if; - (pc2/VT)fif; dS
Determine the corresponding Lagrangian and Hamiltonian densities. 3.9
The equation for the damped vibration of a string is of the form (iJ 21/;/iJt2)
+ 2k(iJif;/iJt)
= C2(iJ 2if;/ iJx 2)
Show that the proper Lagrangian density is L =
I (iJ-Ji/iJt) (iJ1/;/iJt) + k[if;(iJ-Ji/iJt)
- -Ji(iJif;/iJt)] - c2(iJ-Ji /iJx) (iJy; /iJx) I
340
Fields and the Variational Principle
[CH. 3
and determine the equation satisfied by /to Determine the momenta canonical to if; and /t, and find the Hamiltonian density. Discuss the physical significance of the results. 3.10 The steady-state transport equation for anisotropic scattering from very heavy scatterers (see Sees. 2.4 and 12.2) may be put into the form cos 6 (oj/at) = -j(t,6) + (Kj47r)fw(a - ao)f(t,6 0) dfl o where K is a constant, unit vectors a and ao are two directions given by spherical angles 6, I{J and 60 , l{Jo, respectively, and dfl o is the differential solid angle at ao. Show that this equation may be obtained from the variational principle
s 10h dt
J dfl!(t,6) [cos 6 (oj/at) + j - (K/47r)
Show that the equation satisfied by - cos 6 (a!/at) = -!(t,6)
Jw(a -
ao)j(t,60 ) dfl o] = 0
! is
+ (K /47r)Jw(ao
- a)!(t,6 0) dfl o
Interpret these results. 3.11 The diffusion of charged particles under the influence of an external field E is given by ac/at = a2V2c + b(Vc . E) where the assumptions involved are given in Prob. 2.5. Show that the corresponding variational principle is ofJdVdtc[(ac/at ) - a2V2 c - b(Vc ·E)] = 0 Find the equation for c and interpret. 3.12 A pair of integral equations which occur in the theory of the deuteron may be written
10" Go(r/ro)[j(ro)u + g(ro)w] dr« = A 10" G2(rlro)[g(ro)u + h(ro)w] dro
u(r) = A w(r)
where both Go and G2 are symmetric.
Show that the variational integral
IS
10 .. [u2j + 2uwg + w2h] dr - A 10" fo" {[f(r)u(r) + g(r)w(r)]Go(r/ro)[f(ro)u(ro) + g(ro)w(ro)] + [g(r)u + h(r)w]G 2(rlro)[g(ro)u(ro) + h(ro)w(ro)II dr dro 3.13 The equations describing the coupling between mechanical motion and heat conduction in a sound wave are aT/at = a(ap/at) + {jV2T ; a2p /at 2 = -yV2 p + EV2T
CH.3]
Tabulation of Variational Method
where the constants a, f3, -y, and E are given in Prob. 2.3. these equations follow from the variational integral:
341 Show that
IfdV dt{EV2T[(oT jot) - a(opjot) - f3V2T] - a(o"p jot)[(o2pjQt2) - -yV2p - EV2TlI
Show that T, "p satisfy the time-reversed equations if appropriate initial conditions are employed. 3.14 An infinite piezoelectric medium has the properties relating electric field E, polarization P, stress, and strain given in Prob. 2.2. If the z, y, z axes are placed along the three major axes of the crystal, then the coupling relations between E, P, the stress dyadic @5, and the strain dyadic st may be expressed in three sets of three equations, three typical ones being Tn = A;8n + A~Syy A;Szz; etc. T:I:Y = TY:I: = A:I:YS:I:Y + U:I:YP etc. E, = «P, + U:I:yS:I:Y ; etc.
+
Z ;
where the A'S are the elements of the elastic modulus tetradic reduced to its principal axes, the K'S are the reciprocals of the dielectric susceptibilities along the three axes, and the u's are the elements of a nondiagonal "triadic" (changing a vector into a dyadic and vice versa) representing the coupling between strain and polarization. Combine these equations with Maxwell's equations and with the ones for elastic motion for the special case of a transverse wave moving in the z direction, elastic displacement being in the y direction. Show that the result is a pair of coupled wave equations corresponding to two possible shearelectric waves, one with velocity somewhat less than that of pure shear waves (value if the u's were zero), the other with velocity somewhat greater than that of light in the medium. Compute the Lagrange density. For plane shear waves in the z direction (E in y direction) compute the momentum density and the stress-energy dyadic. What is the relative proportion of energy carried by electric to that carried by elastic field for the slow wave? For the fast wave?
Tabulation of Variational Method The Lagrange density L is a function of field variables Y;i(i = 1, 2, ••. , n) and their gradients Y;i8 = (OY;;/O~8) (~l, ~2, ~3 are space coordinates, ~4 = t) . Sometimes L also depends explicitly on the es (through potential functions or charge-current densities, for instance). total Lagrangian integral
.c =
fo bl . .. a1
lb. L dh 04
d~2 d~3 d~4
The
(3.1.1)
342
[cH.3
Fields and the Variational Principle
is an invariant. The requirement that .e be minimum or maximum (i.e., that the first-order variation be zero) corresponds to the LagrangeEuler equations (3.4.1) for the field variables. If L is a quadratic function of the .pi4'S then the canonical momentum density Pi = iJLjiJ.pi4 is a linear function of .pi4. If L is a linear function of the .pi4'S, then Pi and the Hamiltonian are independent of .pi4. The stress-energy tensor sm, having components
contains most of the other important physical properties of the field. For instance, the (4,4) component is the energy density, n
W 44 = H =
L Pi.pi4 -
L
i=1
If Pi depends on .pH, then the terms .pi4 can be eliminated from W 44, obtaining the Hamiltonian density H, a function of the p;'s, the .p;,s
and their space derivatives. In this case the equations of motion may also be written in canonical form, 3
. _
_ iJH. . iJPi _ \ ' iJ (iJH) iJH .p; - .pi4 - iJPi' Pi = at iJ~s iJ.pis - iJ.pi .=1
4
(3.4.2)
which equations only apply when L contains a quadratic function of the .p;4'S. If L contains a linear function of the .pi4'S, H is independent of the P's (see page 315). The field intensity vector S and the field momentum vector P are defined as follows : (3.4.4) 3
P =
L
asW s4 =
.=1
n
L
(grad .pi)
(:~)
(3.4.5)
i=1
3
The rest of
sm
is the three-dyadic U =
L a, W T,' = 1
TSa S,
called the stress
CH.3]
dyadic.
Tabulation of Variational Method
343
The elements of jill satisfy the divergence equations 4
\ ' aWm 8
Lt ar: 8=1 _
aL
-
(3.4 .3)
a~m
where aL/a~m is the rate of change of L with ~m due to the explicit variation of L with ~m (through potentials or charge-currents, etc.) . When L does not depend explicitly on the es then aL/a~ is zero. In terms of the field intensity and momentum this then becomes V· S
+ (aH /at)
= OJ
(U· V)
+ (ap/at)
when L does not depend explicitly on the es. tum density about the origin is
= 0
The field angular momen-
n
M = r X P =
\' et.
Lt a1/;i4 (r X
grad Vti)
i=l
Flexible String or Membrane Field variable 1/; is transverse displacement. Parameters ~8 are x and t for string; x, y, and t for membrane. Lagrange density : L = ip
[(~~y - c
2
grad! 1/;
l
c2 = ;.
Lagrange-Euler equa tion: c2V21/; - (a 2y;jat2 ) = 0 (scalar wave equation) Canonical momentum density: p = p(ay;jat) . Hamiltonian: H = (1/2p)p2 + iT grad" 1/;. Field intensity: S = - T(aif;/at) grad 1/;. Field momentum: P = p(aif;/at) grad if; = -(1 /c 2)S. Compressible, Nonviscous Fluid Field variable if; is velocity potential ; Field velocity Parameters
~8
=
grad 1/;;
excess pressure = - p(a1/;/ at)
are x, y, z and t.
Lagrange density: L = - ip [ (grad 1/;) 2 Lagrange-Euler equation : V21/; -
~ ~:~ =
Yl
~ (~~
c2 =
p;'Y .
0 (scalar wave equation) .
Canonical momentum density; p = (p/c 2 ) if; = - (excess pressurer /c" , Hamiltonian: H = i(1 /pe2)p2 + ip(grad 1/;)2. Field intensity: S = -p(aif;/at) grad if; = (excess pressure) (fluid velocity) .
344
Fields and the Variational Principle
Field momentum: P = (pjc 2)(aif;jat) grad
e
[cH.3
= -(ljc 2)S.
Diffusion Equation
Field variables are temperature or concentration density if; and its "conjugate" if;*. Parameters ~8 are x, y , z, and t. Lagrange density: L = - (grad if;) . (grad if;*) -
-!a 2 (if;* ~~
- if; ate*}
Lagrange-Euler equation for if;, V 2if; = a 2(aif;j at ) (diffusion equation) . Canonical momentum densities: p = --!a 2if;* ; p* = -!a 2if;. . Energy density: U = W 4 4 = (grad if;) . (grad if;*) . Field intensity : S = -~*(grad if;) - (grad if;*)~. Field momentum : P = -!a 2[(grad if;*)if; - if;*(grad if;)] Schroedinger Equation
Field variables are the wave function if; and its conjugate if;*. is the probability density for presence of the particle. Parameters ~8 are x, y, z, and t. Lagrange density
if;*if;
h2 h ( aif; aif;* ) L = - 2m (grad if;*) • (grad if;) - 2i if;* at - lit if; - if;*Vif; . V(X,y,z) is potential energy of particle. Lagrange-Euler equation for if;,
-(h 2j 2m )V 2'"
+
Vif;
=
ih(aif; jat) (Schroedinger equation) .
Canonical momentum densities: p = - (h j2i) if;*; p* = (h j2i)if;. Energy density : U = W 44 = (h 2 j2m)(grad if;*) . (grad if;) + if;*Vif;. Field intensity : S = -(h 2j2m)[(ar jat) grad if; + grad if;*(aif;jat)]. Field momentum : P = - (h j2i) [if;*(grad if;) - (grad if;*)if;]. Current density: J = (ehj2im)[if;*(grad if;) - (grad if;*)if;] where e, m are the charge and mass of the particle. Klein-Gordon Equation
Field variables are the wave function if; and its conjugate if;*. Charge density for particle is particle mass. Parameters
~8
2i~C2 [ate* Yt -
if;*
:t].
where m is the
are x, y, z, and t.
Lagrange density L
=
_.!!!.... [(grad if;*) . (grad if;) _12 (aif;*) 2m c at
+ (mt) if;*if;]
(aif;) at
for the field-free case.
Tabulation of Variational Method
CH.3]
Lagrange-Euler equation for if;, V2if; -
~ (~2t~)
345
(~cy if;
=
(Klein-
Gordon equation) . Canonical momentum densities p =
2~C2 (iJtt*);
2 h (aif;) p* = 2mc2 at .
Hamiltonian : H
=
(2mc2/h 2)p*p
Field intensity : S
=
+ (h2/2m)(grad if;*) • (grad if;) + (mc2/2)if;*if;. -
:~ [ate*
(grad if;)
+ (grad if;*) ~t
1
Field momentum : P = -(1/c 2)S. Current density : J = (eh/2im)[if;*(grad if;) - (grad if;*)if;] where e is the particle charge. Elastic Wave Equation
Field variables if;n are the components of the displacement vector s. Parameters ~s are the coordinates x, y, z, and t. Lagrange density: L = i pS2 - i~:~, where @5 = i(Vs + sV) is the strain dyadic and ~ = XI~13 + 2JL@5 is the stress dyadic for isotropic solids. Lagrange-Euler equation: p(a2s /at 2) = (X + JL) grad (div s) + JLV2S. Canonical momentum density: p = peas/at) . Hamiltonian density : H = W u = (1/2p)p2 + il~· @51. Field intensity: S = - (as/at) • ~. Field momentum: P = p(Vs) . (as/at) . For a nonisotropic solid the stress dyadic ~ = 1:@5, where 1 is a tetradic with coefficients gmnrs which are arbitrary except for the general symmetry requirements gmnrs = gnmrs = gmnsr = grsmn . The Lagrange-Euler equation is then that is;
PSn
=
I
gnmrs
mrs
iJ
a2s.
Xm
aX r
The Hamiltonian density is
H = W 44 = (1/2p)p2
+
(vs) :1: (Vs);
p = peas/at)
The new expressions for S, P, etc ., can be obtained by substituting the new expression for ~ into the equation for S, P given above. Electromagnetic Equations
Field variables are the components of the vector potential A and the scalar potential rp. For simplicity we shall choose the gauge for which rp = 0, so that curIA = B = JLH, aA/at = -cE = -(cD/E), and
346
Fields and the Variational Principle
[CH. 3
div (aA/ at) - (4'll'pc/ E), where p is the density of free charge. eters are x, y, z, and t. .Lagrange density: L =
8;C2Iaa~\2
-
~,u I curl AI2 -+- ~ ]
Param-
. A, where] is
the current density. Lagrange-Euler equation: curl (curl A) -+- (,uE/ C2)(a2A /at 2) = (471",u/c)J Canonical momentum density: p = -(D/471"c). Hamiltonian density: H = W 44 = (27rC 2/E)p2 (1/87r,u)I curl AI2 - (l /c)]· A.
+
Field intensity: S = (c/47r)(E X H). Field momentum: P = -(E/47r)(VA) . (aA/at).
Dirac Equation
«
Field variables and Vtn (n = 1, 2, 3, 4). Probability density for presence of electron is VtiVtl VtN/2 -+- VtN3 VtN4 = 'F*'F. Parameters are x, y , z, t. Wave functions 'F = 'LenVtn and 'F* = 'LtI':e: where the e's are unit vectors in spin space. Operators az , ay, a., ao operate on the e's in a manner given in Eqs. (2.6.53) . Lagrange density : L =
;~ [(grad
+
+
'F*) . a'F - 'F*a ' grad 'F)]
+ ;i [(a:*)
'F -
'F* (a:)] - e'F*(a ' A)'F -+- ec'F*Ip'F - mc2'F*ao'F where A and (/) are the electromagnetic potentials and m the particle mass. Lagrange-Euler equations :
aomc'V -+- a' mc'F*ao -+- ( -
(~grad 'F -+-~A'F) -+- (~aa~
- elp'F) = 0
~ grad 'F* -+- ~A'F*)' a a'F* -+- elp'F*) l e w &
(!:-
=
0
Canonical momentum density : p = - (h/2i)'F* ; p* = (h/2i) 'F.
~ ['F*a ' (grad 'F) - (grad 'F*) . a'F] 2'F*a mc o'F.
Hamiltonian density: H =
e'F*a' A'F - ec'F*Ip'F
+
Field intensity: S = (hc/2i)[(a'F* /at)a'F - 'F*a(a'F/ao] . Field momentum: P = (h/2i)[(grad 'F*)'F - 'F*(grad 'F)]. Current density : ] = ce'F*a'F, where e is the particle charge.
-+-
CR. 3)
Bibliography
347
Bibliography Few books cover the central subject of this chapter in any satisfactory detail, though several deal with some aspects of the subject. The references dealing with calculus of variations in general : Bliss, G. A.: "Calculus of Variations," Open Court, La Salle, 1925. Courant, R., and D . Hilbert : "Methoden der mathematischen Physik," Vol. I , pp. 165 et seq., Springer, Berlin, 1937. Rayleigh, J. W. S.: "The Theory of Sound," pp. 109 et seq., Macmillan & Co., Ltd. , London, 1896, reprinted Dover, New York, 1945. Books on the transformation theory of dynamics, including a discussion of Hamilton's principle : Born, M.: " Mechanics of the Atom ," G. Bell, London, 1927. Corben, H. C., and P. Stehle: "Classical Mechanics," Chaps. 10 to 15, Wiley, New York, 1950. Goldstein, H.: "Classical Mechanics," Chaps. 7, 8 and 9, Addison-Wesley , Cambridge , 1950. Lanczos, C.: "The Variational Principles of Dynamics," University of Toronto Press, 1949. Webster, A. G. : "Dynamics," Chaps. 4 and 9, Stechert, New York, 1922. Whittaker, E. T .:" Analytic Dynamics," Chaps. 9 to 12, Cambridge, New York, 1937. Works discussing appli cation of Hamilton's principle to fields from various points of view: Fermi, E. : Quantum Theory of Radiation, Rev. Modern Phys., 4, 87 (1932). Goldstein , H. : "Classical Mechanics," Chap. 11, Addison-Wesley , Cambridge, 1950. Heitler, W.: "Quantum Theory of Radiation," Oxford, New York, 1936. Landau, L. D., and E. Lifschitz: "Classical Theory of Fields, " Addison-Wesley, Cambridge, 1951. Pauli, W. : Relativistic Field Theories of Elementary Particles, Rev. Modern Phys., 13, 203 (1941). Schiff, L. 1.: "Quantum Mechanics ," Chaps. 13 and 14, McGraw-Hill, New York, 1949. Wentzel, G. : "Quantum Theory of Fields, " Interscience, New York, 1949. Weyl, H .: "Theory of Groups and Quantum Mechanics," Chap. 2, Methuen, London, 1931.
CHAPTER
4
Functions of a Complex Variable
The past two chapters contain a discussion of the connection between physical phenomena and the partial differential equations for fields to represent these phenomena. The next several chapters must be devoted to a discussion of the general mathematical properties of the differential equations and their solutions. We have begun to familiarize ourselves with the possible physical interpretations of field quantities: tensors, divergences, line integrals, and the like. Now we must learn to recognize the different types of equations and their solutions. We must become familiar with the sort of tests which can be applied to tell how a given function will depend on its argument : where it goes to infinity or zero, where it can be integrated and differentiated, and so on. And we must be able to tell what sort of functions will be solutions of given differential equations, how the "singularities" of the equation are related to the singularities of the solutions, and the like. The general properties of functions will be treated in the present chapter, and the interrelation between equations and solutions in the next chapter. To be more specific, we shall devote this chapter to a discussion of functions of the complex variable z = x + iy, where i is the square root of (-1). We have already (pages 73 and 74) shown that such a variable can be represented as a two-dimensional vector, with x the x component and y the y component of the vector; and we have indicated that z can also be considered as an operator, which rotates any other complex number vector by an angle tan- l (y /x) and changes its length by a factor V x 2 + y2. In this chapter we shall continually use the twodimensional vector representation and occasionally use the concept of vector operator. It could be asked why it is necessary to study complex numbers when many parts of physics are interested only in real solutions. One might expect that a study of the real functions of a real variable going from - 00 to + 00 would suffice to obtain knowledge of the physically .int eresting solutions in many cases. The answer is that it is desirable to extend our study to complex values of the variables and the solutions for reasons of completeness and convenience. 348
§4.1]
Complex Numbers and Variables
349
The set of real numbers is not even a sufficient basis for reproduction of the roots of algebraic equations. On the other hand all roots of all algebraic equations can be expressed as complex numbers. In addition, knowledge of the behavior of a function fez) for all complex values of z gives us a much more complete picture of the principal properties of f (even its properties for z real) then does a knowledge of its behavior for only real values of z. The location, on the complex plane for z, of the zeros and infinities of f [i .e., the position of the roots of f = 0 and (l /f) = 0] will tell us a great deal about the behavior of f for all values of z. Often an integral of fez) over real values of z (along the real axis) may be modified into an integral along some elementary path for z in the complex plane, thereby considerably simplifying the integration. It is usually convenient to consider the solution of an equation as complex, to deal with it as a complex number up until the final answer is to be compared with a measured value , and only then to consider the real or the imaginary part of the solution as corresponding to the actual physical problem. But the most important reason for the study of complex functions is the insight we shall obtain into the general properties of fun ctions. For example, the various types of singularities a function might have may be classified. In general these singularities will be related to physical singularities, such as those caused by sources, point electric charges, etc. It turns out to be possible, simply from knowledge of the singularities of a function, to specify the function completely. The corresponding statement in electrostatics is that, once the size and distribution of all the electric charges are given, the electric field at any point can be determined . Because of the close connection between electrostatics and complex variables, it is not surprising that our study will reveal, in addition, a method for generating solutions for the Laplace equation (i.e., will locate possible sets of equipotential lines). We recall from Chap. I that these equipotentials and their normals form the basis of an orthogonal coordinate system. We may therefore say that a method can be developed for generating new coordinate systems, systems which are appropriate to the geometry of the problem at hand.
4.1 Complex Numbers and Variables Perhaps the first use that a student of physics makes of complex numbers is in the expression Ae-i"'t, for a vector rotating with constant angular velocity w, where A gives the length of the vector. This representation is useful also in simple harmonic motion, for its real part is A cos wt while its imaginary part is A sin wt. We have already used this fact several times in the preceding chapters.
350
Functions of a Complex Variable
[cH.4
The connection between vectors and complex numbers is obtained by making a proper definition of the symbol i . Here i is to be considered to be an operator which, when applied to a vector, rotates the vector through 90° counterclockwise. The operator i 2, meaning the application of the
operator twice in succession, results in the rotation of the vector through 180°. Since this yields a vector which is antiparallel to the original vector, we find (4.1.1) i 2 = -1 in agreement with the more usual definition of i. The symbol of i 3 meaning the application of i three times results in a rotation of the original vector through 270° or -900 so that i 3 = - i. Similarly i 4 = 1. We may now differentiate between real and imaginary numbers. We shall plot all real numbers as vectors in the x direction. Thus, we multiply a real number by i to obtain a vector directed along the y axis . Vectors in the y direction are called imaginary numbers. Any vector! f may, of course, be decomposed into its two components u and v along the x and y axis so that we may write f=u+ iv
(4.1.2)
establishing a connection between complex numbers and vectors. The magnitude ' of i. written If I, is equal to the absolute value of the complex 2 number + iv = + v2, while the direction of t. the angle Ip it makes with the x axis, is just the phase, tan- l (v/u), of u + iv . This angle is sometimes called the argument of f. The conjugate of u + iv
u
vu
J=
u - iv
may be obtained from vector f by reflecting it in the x axis. The Exponential Rotation Operator. To obtain the operator for a finite rotation of angle 0, it is necessary only to consider the results for an infinitesimal rotation dO. The operator for the infinitesimal rotation must yield the original vector f plus a vector at right angles to f of magnitude f dO. The new vector is f + if dO = (1 + i dO)f. Thus the change in f , df is df
= ifdO
This equation may be integrated to yield f after a rotation through 0 radians. Let the original (0 = 0) value of f be fo. Then f for 0 = 0 radians is [» = ei 6fo (4.1.3) The operator rotating a vector through 0 radians is thus ei 8 (see page 74). This operator when applied to a vector along the real axis, say of unit length, yields a vector in the direction o. Decomposing this new vector into components and expressing the vector in complex notation, 1
We shall not use the customary boldface type for these complex vectors.
§4.1]
Complex Numbers and Variables
351
we obtain De M oivre's relation ei 8 ..., cos 0 + i sin 0 mentioned on page 74. This agrees with the starting assumption that i yields a rotation of 90° (as may be seen by putting 0 = 7r/2) . A unit vector rotating counterclockwise with angular velocity w is simply e i",t where the unit being operated on is understood, as is customary. Any vector f may now be expressed in terms of its absolute magnitude IfI and the required operator to rotate it, from the z axis to its actual direction, to wit, The angle", is called the phase angle, or argument, of f . Vectors and Complex Numbers. Having established the one-to-one relation between complex numbers and vectors, we shall now explore the relations between various possible combinations of complex numbers and the corresponding combinations of vectors. The law of addition for two vectors, the parallelogram law, is the same as that for the addition of: two complex numbers. However, when two complex numbers are multiplied together, the result, when expressed in vector notation, involves both the scalar and vector product. Consider Jg where f = u + iv and g = s + it: Jg
=
(us
In vector language : Jg = f • g
+ vt) + i(ut + ilf
X gl
vs) (4.1.4)
Thus if two vectors are orthogonal, the real part of their product is zero, while if they are parallel, the imaginary part of their product is zero. It should be noticed that rule (4.1.4) is the same in two dimensions as Eq. (1.6.30), for quaternion multiplication, is for three. This should occasion no surprise, since Hamilton originally wrote down the quaternion algebra in an attempt to generalize the method of complex variables to three dimensions and to three-dimensional vectors. The differential properties of a vector field involve the operator V . Since we are limiting ourselves to two dimensions, the z, y plane, the operator V can be written as V =
i. + i~ ay
ax
(4.1.5)
If now we should operate with Von a vector g, then from Eq . (4.1.4) one obtains the relation, (4.1.6) Vg = div g + ilcurl gl
so that V immediately gives both the divergence and curl of a vector. N ate that V operating on a real function (g along x axis) yields directly from (4.1.6) Vg = ag _ i ag
ax
ay
352
Functions of a Complex Variable
[cH.4
as it should. We thus see that one great advantage of the complex notation is that it affords a means by which one may condense several vector operations into just one. .Some further condensation is possible if we introduce two new variables in place of x and y: ~ -.
z = z
+ iy;
z = x - iy;
x = j(z
+ z);
y = -ji(z - z)
(4.1.7)
where z is just the radius vector to the point (x,y). There is usually some confusion in the reader's mind at this point as to how it is possible to consider z and z as independent variables (no such confusion seems to come up for variables x - y, x + y) for it is often stated that, if z is known, so is z. This is, however, not the case. Given a vector z as a line drawn from the origin to some point, z is not yet determined, for, in addition, the direction of the x axis must be given . Vice versa, if both z and z are known, the direction of the x axis is determined as a line bisecting the angle between the vectors z and z, and tb-en x and y can be found . In terms of these variables and using Eq. (4.1.5), 2 a _ 2 ax a az az ax
+ 2 ay a az ay
_ . lik . 2 a - V, 1 ewise V = az
(4.1.8)
The Two-dimensional Electrostatic Field. Suppose that we have an electrostatic field generated by line charges all perpendicular to the x, y plane. The electric vector E will everywhere lie in the x, y plane, and we need consider only two dimensions. Therefore the electric vector E can be represented by a complex number, say u - iv (the reason for the minus sign will be apparent shortly), where u and v are functions of x and y determined by the distribution of the line charges. We first look at that part of the x, y plane which is free from charge density. In these places Maxwell's equations (2.5.11) state that div E = 0;
curl E = 0
(4.1.9)
Referring to Eqs. (4.1.6) and~(4.1.8), we see that both of these requirements can be written down (in this two-dimensional case only) in the extremely simple form aE jaz = 0 This states that the vector E is not a function of z = x + iy, but only of z = x - iy. Contrariwise the conjugate vector E = u + iv is a function only of z and not of z. Since we usually shall deal with functions of z, we may as well deal with E, from which we can find E, the electric-vector. We have just shown that E is a function of z and riot of Z. By using the equations for E analogous to Eq. (4.1.9) or by writing out the equation 2(aEjaz) = V(u + iv) = 0 in terms of the derivatives
§4.1]
Complex Numbers and Variables
353
with respect to x and y and collecting real and imaginary parts, we find the interesting pair of cross connections between u and v: au /ax = av/ay;
au /ay = - (av/ax)
(4.1.10)
which are called the Cauchy-Riemann conditions. We have derived them here for an electric vector (two-dimensional) in a region free from charge and current, but the way we have obtained them shows that they apply for any complex function f = u + iv which is a function of z only (not z). Any such function, with real and imaginary parts satisfying Eqs. (4.1.10), is called an analytic function of the complex variable z = x + iy. Therefore any analytic function of z can represent a two-dimensional electrostatic field. Such a function may be created by taking any wellbehaved function of a real variable and making it a function of z = x + iy instead [for instance, sin (x + i y ), l /[(x + iy)2 + a 2], log (x + iy) are all analytic functions for all values of z where the functions do not become infinite]. In a region free from charge current, an electric potential V exists such that E = VV = (av/ax) + i(aV/ay), where V is a function of x and y . We may generalize somewhat and allow V also to become a complex function (its real part or imaginary part can be the actual potential). Then we can write E = 2aV/ ai , and since aE/az = 0, we have . (a 2V) _ a 2V a 2v _ (4.1.11) 4 az ai - ax2 + ay2 - 0 which is the Laplace equation for two dimensions. Naturally both real and imaginary parts of V are separately solutions of Laplace's equation, and in fact , combin ing Eqs. (4.1.10) we see that the real or imaginary parts of any analytic function are solutions of Laplace's equation in two dimensions. Therefore either an analytic function can be used to generate an electric vector, or else its real or imaginary parts can be used as a potential function. Contour Integrals.' Integration of complex functions is a natural extension of the process of integration of real functions. The integrand is some analytic function f(z) ; the variable of integration is, of course, z. But since z can move over the complex plane instead of just along the real axis, we shall have to specify along what line the integration is to be performed. This line of integration is called a contour , and if the contour is closed on itself, the integral is called a contour integral and is denoted !ff(z) dz = !ffe i ,!, de, where ds is the magnitude of dz and 'P is its phase. In this two-dimensional generalization of an integral we can no longer give enough details by writing in the upper and lower limits; we must
354
Functions of a Complex Variable
[cH.4
describe, or draw out, the contour followed, as in Fig. 4.1. The expression is analogous to the two-dimensional form of the net outflow integral of Sec. 1.2 and also to the net circulation integral of that same section. As a matter of fact the complex contour integral is a compacted combination of both, as we can see by writing out the contour integral of the electric vector, and using Eq. (4.1.4) .'fE dz = .'fE . ds
+ i.'fIE X ds] =
.'fEt de
+ i.'fEn de
(4.1.12)
where E, is the component of E along ds and En the component normal to ds. Therefore the real part of the contour integral of E is the net circulation integral of E around the y contour, and the imaginary part is B the net outflow integral of E over the sides of a cylinder with axis "perpendicular to the x, y plane, of cross section equal to the contour. (Since x the field in this case is all parallel to the x, y plane, the total net outflow integral over the cylinder is equal to .'fEn de times the length of the Fig. 4.1 Contour integration in the cylinder.) complex plane. In the case of an electric vector in a region devoid of charge density, both outflow and circulation integrals are zero, so that for any contour in such regions .'fE dz
= 0
(4.1.13)
Thistequation is just Cauchy' s theorem, which states that, if f(z) is an analytic function of z at all points on and inside a closed contour, then .'ff(z) dz around this contour is zero. Therefore the two-dimensional electric vector can be represented by an analytic function at all points where there is no charge or current. By a specialization of Eq. (1.2.9) for the cylindrical surface we have set up on the contour, we see that, if the field E is due to a series of line charges each uniformly distributed along lines perpendicular to the x, y plane, the rth line having charge qr per unit length, then, from Eq. (4.1.12) .'fE dz
=
47l"i
1:' qr
(4.1.14) (
r
where the sum is taken over all the lines r which cut the x, y plane inside the contour.
Suppose that we take the case where there is only one line, of charge density ql, which cuts the x , y plane inside the contour at the point Zl = Xl + iYl. The electric field, represented by E, can then be split
Complex Numbers and Variables
§4.1]
355
into two parts: E. , due to the source ql inside the contour; and Eo, due to the sources outside. Elementary integration of the equations for electrostatic fields indicates that E. = (2qtlr)a T , where r2 = (x - Xl)2 + (y - Yl)2 = Iz - Zl!2 is the distance, in the x, y plane , from the source line and a, is a unit vector in the x, y plane pointing away from the source line. In complex notation E = (2/r) [cos", 8
+i
sin ",] = (2/r)e i p
where", is the angle between a, and the x axis.
Therefore (4.1.15)
since reip = z - Zl . Adding Eo to E. we cal}. write
E=
[f(z)]j(z -
Zl)
(4.1.16)
where fez) = Eo(z - Zl) + 2ql is an analytic function within and on the contour (why?) . Thus we have, for any analytic function f(z) , the formula, (4.1.17) which is a more general form of Cauchy's theorem. Therefore Cauchy's theorem is just a restatement, in the notation of analytic functions, of Gauss' theorem for electrostatics. Similarly one can use a function F! of z to represent the magnetic field caused by line currents along lines perpendicular to the x, y plane. A current I along a line cutting the x, y plane at a point Zo inside the contour will produce a field H = 2(1 X aT) /r, which can be represented by the function F! = 2I/i(z - zo) . If there are a number of line currents IT, then, according to Eq. (1.2.11) (4.1.18)
where the summation is over all currents cutting inside the contour. Here we use the real part of the contour integral, but if we substitute for F! its form in terms of (z - zo), we eventually obtain Cauchy's theorem again. Returning tp Fig . 4.1, we note that the integral from A to B along a contour (not closed) gives
JAB Edz =
L.BEtds + i JAB Ends = W = V + iU
(4.1.19)
The real part V of this integral is just the electrostatic potential difference between points A and B . The imaginary part U measures the number of lines of force which cut across the contour between A and B.
356
Functions of a Complex Variable
[cH.4
We note that the family of curves U = constant is orthogonal to the family V = constant, so they can be used to set up orthogonal, twodimensional coordinate systems. If a conducting cylinder is present, with axis perpendicular to the x, y plane, then the field must adjust itself so that it cuts the x, y plane along one of the equipotential lines V = constant. The lines of force are the orthogonal lines U = constant, and the surface charge density on the conductor, per unit length of cylinder, inside the region limited by the points A and B is U(B) - U(A). The function U was called the flow function in Chap. 1 (see page 155). In this section we have related complex variables and electrostatics and have given as examples some electrostatic interpretations of some familiar theorems in function theory. In the remainder of the chapter, we shall develop a more rigorous theory but shall use electrostatic interpretations to give some intuitive meaning to the theorems in much the same way as was done above for the Cauchy theorem and the integral formula.
4.2 Analytic Functions The electrostatic analogue has furnished us with heuristic derivations of some of the fundamental theorems in function theory. In particular we have seen that analytic functions form a restricted class to which most fun ctions do not belong. In this section we shall attempt to understand the nature of these limitations from the point of view of the geometer and analyst. This procedure will have the merit of furnishing more rigorous proofs of the aforementioned theorems. (Rigor in this particular subject is actually useful!) We have taken as a rough definition of an analytic function that it be a function of z only, not a function of both z and z. Thus the study of a function of a complex variable j(z) = u + iv, where u and v are real functions of x and y, is not so general as the study of functions of two variables, for u and v are related in a special way as given by the Cauchy-Riemann conditions (4.1.10). A more precise definition of an analytic function may be obtained by considering the behavior of the derivative of j(z) with respect to z at a point a. The meaning of the derivative is fairly obvious. The function j(z) is a vector. We ask how does this vector change, both in magnitude and direction, as z moves away from the point a in a direction specified by the vector dz. If j(z) is a function of z only (for example , Z2), one would expect the derivative (for example, 2z) to depend only upon the point at which it is evaluated. It is characteristic of a single-valued junction which is analytic at a point a that the derivative at point a is unique , i.e., independent of the direction dz along which the derivative is taken. No matter how
Analytic Functions
§4.2]
357
one moves away from point a, the rate at which f will change with z will be the same . This is not true for an arbitrary complex function u + iv, where u and v are any sort of functions of x and y. It is true only when u and v satisfy Eqs. (4.1.10). This definition of an analytic function has the advantage over the simpler one used earlier in being more precise . Using it, one may determine whether or not a function is analytic at a given point. We see once more how special an analytic function is. Most functions do not possess an "isotropic" derivative. It may be shown that the Cauchy-Riemann equations form a necessary condition only that a function have a unique derivative. To show this, consider the change in f(x) = u + iv as z changes from a to a + .1z: .1f f(a .1z =
+ .1z) .1z
f(a)
[(au/ax)
=
{I
_ (au/ax) + i.(av/ax) 1 + i(.1y/.1x)
+ i(av /ax)].1x + [(au/ay) + i(av /ay)] .1y
.1x + i .1y + i .1y ((av/a y) - i(au/a y))} .1x (au/ax) + i(av /ax)
The last equation shows that, except for exceptional circumstances, the derivatives df/dz = lim (.1f/.1z) will depend upon .1y/.1x, that is, upon Az-O
the direction of .1z. For an analytic function, however, there cannot be any dependence on .1y/.1x . This may be achieved only if av _ i au = au + i ay ay ax au av au ax = ay; ay = -
or
av ax av ax
(4.2.1)
which are the Cauchy-Riemann conditions. These conditions only are necessary. For sufficiency one must also include the requirement that the various derivatives involved be continuous at a. If this were not so, then au/ax, etc., would have different values depending upon how the derivative were (evaluat ed. This would again result in a nonunique first derivative. For example, suppose that
Then lim lim (au/ax) = 1. _0_0
However, lim lim (au/ax) = _0_0
o.
The Cauchy-Riemann conditions show that, if the real (or imaginary) part of a complex fun ction is given, then the imaginary (or real) part is, within an additive constant, determined. We shall later discuss special techniques for performing this determination. A simple example will
358
Functions of a Complex Variable
[CH. 4
suffice for the present. Let the real part be known, and let us try to determine the imaginary part. v
or
=
f =f f (-
v =
dv
(av dx ax Ie!
au dx ay
+ av d ay Y)
+ ax au dY)
(4.2.2)
so that, if u is known , v can be determined by performing the integration indicated in Eq. (4.2.2). For example, let u = In r = i In (x 2 + y2). Then v
=
f (-;2
dX
+ x r~Y) = tarr:' (y jx) + constant
so that In r + i tarr" (y jx) is an analytic function. It may "be more simply written In z. The special nature of an analytic function is further demonstrated by the fact that, if u and v satisfy the Cauchy-Riemann conditions, so do the pair au jax and avjax and similarly au jay and avjay. This seems to indicate that, if f(z) is analytic, so are all of its derivatives. We cannot yet prove this theorem, for we have not shown that these higher derivatives exist; this will be done in the following section. It is a useful theorem to remember, for it gives a convenient test for analyticity which involves merely testing for the existence of higher derivatives. Points at which functions are not analytic are called singularities. We have already encountered the singularity I j(z - a) which represents the electric field due to a point charge located at a. Point a is a singularity because the derivative of this function at z = a does not exist. Some functions are not analytic at any point, for example, IZ/2. This equals zz and is clearly not a function of z only. Another example/is zP/q, where p jq has been reduced to its lowest terms. For example, zi is not analytic at z = 0; its second derivative at z = 0 is infinite. Conformal Representation. Any analytic function f(z) = u + iv, z = x + iy, can be represented geometrically as a transformation of two-dimensional coordinates. One can imagine two complex planes, one to represent the chosen values of z and the other to represent the resulting values of f. Any line drawn on the z plane has a resulting line on the f plane. Of course, many other pairs of functions u, v of x and y can be used to define such a general coordinate transformation. The transformations represented by the real and imaginary parts of analytic functions, however, have several unique and useful characteristics. The most important and obvious of these characteristics is that the transformation is "angle-preserving" or conformal. When two lines drawn on the z plane cross, the corresponding lines on the f plane also cross. For a conformal transformation the angle
Analytic Functions
§4.2]
359
between the lines on the f plane, where they cross, is equal to the angle between the lines on the z plane, where they cross. As shown in Fig. 4.2, for instance, the lines cross at z = a on the z plane and at f(a) on the f plane. An elementary length along line number 1 can be represented by dZ l = Idzllei'Pl and an elementary length along line 2 is given by dZ 2 = Idz 2Ie i'P2. The corresponding elementary lengths on the f plane are dzl(df/dz) and dz 2(df/dz) . If the function f is analytic at z = a, then df/dz is independent of the direction of dz; that is, df/dz at z = a is equal to Idf/dzle ia, independent of the angle !p of dZ l or dz 2• Therefore the elementary length along line 1 in the f plane is Idzl(df/dz)!e i(a+'P!) and the elementary length along line 2 is Idz2(df/dz) lei\a+'P2). The direction of y
v
z Plane
f Plane
2
I
I
I~
dZ
2
~r2
?-'
x
--u
dZ2 f=f(o) Fig. 4.2 Conservation of angles in a conformal transformation. z=a
the two lines is rotated by the angle a from the lines on the z plane, but the angle between the lines, (a + !Pl) - (a + !P2) = !Pl - !P2, is the same as the angle between the corresponding lines on the z plane. Therefore the transformation represented by the analytic function fez) = u + iv is a conformal transform, preserving angles . Likewise, as can be proved by retracing tho/steps of this discussion, if a two-dimensional transform from x, y to u/v is conformal, it can be represented by a function f = u + iv which is an analytic function of z = x + iy, where u and v satisfy the Cauchy-Riemann equations (4.2.1) . We can likewise draw lines on the f plane and see where the corresponding lines fall on the z plane . For instance, the lines u = constant, v = constant, generate a rectangular coordinate system on the f plane; on the z plane ,t he lines u(x,y) = constant, v(x,y) = constant constitute an orthogonal curvilinear coordinate system (orthogonal because right angles stay right angles in the transformation). We note from the discussion above or by using the definitions given in Sec. 1.3 ,_ [see (Eq . 1.3.4)] that the scale factors for these coordinates are equal, hu
=
y(iJu/ax)2
+
(au/ay)2
= h; =
y(av/ax)2
+ (av/Oy)2 =
Idf/dzl
as long as the Cauchy-Riemann equations (4.2.1) are valid (i.e., as long as f is analytic). Therefore any infinitesimal figure plotted on the f plane is transformed into a similar figure on the z plane, with a possible
360
Functions of a Complex Variable
[CR. 4
change of position and size but with angles and proportions preserved, wherever f is analytic. This is an alternative definition of a conformal transformation. The simplest conformal transform is represented by the equation f = zei 6 + C, where the real angle 0 and the complex quantity C are constants. Here the scale factor h-: = h. = Idjjdz I = 1, so that scale ' is preserved. The transform corresponds to a translation given by the complex constant c plus a rotation through the angle O. In other cases, however, scale will be changed, and it will be changed differently in different regions so that the whole space will be distorted even though small regions will be similar. Since, as we have seen, any function of a z Plane
Y f PIane
v
v =2 v =I
x
u
v =-1
-
01
:::>
Fig.4.3
Conformal transform for the equation fez) = (1 - z)j(1
+ z) .
complex variable may be regarded as an example of an electrostatje field, we may say that the effect of the field may be simulated by warping the space and replacing the field by a simple radial electrostatic field E = u + iv. The motion of a charged particle may thus be explained as either due directly to the electrostatic field or to the warping of space by that field. This point of view is reminiscent of the procedure used by Einstein in discussing the effects of gravitation as given by the general theory of relativity. Figure 4.3 shows the conformal transform corresponding to the equation 1- z 1 - x 2 - y2 - 2y . _ 1- j j = 1 + z ; u = (1 + X)2 + y2; v = (1 + X)2 + y2' Z - 1 + j The function is analytic except at z = -1, and the transformation is conformal except at this point. The curves u = constant, v = constant on the z plane constitute an orthogonal set of coordinates consisting of two families of tangential circles. Other cases will be pictured and discussed later (see Sec. 4.7). We note that, at points where the scale factor Idjjdz I is zero, the transform also is not conformal. The region in the neighborhood of such points becomes greatly compressed when looked at in the j plane.
§4.2]
361
Analytic Functions
Inversely the corresponding region on the f plane is tremendously expanded. This suggests the possibility of a singularity of the inverse function to t, z(f), at the point for which f'(z) = O. An example will show that this is indeed the case. The simplest is fez) = Z2 for which 1'(0) = O. The' inverse z = fi has, as predicted, a singularity at this point. Thus the transformation cannot be conformal at this point . This can be shown directly, for if two line elements passing through z = 0 make an angle 1()2 - I()l with respect to each other, then the corresponding lines on the f plane make an angle of 2(1() 2 - I()l) with respect to each other and thus mapping is not conformal at z = O. It is also clear that, whenever f'(a) = 0, the mapping will not be conformal whether fez) behaves as (z - a)2, as in this particular example, or as (z - a)n (n integer). But for the region where f is analytic and where the scale factor Idf/dzl is not zero, the transform is reciprocally conformal. In mathematicallanguage we can say : Let fez) be analytic at z = a, and f'(a) ~ 0, then the inverse of 'fez) exists and is analytic in a sufficiently small region aboutf(a) and its derivative is l /f'(a).
First as to the existence of the inverse function, we note that the inverse to the transfor~ation u = u(x,y) and v = v(x,y) exists if the quantity (au /ax)2 + (au /ay)2 ~ 0. This is indeed the case if f'(a) ~ O. We note that, if /f'ea) were equal to zero, the inverse function would not exist at this point, indicating the existence of a singularity in the inverse function at this point. It now remains to show that the inverse function is analytic, that is, ax /au = ay /av and ax/av = -ay/au . Let us try to express ax /au , etc ., in terms of au /ax, etc. To do this note that ax ax dx = - du +-dv au av
so that 1 = (::)
(~;) + (~~) (:~)
and
°
=
(~~) (~;) + (~~) (:~)
or using Eq. (4.2.1) 1=
(;~) (~;) - e~) (~;)
and 0 = (::)
(~;) + (~~) (~~)
Similarly 1=
(;~) (~;) + (~~) (~~)
and
°= (;~) (~;) -
(~~) (~;)
This gives us four equations with four unknowns, ax /au, ax /av, ay /au, and ay/av, which may be solved . One obtains the derivatives of x: ax au /ax ax -au/ay au = (au/ax)2 (au /ay)2 ; av = (au/ax)2 (au /ay)2
+
+
362
Functions of a Complex Variable
[CH. 4
The derivatives of y turn out to be just as required by the CauchyRiemann conditions ax/au = ay/av, etc . It is also possible now to evaluate dz = ax df au
+i
ay = ax _ i ax au au av
=
1 1 (au /ax) - i(au/ay) = df/dz
proving the final statement of the theorem. We shall devote a considerable portion of the chapter later to develop the subject of conformal representation further, for it is very useful in application. Integration in the Complex Plane. The theory ,of integration in the complex plane is just the theory of the line integral. If C is a possible contour (to be discussed below), then from the analysis of page 354 [see material above Eq. (4.1.12)] it follows that
fe it dz = fe s, ds + i fe En ds ;
ds
=
Idzl
where E, is the component of the vector E along the path of intel?jration while En is the normal component. Integrals of this kind appe~r frequently in physics. For example, if E is any force field, then the integral E, ds is just the work done against the force field in moving along the contour C. The second integral measures the total flux passing through the contour. If E were the velocity vector in hydrodynamics, then the second integral would be just the total fluid current through the contour. In order for both of these integrals to make physical (and also mathematical) sense, it is necessary for the contour to be sufficiently smooth. Such a smooth curve is composed of arcs which join on continuously, each arc having a continuous tangent. This last requirement eliminates some pathological possibilities, such as a contour sufficiently irregular that is of infinite length. For purposes of convenience, we shall also insist that each arc have no multiple points, thus eliminating loops. However, loops may be easily included in the theory, for any contour containing a loop may be decomposed into a closed contour (the loop) plus a smooth curve, and the theorem to be derived can be applied to each. A closed contour is a closed smooth curve. A closed contour is described in a positive direction with respect to the domain enclosed by the contour if with respect to some point inside the domain the contour is traversed in a counterclockwise direction. The negative direction is then just the clockwise, one. Integration along a closed contour will be symbolized by § . One fairly obvious result we shall use often in our discussions: If fez) is an analytic function within and on the contour, and if df/dz is single-valued in the same region,
fe
§(df/dz) dz = 0
This result is not necessarily true if df/dz is not single-valued.
§4.2]
Analytic Funclions
363
Contours involving smooth curves may be combined to form new contours. Some examples are shown in Figure 4.4. Some of the contours so formed may no longer be smooth. For example, the boundary b' is not bounded by a smooth curve (for the inner circle and outer circle are not joined) so that this contour is not composed of arcs which join on continuously. Regions of this type are called multiply connected, whereas the remaining examples in the figure are simply connected. To test for connectivity of a region note that any closed contour drawn within a simply connected region can be shrunk to a point by continuous deformation without crossing the boundary of the region. In b' a
(c)
to -00
from 00
(c')
-00·
from
Fig. 4.4
to
00
Possible alterations in contours in the complex plane.
curve C1 intermediate to the two boundary circles cannot be so deformed. The curve b illustrates the fact that any multiply connected surface may be made singly connected if the boundary is extended by means of crosscuts so that it is impossible to draw an irreducible contour. For example, the intermediate contour C1 drawn in b' would not , if drawn in b, be entirely within the region as defined by the boundary lines. The necessity for the discussion of connectivity and its physical interpretation will become clear shortly. Having disposed of these geometric matters, we are now able to state the central theorem of the theory of functions of a complex variable. Cauchy's Theorem. If a function f(z) is an analytic function ; continuous within and on a smooth closed contour C, then §f(z) dz = 0
(4.2.3)
For a proof of Cauchy's theorem as stated above, the reader may be referred to several texts in which the Goursat proof is given. The simple proof given earlier assumes that 1'(21) not only exists at every point within C but is also continuous therein. It is useful to establish
364
Functions of a Complex Variable
[cH.4
the theorem within a minimum number of assumptions about fez), for this extends the ease of its applicability. In this section we shall content ourselves with assuming that C bounds a star-shaped region and that f'(z) is bounded everywhere within and on C. The geometric concept of "star-shaped" requires some elucidation. A star-shaped region exists if a point 0 can be found such that every ray from 0 intersects the bounding curve in precisely one point. A simple example of such a region is the region bounded by a circle. A region which is not star-shaped is illustrated by any annular region . Restricting our proof to a star-shaped region is not a limitation on the theorem, for any simply connected region may be broken up into' a number of star-shaped regions and the Cauchy theorem applied to each . This process is illustrated in Fig. 4.4c for the case of a semiannular region . Here the semiannular region is broken up into parts like II and III, each of which is star-shaped. The Cauchy theorem may then be applied to each along the ind icated contours so that ¢fdz II
+ ¢fdz
=0
III
However, in the sum of these integrals, the integrals over the parts of the contour common to III and II cancel out completely so that the sum of the integrals over I , II, and III just becomes the integral along the solid lines, the boundary of the semiannular contour. The proof of the Cauchy theorem may now be given . Take the point 0 of the star-shaped region to be the origin. Define F(X) by F(X)
=
X.Ff(Xz) dz ; 0::; X ::; 1
(4.2.4)
The Cauchy theorem is that F(l) = O. To prove it, we differentiate F(X) : F'(X) = .Ff(Xz) dz + X.!fzf'(Xz) dz Integrate the second of these integrals by parts [which is possible only if f'(z) is bounded] : F'(X)
=
¢f(Xz) dz
+ X {[Zf~z)J
-
~¢f(Xz) dZ}
where the square bracket indicates that we take the difference of values at beginning and end of the contour of the quantity within the bracket. Since zf(Xz) is a single-valued function, [zf(Xz)/X] vanishes for a closed contour so that F'(X) = 0 or F(X) = constant To evaluate the constant, let X = 0 in Eq. (4.2.4), yielding F(O) = 0 = Therefore F(l) = 0, which proves the theorem. This proof, which appears so simple, in reality just transfers the onus to the
F = F(X) .
§4.2]
Analytic Functions
365
question as to when an integral can be integrated by parts. The requirements, of course, involve just the ones of differentiability, continuity, and finiteness which characterize analytic functions. Cauchy's .theorem does not apply to multiply connected regions, for such regions are not bounded by a smooth contour. The physical reason for this restriction is easy to find. Recall from the discussion of page 354 that the Cauchy theorem, when applied to the electrostatic field, is equivalent to the statement that no charge is included within the region bounded by the contour C. Using Fig . 4.4b' as an example of a multiply connected region, we see that contours entirely within the region in question exist (for e1ample, contour Ci in Fig. 4.4b') to which Cauchy's
Fig. 4.6
Contours in multiply connected regions .
theorem obviously cannot apply because of the possible presence of charge outside the region in question, e.q., charge within the smaller of the two boundary circles. The way to apply Cauchy's theorem with certainty would be to subtract the contour integral around the smaller circle; i.e.,
,r.
r-.' j dz _,r.r
jdz = 0 0,
(4.2.5)
This may be also shown directly by using crosscuts to reduce the multiply connected domain to a single-connected one. From Fig. 4.5 we see that a contour in such a simply connected domain consists of the old contours -C. and C2 (Cl described in a positive direction, C2 in a negative direction) plus two additional sections C3 and C 4• Cauchy's theorem may be applied to such a contour. The sections along C3 and C4 will cancel, yielding Eq. (4.2.5). Some Useful Corollaries of Cauchy's Theorem. From Cauchy's theorem it follows that, ij j(z) is an analytic junction within a region (" j(z) dz, along any contour within C bounded by closed contour C, then jz, depends only on Zl and Z2. That is, fez) has not only a unique derivative
but also a unique integral. The uniqueness requirement is often used as motivation for a discussion of the Cauchy theorem. To prove this, we
366
[cH.4
Functions of a Complex Variable
compare the two integrals ( and ( , in Fig. 4.6, where C 1 and C2 are
Ic,
lOI
two different contours starting at
ZI
I
and going to
Z2 .
According to
Cauchy's theorem ( f(z) dz - ( f(z) dz = J.. f(z) dz, is zero, proving . lOI Io, r the corollary. It is a very important practical consequence of this corollary that one may deform a contour without changing the value of the integral, provided that the contour crosses no singularity of the integrand during Ihe deformation. We shall have many occasions to use this theorem in the
Fig. 4.6 Independence of integral value on choice of path within region of analyticity.
evaluation of contour integrals, for it thus becomes possible to choose a most convenient contour. Because of the uniqueness of the integral ( .. f dz it is possible to lZI define an indefinite integral of f(z) by F(z)
= (z f(z) dz l ZI
where the contour is, of course, within the region of analyticity of f(z). It is an interesting theorem that, if f(z) is analytic in a given region, then F(z) is also analytic in the same region : Or, conversely, if f(z) is singular at Zo, so is F( zo). To prove this result, we need but demonstrate the uniqueness of the derivative of F(z), which can be shown by considering the identity F(z) - F(f) . H L'
z-f
. -
f(r) =
f
[f(z) - f(f)] dz
~-_._-
z-f
Because of the continuity and single-valuedness of f(z) the right-hand side of the above equation may be made as small as desired as z is made to approach f. Therefore in the limit lim [F(Z) - F(r)] z->i
z-
r
= f(t)
Analytic Functions
§4.2]
367
Since the limit on the left is just the derivative F ' (r), the theorem is proved. We recall from Eqs. (4.1.19) et seq. that, if f(z) is the conjugate of the electrostatic field, then the real part of F(z) is the electrostatic potential while the imaginary part is constant along the electric lines of force and is therefore the stream function (see page 355). Therefore the two-dimensional electrostatic potential and the stream function form the real and imaginary parts of an analytic function of· a complex variable. Looking back through the proof of Cauchy's theorem, we see that we used only the requirements that f(z) be continuous, one-valued, and that the integral be unique, with the result that we proved that F(z) was analytic. We shall later show that, if a function is analytic in a region, so is its derivative [see Eq. '(4.3.1)]. Drawing upon this information in advance of its proof, we see that, once we have found that F(z) is analytic, we also know thatf(z) is analytic. This leads to the converse of Cauchy's theorem, known as Morera's theorem: If f(z) is continuous and single-valued within a closed contour C, and if !ff(z) dz = 0 for any closed contour within C, then f(z) is analytic within C.
This converse serves as a means for the identification of an analytic function and is thus the integral analogue of the differential requirement given by the Cauchy-Riemann conditions. Since the latter requires continuity in the derivative of f, the integral condition may sometimes be easier to apply. The physical interpretation of Morera's theorem as given by the electrostatic analogue will strike the physicist as being rather obvious . It states that, if f(z) is an electrostatic field and the net charge within any closed contour [evaluated with the aid of f(z)] within C is zero, then the charge density within that region is everywhere zero. Cauchy's Integral Formula. This formula, a direct deduction from the Cauchy theorem, is the chief tool in the application of the theory of analytic functions to other branches of mathematics and also to physics . Its electrostatic analogue is known as Gauss' theorem, which states that the integral of the normal component of the electric field about a closed contour C equals the net charge within the contour. In electrostatics the proof essentially consists of separating off the field due to sources outside the contour from the field due to sources inside . The f;,- a, then the
q(x) dx
+ fa:& q(x) dX}
369
Analytic Functions
§4.2]
In terms of the electrostatic analogue, the largest values of an electrostatic field within a plosed contour occur at the boundary. If f(z) has no zeros within C, then [l/f(z)) will be an analytic function inside C and therefore Il/f(z) I will have no maximum within C, taking its maximum value on C. Therefore If(z) I will not have a minimum within C but will have its minimum value on the contour C. The proof and theorem do not hold if f(z) has zeros within C. The absolute value of an analytic function can have neither a maximum nor a minimum within the region of analyticity. If the function assumes either the maximum or minimum value within C, the function is a constant. Points at which f(z) has a zero derivative will therefore be saddle points, rather than true maxima or minima. Applying theseresults to the electrostatic field, we see that the field will take on both its minimum and maximum values on the boundary curve . These theorems apply not only to If(z) I but also to the real and imaginary parts of an analytic function and therefore to the electrostatic potential V. To see this result rewrite Eq. (4.2.7) as 27fif(a)
= 27fi(u
+ iv) = i 1021a n
-
1
---
z - an
Therefore b; = -1. Next we set up a sequence of circles R p • These are circles whose radius is p7r (p integer). On these circles, tan z is bounded for all values of p, satisfying the requirements of the theorem. Therefore .,
1: -1: 1:
tan z = -
[z -
i(2~ + 1)71' + i(2n ~ 1)71']
[z -
i(2~ + 1)71' + i(2n ~ 1)7I'J
.,
o .,
tan z
=
[z o ., ~ '-' [i(2n n=O
+ i(2~ + 1)71' 2z
+ 1)71']2 -
Z2
i(2n
~ 1)71'J (4.3.7)
Equation (4.3.6) may be also used to obtain an expansion of an integral junction j(z) into an infinite product, since the logarithmic derivative of an integral function, j'(z) /f(z), is a meromorphic junction. Its only singularities in the finite complex planes are poles at the zeros
§4.3]
Derivatives of Analytic Functions, Taylor and Laurent Series
385
an of fez) . Suppose, again for simplicity, that these poles are simple poles, that is, fez) - - - 7 constant (z - an)
.
z-+a n
[f'(z) lf(z)] = [f'(O)lj(O)]
+~ \' [_1_ +..!.J z - an an
or
In [fez)]
= dd In fj(z)]
z
.
n=1
(1 - :) + :J . j(z) = j(O)e[/'(O)/I(O)] z n(1 - :,,) e (4.3.8)
= In'[f(O)] +
[!,(O)lj(O)]z
+
2:
[In
n=1
or
Z/ G n
n=1
For this formula to be valid it is required that j(z) be an integral function, that its logarithmic derivative have simple poles, none of which are located at 0, and that it be bounded on a set of circles R; etc . Let us find the product representation of sin z, We shall use it often in our future work. Now sin z satisfies all our requirements except for the fact that it has a zero at a = O. We therefore consider (sin z)/z. The logarithmic derivative is cot(z) - liz, a function which satisfies all our requirements. The points an are n7l', n ~ 0, so that
si~ z =
.
.
Il [1 - :7l'Jez/n~ = Il [1 - (:7l'YJ -
00
(4.3.9)
n= 1
(n;o'O)
Similar expansions may be given for other trigonometric functions and for the Bessel functions J n(Z). Behavior of Power Series on the Circle of Convergence. In many problems it is impractical (or impossible) to obtain solutions in closed form, so that we are left with only power series representations of the desired solutions. Such series will generally have a finite radius of convergence, which may be determined if the general form of the series is known or may be found from other data available on the problem. The power series is not completely equivalent to the solution, of course; inside the radius of convergence it coincides exactly with the solution, but outside this radius there is no correspondence and another series must be found, which converges in the next region, and so on. As mentioned earlier, it is as though the solution had to be represented by a mold of many pieces, each series solution being one piece of the mold, giving the shape of a part of the solution over the area covered by the series but giving no hint as to the shape of the solution elsewhere . In order to ensure that the various pieces of the mold "join" properly, we must now inquire as to the relation between the solution and the power series on its circle of convergence.
386
Functions of a Complex Variable
[CR. 4
A priori, it is clear that the investigation of the behavior of a power series on its circle of convergence will be an extremely delicate matter, involving rather subtle properties of analytic functions. Consequently, the proofs are involved and highly "epsilonic." Fortunately, the theorems are easy to comprehend, and the results, as stated in the theorems, are the relevant items for us in this book, not the details of mathematical machinery needed to prove the theorems. We shall, therefore, concentrate in this section upon a discussion of the aforementioned theorems, omitting most of the proofs . For these, the reader is referred to some of the texts mentioned at the end of the chapter. Suppose, then, that the series representing the solution is written in the form (4.3 .10) n
The first question is to determine its radius of convergence R . If the general term an of the series is known, the Cauchy test yields the radius as
R = lim
n~ ao
[~] a +l
(4.3.11)
n
Finite radii of convergence occur only if the limit of the ratio [an /an+d is finite. In this case we can reduce the series to a standard form, with unit radius of convergence, by changing scale. Letting r = zR, we have f(z) = ~bnzn; b; = anRn (4.3.12) It is convenient to normalize the series to be discussed to this radius. A great deal of care must be exercised in arguing from the behavior of b; for n large to the behavior of f(z) at a given point z on the circle of convergence. If series (4.3.12) diverges or converges at z = e», it is not correspondingly true that the function is, respectively, singular or analytic at z = e« . For example, the series ~(-z)n, which represents 1/(1 + z) for Izi < 1, diverges at z = 1, but 1/(1 + z) is analytic there. On the other hand the function -
foz In (1 -
w) dw = 1
+
(1 - z)[ln (1 - z) - 1] is singular (but finite) at z = +1, but the corresponding series, ~(z)n+l/n(n + 1), converges at z = +1. Other series can be given where b; ~ 0 but which diverge at every point on the unit circle, and still others which converge at z = 1 but at no other point on the unit circle. With this warning to the reader to refrain from jumping to conclusions, we shall proceed to examine what it is possible to say about solutions and the corresponding series on their circles of convergence. The first matter under discussion will be the available tests determining whether or not a point on the convergence circle is or is not a
§4.3]
Derivatives of Analytic Functions, Taylor and Laurent Series
387
singular point. We first transform the point under examination to the position z = 1. Then there are two theorems which are of value. The first states that, if f(z) = '1:.b nzn and g(z) = '1:. Re b« zn have radii of convergence equal to 1 and if Re bn ~ 0, then z = 1 is a singular point of f(z) . [Re f = real part of f.l In other words, if the phase angle of z on the circle of convergence is adjusted so that the real part of each term in the series is positive, this phase angle locates the position of the singularity on the circle of convergence. For example, when this theorem is applied to f = '1:.z n+l /n(n 1), the series mentioned earlier, we find that z = 1 is a singular point of f in spite of the convergence of the series at z = 1. All the terms are positive, and the series for the derivative diverges, so we might expect complications at z = 1. We note that, even if f(z) = '1:.b nz n does have a singularity at z = 1, according to this test, the series does not necessarily have a singularity for z = e», where", is small but not zero; for we can write z = Zei
.
'( n~ m ) ., b-; the necessary and sufficient
m. n
condition that z = 1 be a singularity for '1:.b nzn is that the quantity (C n)- lIn never gets smaller than i as n goes to infinity. For example, for the series '1:.( -z)n, b; = (-I)n, the point z = 1 is not a singular point. For the series '1:.z n on the other hand, C; = 2 n, lim (2 n)- lIn = i so that z = 1 is a singular point. A more dramatic case is provided by the series '1:.(n + 1)(n + 2)( -z)n [which represents the function 2/(1 + Z)3] with b« = ( -I)n(n l)(n 2); then for n 2, C; = 0, so that in spite of the strong divergence of the series, the function f is analytic at z = 1. Having determined the radius of convergence and the positions of the singularities on the circle of convergence, it is obviously useful to be able to estimate the behavior of f(z) on the whole circle of convergence, particularly the singular part. A tool for doing so is provided by the following theorem: If f(z) = '1:.b nzn and g(z) = '1:.Cnzn, where Re b« and Re C; ~ and where 1m o; and 1m Cn ~ 0, and if as n~ co, bn ~ DC n (D constant), then as Izi ~ 1, f(z) ~ Dg(z) . (1m f is the imaginary part of f.) It should be noted that the condition Re o.; Re C; ~ 0, by a slight extension, may be relaxed to require that Re bn , Re Cn eventually maintain a constant sign ; similarly for 1m bn , 1m Cn. This theorem simply states that, if two expansions have a similar behavior for large n, then the two functions have the same singularities . By using the theorem, some statement about the asymptotic
+
°
+
. [cH,4
Functions of a Complex Variable
388
behavior of the coefficient bn may be made, For example, we may assert that, if b« ~ D nP-1/(p - I)!, then fez) - D/(1 - z)p as Izl- 1. n-
00
To demonstrate this! we need only show that the coefficient C; in the expansion "1:.C nzn = (1 - z)-p is asymptotically nP-l/(p - I)!. We note that _p _ ~ (n + 1) , , , (n + p - 1) n (1 - z) (p _ I)! z
Lt
en
from which the required asymptotic behavior of follows, The hypergeometric function provides another example. This function, which we shall discuss at great length in Chap. 5, is given by
F( a, bl cIz)
=
1
+ ab + (a)(a + 1)(b)(b + 1) Z2 + ' , , c z
(c)(c
+
1)
2!
.
It is known to have a singularity at z = 1. The general coefficient b; is b = -'-(a-'.-)('--a_+'-------'1),--' _...,..".(a. . ,. . . . :-+-:-n--::--.,---I.!. . C(),--,b),-o(b-----;-+_I...:....)_'---:'__'_(,--b---,+_n_---.:1) n n!(c)(c + 1) , . . (c + n - 1)
+
Assuming a, b, c integers and a bc of b; may be determined as follows :
~
b, the asymptotic behavior
b = (c-l)! [(a+n-l)!(b+n-l)!] n (b - 1)!(a - I)! nl(c + n - 1)!
= (b -
(c-l)! [(n+l)'" (a+n-l)] 1) l(a - 1) I (b + n) .. . (c + n - 1) ---->
Thus one may conclude that, when c
F(a,b/clz) ~
(c - 1) •I
(b - 1)!(a - I)!
n a +b-
c- 1
< a + b,
+ b - c - I)! 1 (b - 1) !(a - I)! (1 - z)a+b-c
(c - 1) l(a
We noted earlier a case f = "1:.z n+1/ n(n + 1), in which a series converged at a point z = 1 at which f was singular but finite. We might ask whether the convergent sum s = "1:.1 /n(n + 1) is equal to f at z = 1. The answer to this query is given by a theorem due to Littlewood, which states: If fez) - s as z ----> 1 along a smooth curve and lim (nan) is bounded, then "1:.a n converges to s. In the example quoted above, "1:.1 /n(n + 1) = lim {I (1 - z)[ln (1 - z) - III which is, of course, unity, as can be _1
+
verified directly from the series itself. From the above theorems, it is clear that knowledge of the power 1 The proof holds only for p integer. A suitable generalization for noninteger p in which the factorials are replaced by gamma functions (to be discussed later in this chapter) is possible .
§4.3]
Derivatives of Analytic Functions, Taylor and Laurent Series
389
series representation of a function yields fairly complete information about the function on the circle of convergence. Can the power series be used to determine properties of the function it represents outside the circle of convergence? Under certain fairly broad restrictions, the answer is yes. We shall discuss this point in the next subsection. Analytic Continuation. It is often the case that knowledge of a fun ction may be given in a form which is valid for only limited regions in the complex plane. A series with a finite radius of convergence yields no direct information about the function it represents outside this radius of convergence, as we have mentioned several times before. Another case, which often occurs, is in the representation of the function by an integral which does not converge for all values of the complex variable. The integral e:" dt
fo
00
represents l iz only when Re z > O. It may be possible, however, by comparing the series or integral with some other form, to find values of the function outside of the region in which they are valid. Using, for example, the power series f(z) = 1 + z + Z2 + ... which converges for Izi < 1, it is possible to find the values of f(z) for Izi < 1 and to identify f with 1/(1 - z) which is then valid for Izl > 1. The process by which the function is extended in this fashion is called analytic continuation. The resultant function may then (in most cases) be defined by sequential continuation over the entire complex plane, without reference to the original region of definition. In some cases, however , it is impossible to extend the function outside a finite region. In that event, the boundary of this region is called the natural boundary of the function, and the region, its region of existence. Suppose, as an example, that a function f is given as a power series about z = 0, with a radius of convergence R, a singular point of f being on the circle of convergence. It is then possible to extend the function outside R. Note that at any point within the circle (izi < R) it is possible to evaluate not only the value of the series but also all its derivatives at that point, since the derivatives will have the same region of analyticity and their series representation has the same radius of convergence . For example, all these derivatives may be evaluated at z = zoo Using them, one can set up a Taylor series:
f = \' j O. If on the other hand C was in the other branch, the image of cd would be the dashed CD. In either case crossing the branch line on the z plane takes j from one of its branches to the other. This fact may also be derived if we make use of the theorems developed in the section on analytic continuation. For example, consider circle ejg in Fig. 4.13. If the function zi is continued along this path from e to g, then the value obtained at g should equal the value at e inasmuch as no singular point of the function has been enclosed. Pictorially this is exhibited by the transformed circle EFG. On the other hand if zi v
z Plane
u
--90° Fig. 4.14 Transformation f formality at z = o.
=
0 , showing
lack of con-
is continued along the circle a ------7 b, we know that the value obtained at b will not equal the value at a. From our discussion of analytic continuation we know that there must be a singularity enclosed. Since circle a ------7 b is of arbitrary radius, the singular point is z = 0 as may be verified by direct examination of the function z at z = O. .This type of singular point (see page 391) is called a branch point. We note a few additional properties of branch points. The value of j(z) at the branch point is common to all the branches of the function f. Most often it is the only point in common, though this is not always the case. Second, the transformation defined by j(z) is not conformal at the branch point. This is illustrated in Fig. 4.14 where, for the function zi, the angle between the transforms of two radial lines emanating from z = 0 is half of the angle between these two radii. For a function ev«, the angle will be reduced by a factor of a. Finally, it is important to note that branch points always occur in pairs and that the branch lines join the branch points. For example, in the case of zi , Z ------7 00 is also a branch point This may be verified by making the substitution z = 1!r, zi = r-i , which is a multivalued function of r and has a branch point at r = 0 ; that is, Z = 00 .
§4.4]
M ultivalued Functions
401.
The branch line we have used ran from the branch point at z = 0 to the branch point at z = 00 along the negative real axis. Any curve joining the branch points 0 and 00 would have done just as well. For example, we could have used the regions 0 < r/J < 27r and 27r < r/J < 47r as the defining regions for the first and second branch. On the f plane these two would correspond to v > 0 and v < 0, respectively. (The branch line in this case is thus the positive real axis of z.) This division into branches is just as valid as the one discussed on the previous page. We therefore may choose our branch line so that it is most convenient for the problem at hand. Riemann Surfaces. The notion, for the case zt, that the regions -7r < r/J < 7r and 7r < r/J < 37r correspond to two different regions of
Branch Line
Sheet2
z@jt=§_· ff' Branch Line -
Sheet 2
Fig. 4.16 Joining of Riem ann surfaces along branch lines for the functions f = and f = vi Z2 - 1.
vz
the f plane is an awkward one geometrically, since each of these two regions cover the z plane completely. To reestablish single-valuedness and continuity for the general purpose of permitting application of the various theorems developed in the preceding section, it is necessary to give separate geometric meanings to the two z plane regions. It is possible to do all this by use of the notion of Riemann surfaces. Imagine that the z plane, instead of being one single plane, is composed of several planes, sheets, or surfaces, arranged vertically. Each sheet, or ensemble of z values, is taken to correspond to a branch of the function, so that in the simple case of zt each sheet is in one-to-one correspondence with a part of the f plane. For the function zt only two sheets are needed, for there are only two branches, sheet 1 corresponding to -7r < r/J < 7r, sheet 2 to 7r < r/J < 37r. These sheets must be joined in some fashion along the branch cut, for it should be recalled that it is possible to pass from one of the branches to the other by crossing the branch line. The method is illustrated in Fig. 4.15. Each sheet is cut along a branch line, also often called a branch cut as a result. Then the lip of
Functions of a Complex Variable
402
[cH.4
each side of the cut of a given sheet is connected with the opposite lip of the other sheet. A closed path ABCDE on both sheets is traced out on the figure. Path AB is on sheet 1, but upon crossing the branch cut on sheet 1 at B we pass from sheet 1 to sheet 2 at C. Path CD is on sheet 2. Beyond D we cross the branch cut on sheet 2 and so pass on back through A to point E on sheet 1. It now becomes apparent what is meant by a closed contour. Path AB is not, while path ABCDA is. The application of Cauchy's theorem and integral formula now becomes possible. We shall look into this y matter in the next section. Right now let us continue to examine z Plane the behavior of multivalued functions. v -O v- O We turn to a more complicated x z=-l example. Consider the function Branch line
f Fig. 4.16 tion f =
Branch line for the transforma1.
V 22 -
=
yz
2 -
1
This function has branch points at z = ± 1. The point at z = 00 is
not a branch point, for letting z = lit, f = y(11r 2 ) - 1 = ~/t -----t lit. The point at infinity is therefore a simple pole. The branch bO line for f runs therefore from z = 1 to z = -1. One may go directly from -1 to 1 as in Fig. 4.15. One could also have gone between these points by going along the negative x axis from -1 to x = - 00 and along the positive x axis from x = 00 to x = 1. There are , of course, many other possibilities. Along the real axis for Ixl > 1, ~ = VX2="1 is real, so that [z] > 1 on the real axis corresponds to Im f = v = 0. As soon as Ixl < 1, that is, along the branch line, the phase of f becomes uncertain ; it may be either 11"/2 or -11"/2. Resolving this uncertainty may be reduced into deciding where the branch lines of the constituent factors -vz=F1, yz - 1 are to be placed. Different choices will lead to different branch lines for the product. To obtain the branch line of Fig. 4.15, we let the branch line VZ + 1 extend from -1 to - 00 ; for ..yz=-I from z = 1 to - 00 . The phase of the product function ..yZ2 - 1 is then given by ·i(T+ + L), where T+ is the phase of z + 1 and T_ the phase of z - 1 as shown in Fig. 4.16. Consider now a point just above the real axis between +1 and -1. Here T + = 0, T_ = 11", so that the phase of f is +11"/2. At a corresponding point just below the x axis, T+ = 0, L = -11" so that the phase of f = -11"/2. Therefore above the line u = 0, v > 0, while below the line u = 0, v < 0, again demonstrating the discontinuity which exists at a branch line. Now consider
§4.4]
Mullivalued Functions
403
the y axis. For any point on the positive y axis, T + + T _ = 71", so that the phase is 71"/2 and u = O. Below the axis, the phase is -71"/2. With this information available we may now attempt to sketch the lines of constant u and constant v on the z plane . This is shown in Fig. 4.17. Note that these lines must be mutually orthogonal except at the branch points, where the transformation from the z plane to the f plane is not conformal. One example of this has already been found at points z = ± 1 where the lines u = 0 and ,v = 0 meet. For large values of z(lzl » 1) we may expect that f ~ z, so that asymptotically the constant u and constant x lines are identical, and similarly for the constant v 1.6 v'I.6 +IH-1"=--I-=ft----I--+---\--4--=~=RH
1.2 0.8 0.4
v'-0.4 -0.8
y=-IC:±::::F:==--t'""'=4J--------i------jI---+--P'-f.-=±-+-J
-1.2
'"="'-;:;;-------l;--------!I;----J,-----!-J -1.6
and constant y lines. We note in Fig. 4.17 that the contour u = 0 is the y axis, plus that part of the z axis between x = -1 and x = + 1. Between z = -1 and z. = 0, v varies from 0 to 1 above the axis, while for points below the axis it varies from 0 to -1. For Ivl > 1 we must turn to the yaxis, for constant v lines for v > 1 (or v < -1) intersect the y axis at right angles, whereas for Ivl < 1 v = constant lines intersect the x axis between 0 and -1. From this discussion, it becomes possible to sketch the v = constant lines. The u = constant lines may be drawn in from the requirement of orthogonality to the v = constant lines. We need to determine the sign of u to which they correspond. This may be done by asking for the phase of f along the z axis (x < -1). Here, just above the x axis, T + = T _ = 71" so that the phase of f is 71". For z just below, the same argument leads to a phase of -71", which, of course, is the same as 71". We see then that the u = constant lines which are on the x < 0 half-plane are negative, those in the x > 0 half-plane are positive.
Functions 01 a Complex Variable
404
[CH. 4
We have drawn in Fig. 4.17 the u and v contours corresponding to sheet 1 of the Riemann surface. The v > 0 lines, for example, must join with corresponding v > 0 lines in the lower half plane of the second sheet and vice versa. Aside from this change in phase the first and second sheets will be identical. To verify this, consider the phase at points A and B if we multiply the values of vT+Z on its first sheet and YZ='l. on its second sheet. For A , T+ = 0, T_ = 371" so that the phase at A is 371"/2 or -71"/2. At B, T+ = 0, T_ = .". so that the phase of 1 at B is 71"/2. We could also have taken values of "vz:FT and .yz-=-I from their second sheets only to find that this would lead to the same situation as
+t I
+
I
-
I
+
~
i
t+
Fig. 4.18 Configurations of boundary surfaces for which the transformation of Fig . 4.17 is a ppropriat e.
before . For example, at A , T+ = 271", T- = 371", and the phase of 1 = 71"/2. We now see in addition that there are only two sheets for the two branches of ~. These two sheets are joined through the branch line extending from -1 to 1. As a final part of this discussion it is instructive to point to possible physical applications of the function ~ in order to observe the correlation between the mathematical branch cuts and the barriers of the corresponding physics problem. For example, the lines u = constant represent the equipotentials, the lines v = constant the lines of force for an infinitely conducting plate of shape shown in Fig. 4.18a, placed in a uniform electric field in the x direction. In hydrodynamics, u = constant would be the lines of flow about the same boundary. The lines u = constant may also be used as lines of force for another configuration illustrated in Fig. 4.18b, and the lines v = constant correspond to the equipotentials. The two plates are kept at a negative potential with respect to Iyl » 1. An Illustrative Example. As a final example of the behavior of multivalued functions, consider the function z = (tanh 1)/1
(4.4.4)
M ultivalued Functions
§4.4]
2
405
4
u Reol Port of f(z)
Fig.
4.19
Conformal mapping of transformation = z = rei 4> on the f plane ; contours of constan-t magnitude r and phase angle '" of z, The first three branch points are shown. (l /f) tanh f
This function occurs in the solution of the wave equation (see Chap. 11). The function z has simple poles at f = [(2n + 1)j2]1ri (n = integer) . Its branch points occur at the points at which z has a zero derivative. At such a point, say a, the transformation defined by (4.4.4) is not conformal. On page 361 we pointed out that this results from the fact that (z - a) has a multiple zero at a so that f - f(a ) in the neighborhood of
[cH.4
Functions of a Complex Variable
406 10
1
I--
6 4
1+8=9~V"'
'\
rr-:
r-.
/ ~/ ~ "":"' V0. 0 V 1\"tP g"
"
'~ I\
~n.J
rrr
2 N
"
~j
"~ 'c o '"
\
\
K"
R=
rr o\
~0.6 f----- {3r1.5
10 " CI:>
{3=2
co
7
,
120'
160·
of z Fig. 4.20 Conformal mapping of the first sheet of the trans form ation (1/f) tanh f = z = etD (f = ..-(3ei 9) on the w plane. One branch point is at a. Phose Angle
z = a is a fractional power of (e - a) . The branch points are therefore solutions of [(sech 2 f) /f] - [(tanh f) IP] = 0 or 2f = sinh 2f
(4.4.5)
A table of solutions of (4.4.5) is presented below (in addition to z = ao = 1).
f =u
+ iv
z
=
re '
Branch point
al
a2 aa
a,
u
v
r
1.3843 1.6761 1.8584 1. 9916
3.7488 6 .9500 10.1193 13.2773
0 .23955 0 .13760 0 .09635 0 .07407
'"
63 .00' 72 .54°
76 ,85· 79.36'
407
Multivalued Functions
§4.4]
+
The branch points for large values of v, an(n» 1) are an = u iv, u ~ In (4n 1)7r, v ~ - (n i)7r, as may be ascertained by substitution in Eq. (4.4 .5). We see that we have an infinite number of branch points and a corresponding infinite number of sheets. One simplification should be noted, howeve r, that at each branch point f has the behavior [z - all, for zll[f( a)] rf O. As a result the type of branch point is that
+
+
10
6 1-8 =90·
4
1
Results (4.5.3) may be combined into one by writing I = 271'/I p 2 - 11 for p real. Turning next to integrals of type 2, we consider the integral of f(z) along a closed contour consisting of the real axis from - R to R and a semicircle in the upper half plane, as in Fig . 4.22. By virtue of the assumptions on f(z), the integral along the semicircle will vanish in the limit R ~ co. Then
+
J-.. . f(x) dx = 2m Lresidues of f(z) in the upper half plane
(4.5.4) '
410
Functions of a Complex Variable
[cH.4
As a simple example, we compute I -
-
f"_" 1 +
dx
x2
which is known to equal n: From Eq. (4.5.4), I = (27ri) (residue at z = i) for z = i is the only pole of 1/(1 + Z2) = 1/(z + i)(z - i) in the upper half plane. The residue is 1/2i, and the value of I will therefore be 7r. Less simple examples will be found in the problems. Integrals Involving Branch Points. Finally we consider type 3 with j having no poles on the positive real axis. Here we start with the contour integral y
¢ (-z)I'-!.j(z) dz around a contour which does not include the branch point of the integrand at z = 0 and thus remains on one sheet of the associated Riemann surface. We choose the contour illustrated in Fig . 4.23, involving a small circle around z = 0 whose radius will eventually be made to approach zero, a large circle whose radius will evenFig. 4.23 Contour for evaluation of tually approach infinity, and two integrals involving a branch point at integrals along the positive real axis e = o. described in opposite directions and on opposite sides of the branch cut. Since the function is discontinuous along the branch cut, these two integrals will not cancel. Since Ji. is in general any number, the phase of ( -Z)I'-l may be arbitrarily chosen at one point. We choose its phase at E to be real. At point D, achieved by rotating counterclockwise from E around z = 0, the phase factor is (e..i) 1'-1 so that the integrand along DC is er i(l'-l)xl'-!.j(x) . To go from point E to A one must rotate clockwise so that the phase of the integrand at A is -7ri(Ji. - 1). AlongAB, the integrand is e:..i (l'- l)xl'- !.j(x) . The integrals along the little and big circles vanish by virtue of the hypotheses for type 3 so that in the limit of an infinitely large radius for the big circle and zero radius for the little circle
¢ (-Z)I'-lj(Z) dz = - fa " e
r i(l'-l)xl'-!.j(x)
dx.+
fa" e-";(I'-l)XI'-!.j(x) dx
where the first of the integrals on the right is the contribution of the integral along DC while the second is the contribution from path AB. Combining, we obtain
¢ (-z)I'-!.j(z) dz = 2i sin J" XI'-lj(X) dx 7rJi.
§4.5]
Calculus of Residues; Gamma and Elliptic Functions
411
Applying Cauchy's integral formula to the contour integral we obtain
h'' ' xl'-1f(x) dx = 7r csc(1l"JL) Lresidues of (-Z)I'-lj(Z) at all poles of j (4.5.5) As a simple example consider
J'" [xl'-l/(1 + x)] dx.
From the theorem
. (4.5.5) this has the value 1l" csc (7rJL) if 0 < JL < 1. Finally we should point out that occasionally we encounter poles of the integrand of the second or higher order. Such cases can be calculated by using Eq. (4.3.1). Suppose that there is a pole of nth order in the integrand j(z) at z = a. We set the integrand j(z) = g(z)/ (z - a)n where g(z) is analytic near z = a (that is, of course, what we mean by a pole of nth order at z = a). Then by Eq. (4.3.1), the contribution to the integral due to this nth-order pole at z = a is 1_ 27ri(n
1) !
[~nn-=-~g(Z)l_
The same result could be obtained by considering the nth-order pole to be the limiting case of n simple poles all pushed together at z = a. From this result we can extend our discussion of the term residue at a pole. We say that the residue of a fun ction j(z) at the nth-order pole at z = a [i.e., if j(z) = g(z)/(z - a)n near z = a] is 1
[
dr:!
(n - 1)! dz n- 1 g(z)
]
z=a
(4.5.6)
This extension allows us to use the prescriptions of Eqs. (4.5.3) and (4.5.4) for a wider variety of functions f . Inversion of Series. The problem of inverting a function given in the form of a power series '" W = j(z) = Wo + an(z - zo)n; al ~ 0 (4.5.7) n=l
L
is often encountered. We ask for the inverse function z = z(w), such that j(z) - W = O. From Eq . (4.5.7) we see that the inverse function may.be expressed in terms of a power series z(w) = Zo
+
L bn(w n=l '"
wo)n
(4.5.8)
by virtue of the theorem on page 361. The coefficients b; may be expressed in terms of an by introducing Eq. (4.5.8) directly into Eq. (4.5.7). However, it is possible to' derive the coefficients more elegantly by employing Cauchy's formula. We shall first devise a contour integral whose residue is just the function z(w) . Using j(z) as the independent
412
Functions oj a Complex Variable
[CH. 4
variable, this integral is (1/211"i).f[z dj/(f - w)], for it has the required value when the contour is taken about the point j = wand it includes no other zeros of (j - w). In terms of z the contour integral can be written as z(w) = ~.1. zj'(z) dz (4.5 .9) 2m 'f j(z) - w Differentiating this with respect to wand then integrating by parts, d dw z(w) =
1 1.
dz - w
(4.5.10)
2m 'f j(z)
In this form, it is possible to evaluate the integral.
Writing
., 1 1 [ fez) - w = j(z) - Wo 1
'\' (w - wo)n ] wo]n
+ n-l Lt [fez) -
and referring to the series (4.5.8), differentiated with respect to w, we see that nb = _1_1. dz n 2m 'f [fez) - wo]n The value of the integral is
or
The derivative may be evaluated explicitly through use of the multinomial theorem (1
'\' [-,---,,,,P;-!_-J arb' . + a + b + c + . . ')1' = Lt r!s!t!· ··
• • • ,
T,8 ,t , .. •
where
r+s+t+ ·· · =P
Introducing the multinomial expansion and performing the required differentiation yields
»; =
_1_ '\'
Lt
na l
(4.5.12)
8,t,U, • . •
. where
(_1)8+1+1'+ . . .•
(n)(n
+ 1)
+ + t + u + .) (aa21) (aa + ... = n - 1
.. . (n - 1 s s!t!u! . . s 2t 3u
+ +
8
3
1)
I ••
•
Calculus of Residues; Gamma and Elliptic Functions
§4.5]
413
We list the first few bn's, indicating their derivation from (4.5.12) 1
b1 = al
1 a2 b2 = - - - = a~ al
ba =
3~i [3;1
b4 =
~
=
4at
4
a2
ai
(::Y -:,(::)]
[_ 4 . 5 . 6 (a 2) 3! al
3
=
~i [2 (::Y - (~:)]
+ ~ (a
2)
1 !l ! al
~t [ -5 (::Y + 5 (::) (::) -
(a 3)
_
al
(4) (a 4) l! al
(4.5.13) ]
(::)]
When this is inserted into Eq. (4.5.8) we have the required inverted series. Summation of Series. The next application of the Cauchy formula to
L fen). 00
merit discussion is the summation of a series of the form
n=-
The
00
device employed to sum the series first replaces the sum by a contour y Poles of f(z) Poles of
rr cot (17" Z)
( (
-3
o
I
2
3
x
C1
' - - - - - - - t - - - - - - l C3
Fig. 4.24
Poles for integrand equivalent to series.
integral. To do this we require a function which has simple -poles at Z = n and in addition is bounded at infinity. One such function is 17" cot (17"z), which has simple poles at n = 0, ± 1, ±2 ± . . . , each with residue 1. Moreover it is bounded at infinity except on the real axis . Another function of this type is 17" CSC (17"z) which has poles at Z = 0, ± 1 ± . . . with residue (-I)n. The contour integral jJ17"f(z) cot(17"z) dz around the contour C 1 shown in Fig. 4.24 is just 217"i times the residue of the function 7rf(z) cot(7rz) at
Functions of a Complex Variable
414 Z
[cH.4
= 0, which is f(O). The integral about contour C2 is 2ril!(0)
+ f(l) + f( -1) + residue of [7rf(z)
cot(7rz)] at ad
and so on. Finally, for a contour at infinity, the integral must be 27ri {
L'"
fen)
+ residue
of [7rf(z) cot(7rz)] at all the poles of fez) }
if fez) has no branch points or essential singularities anywhere. If in addition Izf(z)l-+ 0 as Izl-+ 00, the infinite contour integral will be zero, so that in this case
L'" fen) -
= -
Lresidues of 7rf(z) cot (7rz) at the poles of fez)
,(4.5.14)
'"
If 7r csc(7rz) is employed, one obtains
- Lresidues of 7rf(z) csc(7rz) at the poles of fez)
L(d ~1~~ 2' '"
As a simple example consider
Then fez) = (a
~ Z)2 with
- '"
a double pole at z = -a. The residue of 7rf(z) csc(7rz) at -7r 2 csc(7ra) cot(7ra) [see Eq. (4.5.5)] so that
L(d ~1~~2
Z
= -a
is
'"
= 7r 2csc(7ra) cot(7ra)
-'" This method for summing series also illustrates a method for obtaining an integral representation of a series. We shall employ this device when we wish to convert a power-series solution of a differential equation into an integral (see Sec. 5.3). Integral Representation of Functions. We shall often find it useful to express a function as an integral, principally because very complex functions may be expressed as integrals of relatively simple functions. Moreover by changing the path of integration in accordance with Cauchy's theorem one may develop approximate expressions for the integral. In the chapter on differential equations we shall employ such techniques repeatedly. As a first example consider the integral u(x)
i
i kz
1 e- dk = -2' -,7rt c IC .
(4.5.16)
§4.5]
Calculus of Residues; Gamma and Elliptic Functions
415
The contour is given in Fig. 4.25. The function u(x) occurs in the theory of the Heaviside operational calculus (see Sec. 11.1). It may be evaluated exactly, employing Cauchy's integral formula. If x > 0, we may close the contour by a semicircle of large (eventually infinite) radius in the lower half plane. The contribution along the semicircle vanishes. Therefore u(x) = Res [e-ikx/k] at k = 0 so that u(x) = 1 for x > O. If x < 0, the cont our may be closed by a semicircle of large radius in the upper half plane. Again the cont ribut ion along the semicircle vanishes. 1m k Re k Contour
Fig. 4.26
C
k =0
Contour for unit step function u(x).
Since [e-ikx/k] has no poles within this contour, u(x) = 0, x < O. Therefore we find u(x)
= {I; x> 0 0;
x
(4.5.17)
0, we may close the contour by a circle of large radius in the upper half plane. As we permit the radius of the semicircle to become infinite, the contribution of the integral along the semicircle tends to zero. The value of the integral is then (211"i) (residue of the integrand at k = K). Therefore Gk(x - x') = (i/2K)e i K ( x- x' ) ;
x - x'
> 0; contour
C1
(4.5.19)
If x - x' < 0, the contour may be closed from below. The value of the integral is now equal to (- 211"i) (residue at k = - K) . Gk(x - x')
=
(i/2K)e- i K ( x-r
);
x - x'
< 0;
contour C 1
(4.5.20)
416
[CR. 4
Functions of a Complex Variable
Combining (4.5.19) and (4.5.20) we obtain Gk(x - x')
= (i/2K)e iK1"-""I; contour C1
(4.5.2])
agreeing with page 125. On the other hand if contour C2 is used, Gk(x - x')
= (-i/2K)e- iK1",-z'l; contour C2
(4.5.22)
This result is no surprise, since contour 2 is just the reflection of contour 1 with respect to the real axis. Contour 1 gives us the Green's function satisfying the boundary condition requiring diverging waves; i .e., the point x' acts as a point source. Using contour 2, one obtains the expression for a converging wave; i .e., the point at x' acts as a sink . The 1m k k=K
Re k
-~ontour ~-...,-Fig. 4.26 Alternate contours for source function for string.
functions represented are continuous but have discontinuous first derivatives. By proper manipulation it is possible for each type of contour to express Gk(x - x') directly in terms of the step function u(x/ - x) . For example, for contour 1 Gk
= -1- { eiK(z-z') 47rK
1 o
e-i('-K)(",'-Z) dk k - K
+ e- iK(",-",')
1 c
e-i(k+K) (",'-z) } dk k+K
The first of these integrals has a singularity at k = K only, so that no detour around k = -K is necessary. Similarly in the second integral only the detour around k = - K is required in the contour. Comparing with (4.5.16) and Fig. 4.25 one obtains GJc(x - x') = (i/2K)le iK(z-z')[1 - u(x ' - x)]
+ e-iK(z-z')u(x'
- x) I
which agrees with (4.5.21). Integrals Related to the Error Function. So far we have dealt with integrals which could be evaluated, and thus expressed directly, in terms of the elementary transcendental functions. Let us now consider a case in which the integrals cannot be so expressed . An integral solution of the differential equation (d?,J;~/dz2) - 2z(d>/l~/dz) + 2A>/I~ = 0 is 1
~ = 27ri
(e- I '+ 2I Z
[c f+l dt
(4.5.23)
§4.5]
Calculus of Residues,' Gamma and Elliptic Functions
417
where the contour is shown in Fig. 4.27a. We have chosen the branch line for the integrand to be the positive real axis . We now evaluate the integral for various special values of Xand for z small and z large. First, for X an integer n, the origin is no longer a branch point, so that the contour may be deformed into a circle about the origin. Then by Eq. (4.3.1)
\bn = -\ n. \b n
or
[d: dt
(e-t'+2/Z)] 1-0
=..!.. n! eZ' [!!!:.... dzn e-
, X=
z, ] •
~
(4.5.24)
The resulting polynomials are proportional to the Hermitian polynomials (see table at the end of Imt Chap. 6). For X < 0, the contour integral Contour C may be replaced by a real integral Re t B as follows : The contour may be broken up into a line integral from 00 to 0, a circle about zero, and an (0) Imt integral from to 00. For X < 0, the value of the integral about the Re t circle approaches zero as the radius approaches zero, so that we need (b) consider only the two integrals exFig. 4.27 Contours for error integral and tending from to 00 . By virtue gamma fun ction. of the branch point at t = 0, we must specify what branch of the multivalued integrand is being considered. This is done by choosing a phase for r(Hl) . Since X is arbitrary, the phase (see page 398) may be chosen to suit our convenience, which is best served by making it zero at point A . Then the integral from infinity to A is 1 e- I'+2tz • _ . t>'-jol dt, where t is real 21I"t .,
°
°
fO
The phase of the integrand at D is obtained from that at A by rotating counterclockwise from A to D by 211". Thus the phase is -2ri(X + 1). The integral from D to infinity is thus e-2ri(>'+I)
271"i
{., e- t'+2tz
}0
•
tM=l dt,
t real
We thus obtain, for the integral (4.5.23), when X < 0, the new form
\b>.
1
= - e- r i \ }.+ I ) sin 1I"(X 11"
e- '+ 2tz + 1) ~., ----'-----+1 dt 0 t~ t
(4.5.25)
Since the infinite integral is real and does not vanish (or have a pole) when Xis a negative integer, the function \b>. must vanish for these values
418
Functions of a Complex Variable
[CR. 4
of A. This is not surprising, since then the value of the integral along the upper half of the contour is equal and opposite to the value along the lower half of the contour. We also see that it would have been more satisfactory (perhaps!) for us to have chosen the integrand to be real at point B on the contour, for then the exponential factor would not have remained in Eq. (4.5.25) . To investigate the behavior of ifix as z --> 0, we expand e21Z in a power series :
.
ifix
=
\ ' (2z) n {~ '-' n! 21l"t
n=O
r e-t'tn-
Jc
X- I
dt}
(4.5.26 )
The integral may be expressed in terms of a gamma function. (The integral representation required is derived in the following section.) We shall therefore relegate the evaluation of (4.5.26) to a problem and shall content ourselves with noting one interesting point. If A is an integer p > 0, the contour integral will vanish for all n > p. Thus ifip is a finite polynomial of order p. This is, of course, confirmed by further examination of the explicit formula for these cases, given in Eq. (4.5.24) . Finally we investigate the values of ifix for z large. (The advantage of an integral representation of ifix should be by now manifest. It contains all the information, all the details of the function's behavior in its most tractable form . It enables us to establish a correspondence between its values for z --> 0 and those for z --> 00.) For Izl--> 00 and for Re z < 0 the simplest procedure involves the substitution 2tlzl = u. - (21z1)X ifix - -2-'1l"t
1
e- (u/ 2Iz
c
U
j) 'e- u
HI
du
We now expand the exponential in a power series : .1., 't' A
-1-1
= (2Izl)X \ ' (-l)n u2n-X-Ie-u du: (21zl) 2n 21l"i c ' '-'
Re z
> 0 (4527) . .
Again the integral may be expressed in terms of gamma fun ctions [see Eq. (4.5.36)]. For very large values of z, only the first term is important and ifix --> Alzlx. When Re z > 0 this pro cedure must be changed. Let u = -2tlzl - (2Izl? ifix - 21l"i
1 c
e- (u/2!zj)'e- U (-u)HI du
The path of integration is obtained by performing the same transformation for the path, with the result illustrated in Fig. 4.27b. We may now expand e-(u/2!zj)': ifix = (2lzl)x
2:
(2Izl)-2n 2~
1-
e- U ( _u)n->'-l du
(4.5.28)
,
§4.5]
Calculus of Residues; Gamma and Elliptic Functions
419
',As we shall see in the section immediately following, the contour integral is just a gamma function [see Eq. (4.5.36)]. Gamma Functions. Earlier [Eq. (4.3.14)] the gamma function was -defined by the infinite integral I'(s)
=
t:
e-lt,-l dt ; I'(n)
=
(n - I)!
(4.5.29)
In order that this integral converge , the real part of z must be positive. When z is not an integer, z = 0 is a branch point for the integrand. We take the positive real axis for the branch line. A number of equivalent forms obtained by transformation will prove useful : r(z) = 2 r(z)
=
10" e-
2'10
1
I't 2H
dt;
[In (l/t)]'-1 dt ;
Re z
>0
Re z
>0
(4.5.30)
As discussed in Sec. 4.3, it is possible to extend definition (4.5.29) to negative values of real z by means of the recurrence relation zr(z) = I'(z
+ 1)
(4.5.31)
Indeed this relation immediately yields considerable information about the singularities of T'(z} . Since the integral converges for Re z > 0, T'(z) is always finite for Re z > O. Moreover since the derivative r '(z) may be evaluated directly by differentiating under the integral sign, it follows that I'(s) is analytic for Re z > O. To explore the behavior of r(z) for Re z < 0, we find an integer n, for any given value of z, which will be sufficiently large so that Re (z + n + 1) > O. Let n be the smallest possible integer of this sort. Then, of course,
+ n)(z + nr~ ~(; : ~) _ so that I'(z) is now defined in terms of I'(z + n) r(z) = (z
2) .. . (z)
(4.5.32)
where Re (z + n) > O. We note that r(z) is generally finite, with a defined derivative for z, except when z is zero or a negative integer. Near these points we can set z = - n + E, where lEI «1 , and use Eq. (4.5.32) to give us r( -n + E) = (-l)nr(l + E) (n - E)(n - E - 1) . .. (1 - E)E which has a simple pole at E ---+ O. Consequently the function I'(a) is analytic over the finite part of the z plane, with the exception of the points z = 0, -1, -2, -3, . .. , where it has simple poles. The residue at the pole at z = -n is (_l)n /n! [since r(I) = 1]. The regular spacing of these poles is reminiscent of the regular spacing of the poles of the trigonometric functions csc (7l'z) or cot (7l'z) . How-
420
Functions oj a Complex Variable
[cH.4
ever, the latter also has poles along the positive real axis for the positive integers. We may, of course, find a function which has poles there, namely, I'( -z) . Thus the product r(z)r(1 - z) has poles at z = n, where n is an integer (positive or negative). It has poles nowhere else. Thus sin(n"z)r(z)r(l - z) is analytic everywhere in the finite z plane. [We cannot use cot (1l"z) in place of csc(1l"z) because of its zeros at z = t(2p + 1)11", P integral.] In fact we y shall now show that r(z)r(l - z) -4
-3
-2
-I
= 11" csc(1l"z) (4.5.33)
o
r(a)r(l - a) = 4
fa" J"
e- \z'+II')x 2,,-l y-(2,,-I)
We shift now to polar coordinat es x = r cos 0, y rlr/2
r(a)r(1 - a) = 4}0
t- re:" dr
(cot 0)2"-1 dO}o
dx dy
= r sin 0 so that =
(..-/2
2}0
(cot 0)2"-1 dO
To evaluate the final integral, use the substitution cot 0 = s, then r(a)r(1 - a) = 2
fa" (s2-1)/(1 + S2) ds.
It will be recognized that
for 0 < a < 1, this integral is in a form which falls under formula (4.5.5). Then r(a)r(1 - a) = 2 csc(21l"a) {~ residues of [( -s)2,,-IJ![1 + S2] at s = ±i} if 0 < a < 1. A little manipulation yields Eq. (4.5.33). As mentioned, analytic continuation will extend the range of the formula. We shall employ this result to establish two results. First by letting z = t, we find r 2 (t) = 11" or ret) = -V;;:. Second, we shall establish that [r(z)]-l is an integral junction. Since the singularities of r(z) are poles, it is only necessary to show that r(z) has no zeros in the finite complex plane. This is a consequence of Eq. (4.5.33), for if r(z) is zero at some point, I'(I - z) must be infinite there. However, the infinities of I'(I - z) are known and r(z) is not zero at these points, so it cannot be zero anywhere. Contour Integrals for Gamma Functions. We now have a good idea of the general behavior of I'(e), and we may now proceed to find closed expressions for it which are valid over a greater range in z than (4.5.29). One such representation is obtained when the direct solution of the difference equation (4.5.31) is attempted. Care must be exercised, however, to choose a solution which reduces to (4.5.29) when Re (z) > 0,
§4.5]
Calculus of Residues; Gamma and Elliptic Functions
421
for any solution of (4.5.31) when multiplied by a periodic function with period 1 yields another solution of (4.5.31). The form of (4.5.29) suggests that it may be rewritten as a contour integral r(z) =
fc v(t)t
z-
I dt
where the contour C and the function vet) are still to be determined. Substituting in (4.5.31) yields
i
v(t)t z dt =
i
v(t)zt z- I dt =
f
vet) :t (tz) dt
Integrating the second of these by parts yields
i
[V(t)
+ ~~] tz dt =
[v(tW]
where the expression on the right is to be evaluated at the ends of the contour. We shall pick the contour so that [v(t)tz] vanishes. Hence v may be taken as the solution of dv vet) + dt = 0;
v = (constantjer"
r(z) = A fe (e- I)( -t) z-I dt
so that
(4.5.34)
where A is a constant. The contour chosen is illustrated in Fig. 4.27a. Next we evaluate the integral for Re z > 0, and then choose A so that Eqs. (4.5.34) and (4.5.29) are identical. Choose B as the point at which the phase of (-t) z-I is zero. Then for Re z > 0 r(z) = A =
{f: e- 1e-ri(z-1) (t)z-I dt + fo" e- 1e"i(z- l)tz-
2iA sin [1r(z - 1)]
fa .. e-1t
z- 1
1
dt}
dt
Therefore A = -1 /(2i sin 1rz) and T'(z) = - 2' .1 ( e-I(-t)z-ldt t sin 1rZ [o
(4.5.35)
This representation is valid for all z. Combining it with the relation (4.5.33) yields another integral representation of I'(z} :
- 1 = - - 1 . ~ e- I ( -t)-z dt I'(z)
21rt
c
(4.5.36)
The behavior of I'(z) when z is an integer may be found directly from Eq. (4.5.35) or (4.5.36). Infinite Product Representation for Gamma Functions. Another useful representation of I'(s) may be obtained by applying the formula
422
[CR. 4
Functions of a Complex Variable
developed for the product representation of an entire function. The function [fez + 1)]-1 is an entire function, with zeros at z = -1, -2, Using Eq. (4.3.8), one obtains
_1 fez)
=
n(1 .,
zev
=)n «v»
+
(4.5.37)
n=!
The constant 'Y = - f'(l)jf(1) equals 0.5772157 and is known as the Euler-Mascheroni constant. The value of 'Y may be obtained by placing z = 1 and taking logarithms of both sides of Eq. (4.5.37), yielding .,
2: [~ -
=
'Y
log ( 1
n=!
+ ~) ]
.(4.5.38)
This series may be easily evaluated. Other useful forms for 'Y may be directly derived from (4.5.38): 'Y
= lim
M-+.,
(1 + ~ + ~ + . . . + ~ - M) M
and
'Y
= lim {'\' M-+.,
t : e-
nq
~)o
dq _
n=!
or
'Y
=
t : e-
)0 .
In
23M
q [
q -
q
e-
-~] dq
1 _
1 - e
t: e-
)0
d q}
(4.5.40)
q
q
M q
(4.5.39)
Derivatives of the Gamma Function. Of considerable importance to subsequent applications is the logarithmic derivative of the gamma function 1/;1(Z) = dIn fez) f'(z) (4.5.41 ) dz = fez) From (4.5.37) ., _ 'Y _
~+ z
When z is an integer N,
'\' [~ __ 1 ]
~ n
n=!
n+z
(4.5.42)
N-!
1/;1(N) =
-'Y
+ 2:~;
1/;1(1)
=
-'Y
(4.5.43)
n~!
The derivative of 1/;1 is (4.5.44)
423
Calculus of Residues; Gamma and Elliptic Functions
§4.5]
In general (4.5.45) These functions have been tabulated for real values of z, permitting the ready evaluation of sums, each term of which is a rational function of the summation index. For example, the sum
.
S _ ~
1
- n=O .4 (n + 1)2(n + a)2
may be written
..
S 1 - (a - IF
so that
~
.4 n=O
S = (a
.
{[
~
1
+ IF
(n
IF {[¥t2(1)
+
(n
] ] __ 2 [_1 _ _ 1 ]} + a)2 a-I n + 1 n + a
+ ¥t2(a)]
~ 1 [-¥ti(l) + Vti(a)]}
- a
The polygamma functions, as the ¥tp functions are called, satisfy a simple recursion formula with respect to their argument.
.
Vtp(z
+ 1) =
(-I)pp!
2:
(n
+ f + z)p =
.
(-I)pp!
n~O
so that
¥tp(z
+ 1) = ¥tp(z)
2:
(n
~ z)p
n=i
- [(-I)pp!l/zP
(4.5.46)
Integral representations for the polygamma function may be found by employing the device which led to Eq. (4.5.40) for 'Y. From Eq. (4.5.42) one has ¥ti(Z) =
-'Y
+ ~ ..
[1 ~-:-q 1~:;_q] -
dq
Introducing Eq. (4.5.40) for 'Y, we find that (4.5.47) Differentiating with respect to z yields the other polygamma functions : ¥tp(z) = (-I) p
Jot : r-ie-~q - e dq q
(4.5.48)
These formulas may be used to obtain the behavior of ¥tp(z) as z ~ IX); thence, by integration with respect to z of ¥ti = [d In f(z)l/dz and, finally , by another integration, the behavior of fez) for large z, The determination of the consta nt s of integration is rather involved so that
424
[CH. 4
Functions of a Complex Variable
we shall postpone discussion of the behavior of I'(s) for large values of z until the next section. For reference, the behavior is given by the Stirling approximation In [r(z)] - - - > -z z-> eo
+ (z -
!) In z + !
In (211")
+ O(l /z)
+ ...);
z -->
(4.5.49)
when z is in the first quadrant, or r(z)
-->
e-zz.-! .y2; (1
+
(l/12z)
00
(4.5.50)
The Duplication Formula. Another application of the infinite product (4.5.37) is in the derivation of the duplication formula 2 2 Z- 1r (z)r (z
+!)
y'; r(2z)
=
(4.5.51)
To prove this consider the ratio of I'(s) to r(2z);
Il r(z) r(2z)
or
r(z) = 2e'Y z(1 r(2z)
=
2
+ 2z)e- 2z
[1 + (2Z/n)]e-2./1O
'Yz ::.10=_1=-
e
_
..
Il
.
[1
+ (z/n)]e-
10=1
n(1 +
~).e-2Z/(2P+l) 2p
p=1
= 2e'l"~1 + 2z)e-2z
z n /
n [1 +
(z
+1
+ i)/p]e-(z+i)/p
"-p=_1=---. .
Il
_
[1
+ (1/2p)]e-ip
p=1
p=1
Utilizing the product representation of the gamma function (4.5.37), we may write 2Z r(z) 2r(i)e[~ r(2z) = r(z + -!) exp z P- p + ! (4.5.52)
..
4
(1
l)J
p-1
To evaluate the exponential, let z = 1 in (4.5.52).
..
exp
[2: 0- ~ !) J = ~ p
Then one obtains (4.5.53)
p-l
Introducing (4.5.53) into (4.5.52) yields the duplication formula (4.5.51).
Calculus oj Residues; Gamma and Elliptic Functions
§4.5]
425
Beta Functions. We conclude our discussion of the gamma function with an investigation of the beta function defined as
B(p,q)
=
101 tP-1(1 -
t)q-l dt
(4.5.54)
or alternately by B(p q) ,
=
~
0
..
tp -
(1
1
dt· + t)p+q'
B(p q) ,
= 2
~"/2 0
sin 2 p-l 8 COS 2 q- 1 8 d8 (4.5.55)
We shall content ourselves with establishing the formula B(p q) ,
Using Eq. (4.5 .30) r(p)r(q) = 4'
= r(p)r(q) r(p
(4.5.56)
+ q)
t: e-'"'X2p- dx fo" e1
1I'y2q-l
dy
(4.5.57)
and transforming to polar coordinat es, x = r cos 8, y = r sin 0, we find r(p)r(q) = 4
t: e- 'r2P+2 q-l dr 10"/2 sin r
2p - 1
8 COS 2q-l 0 d8
Employing (4.3.30) again , and (4.5.55) , we eventually obtain Eq. (4.5.56). Periodic Functions. There are many times when we shall encounter or need functions with the property of periodicity, of repeating themselves over and over again. Mathematically this is expressed by saying that, for the periodic function f(z) , there is a certain complex number a such that f(z
+ a)
= f(z)
for all values of z for which f is analytic. The constant a is called a period of f . After z goes from Zo to Zo + a, the fun ction then starts in again and repeats its former behavior over the next period, from Zo + a to Zo + 2a. Of course, if a is a period for t. 2a is also a period and any integer (positive or negative) times a is likewise. Sometimes -ja is a period, or ia, etc. But the subdivision cannot go on forever, for a bit of the simple but close reasoning one uses in modern point-set theory will persuade one that there cannot be an infinitesimally small period (unless the periodic function is a constant!) In fact , if one arranges all the periods of a given periodic function in order, one soon finds that they all can be represented as integral multiples of a fundamental period r, characteristic of the fun ction f and such that the equation
+ r)
= f(z) (4.5.58)' holds, but no similar equation holds for f for any period smaller than r, f(z
The functions sin z, tan z are periodic fun ctions with fundamental period 211"; the function ez is a periodic function with fundamental period
426
Functions of a Complex Variable
[CR. 4
2m ; and so on. These functions may be characterized by series or by infinite product expressions, which display the fact that their poles (or zeros) are equally spaced along a line (or lines ) in the z plane, with spacing equal to T , such as the following 1 1 -=-+ sin z z
l:.
2( -1 )nz Z2 -
.
(4.5.59)
(n1l")2'
n= l
which have been discussed earlier. From these expressions, by algebraic manipulation, one can demonstrate the periodicity of the sine function . We can, however, characterize the functions by their differential equations. For instance, the equation (dy jdz)2 = 1 - y2
' (4.5.60)
or the second-order equation, formed from this by differentiation, (4.5.61) may be used to charac terize the periodic fun ction y = sin z. To show that the function is periodic we use Eq. (4.5.61) to follow the behavior of y as z is increased from zero along the real axis. We start off by saying that y is the solution of Eq. (4.5.60) which is zero at z = 0; the equation then shows that its slope is unity at z = 0 (it could be ± 1, but we choose + 1). As z is increased, y increases until y = 1, when dyjdx is zero, but d 2yjd x 2 is -1, as Eq. (4.5.61) shows . Consequently the maximum value of y along the real axis is + 1. As z is increased further, y diminishes, dyjdz gets more and more negative, until, when y is zero again, dy jdz is -1. Increasing z still further results in an excursion of y to -1 and then back on ce more to zero, when it has a slope of +1 , as at the beginning. At this value of z, y has gone through a full cycle and is ready to start again. We note that due to the symmet ry of the equation this full period is divided into four equal parts, the first covering the distance over which y goes from zero t o 1, the second over which y goes from 1 to 0, and so on. Going back to Eq. (4.5.60) we see that the solution whi ch goes to zero at z = 0 can be formally written Z
=
~o
1I
du
Vf=U2
. 1y ; that at iis, z = sm"
This gives us an inverse sort .of solution, that is, z as a function of y, but it enables us to compute the period easily. For fr om the discussion of the previous paragraph we see that one-quarter of the total period T is
iT =
t .----,=d=u==
Jo VI - u
2
§4.5]
Calculus of Residues; Gamma and Elliptic Functions
427
Calculation of this integral shows it to be equal to p so that the period of the function y we are discussing is T = 211". In .books on analysis the periodic properties of the trigonometric function are discussed in more detail and with much more rigor than we have space for here, particularly sin ce we are using this only as an introduction to a more general type of periodic fun ction. For we might ask whether there exist functions of the complex variable z which have two periods, al and a2, such that y(z al) = y(z) and also y(z a2) = y(z) . This pair of statements would not introduce anything particularly new if al and a2 were in the same direction in the complex plane (i.e., if al/a2 were a real number); for with simply periodic functions, if al is a period, 2al or 3al is also a period. Of course we might ask whether Tl and T2 could be in the same direction but be incommensurate. The same close reasoning which states that there ca nnot be periods of infinitesimal size (except when the function is a constant ) shows that this cannot be. [If T 2 were equal to AT1, where A is real but not a rational fraction, then among the set (m + An)Tl we would find values as close together as we wish and as small as we wish, which is impossible.] But if al is in a different direction in the complex plane (i.e., if al/a2 is not a real number) then we have a situation specifically different from and more complicated than for the simply periodic functions . For one thing, instead of a one-dimensional sequence of periods, there will be a two-dimensional lattice of parallelograms, with the function repeating, in each parallelogram, Its behavior in every other parallelogram. The smallest unit within which the function goes through all its behavior is called the unit cell for the function; each side of a unit cell is one of the fundamental periods for the fun ction, Tl or T2 . The behavior of f in the (m,n)th cell, at the point z mTl nr2 (m, n = 0, ± 1, ±2, .. .) is the same as the behavior of f in the central cell, at the point z. The points z and z mTl nr2 are ca lled congruent points for f . We might ask if there are functions of the complex variable z which have more than two independent periods. The reasoning which proves that this is impossible is similar to that which proves that there cannot be two independent periods having a real ratio and depends basically on the fact that the complex plane is two-dimensional. Consequently, we can have singly and doubly periodic functions of z, but not triply or n-tuply periodic ones. Fundamental Properties of Doubly Periodic Functions. Before we ever set down a formula for a doubly periodic function, we can use contour integration to prove certain of their properties. We make the contours go around the boundary of a unit cell, so that the periodicity ensures that the integral is zero. The doubly periodic function j(z) , with fundamental periods Tl and T2 , may have poles and zeros. If it
+
+
+
+
+
+
428
Functions of a Complex Variable
[CR. 4
does, it must have the same number in each cell (we assume that we have drawn our unit cell so none of the poles or zeros is on the boundary). If it has branch points, they must come in pairs in each cell, so no branch lines need cut the cell boundaries. Suppose that we concentrate on those doubly periodic functions f(z) which have no branch points, only poles and zeros of integral order [where the function is of the form (z - Zi)n'g(Z) where ni is an integer (positive or negative) the order of the zero, and g is analytic at z = z;] . Such functions are called elliptic functions. According to page 408, the contour integral (I /21l"i) §f(z) dz for such an f is equal to the sum of the residues of f inside the closed contour. But if the contour is the boundary of a unit cell, such a contour integral must be zero, so we have T~ sum .of the residues of all the poles of an elliptic funct ion, in one} unit cell, ts zero.
(4.5.62)
If the elliptic function has only simple poles, it cannot therefore have one simple pole per cell; it must have at least two, so the residues can cancel out. The elliptic function could , of course, have one pole of order two per cell, if the residue of this double pole were zero. (Of course the function may have no poles per cell, in which case it is a constant. Why?) We also see that, if f(z) is an elliptic function, f + c, Ilf, 1', fn, 1'lf are all elliptic functions. If f(z) is an elliptic function, then [Ilf(z)] is likewise an elliptic function, and a considerat ion of the integral §(llf) dz around the boundary of a unit cell shows that the sum of the residues of all the poles of [Ilf(z)] at all the zeros of f(z) in one unit cell is zero. Therefore simple zeros of f must also come in pairs . Since, by definition, an elliptic function cannot have an essential singularity in the finite part of the complex plane, it cannot have an infinite number of poles in a unit cell, for this would result in an essential singularity. Consequently an elliptic function can have only a finite number of poles in each unit cell and, correspondingly, only a finite number of zeros. We next look at the contour integral §[f'(z) lf(z)] dz, where f' = df I dz and f has a finite number of simple poles and zeros inside the contour. Near the ith pole, which is of the nith order, the function I :: (z - Zi)-n'gi(Z), where gi is analytic (and not zero) at z = Zi. Therefore the function f' lf has a simple pole at Zi, with residue -ni. Near the jth zero, which is of the mjth order, the function f -7 (z - zj)m;giz), where gj is analytic (and not zero) at z = Zj. Therefore the function f'lf has a simple pole at Zj, with residue + mj. Therefore the contour integral 1 ,r..1'(z) (4.5 .63) 21l"i 'f f(z) dz =
.§4.5]
Calculus of Residues; Gamma and Elliptic Functions
429
In other words, this contour integral equals the sum of the orders of all the zeros, minus the sum of the orders of all the poles of f inside the contour. If the poles and zeros are all first order, then this integral equals the difference between the numbers of zeros and the numbers of poles. Applying this to the contour around the unit cell for an elliptic function , we see that: The sum of the orders of all the poles in one cell of an elliptic fun ction equals the sum of the orders of all the zeros in the cell. If the poles and zeros are all simple, then the number of poles equals the numb er of zeros.
I
(4.5.64)
By considering the integral .f[f' / (f - C)] dz around the boundary of a unit cell we see that The numb er of roots of the equation f(z) - C in a unit cell, where f is an elli ptic fun ction and C is any 'constant, is equal to the sum of the orders of all the poles of f in a un it cell.
I
(4.5.65)
The sum of the order of the poles of f in a unit cell (which equals the number of poles in a unit cell if all the poles are first-order) is called the order of the ellipti c function. By (4.5.62) we have seen that there cannot be an elliptic function of first order. An elliptic function of zero order is a constant, and the simplest elliptic function with nontrivial properties is one of second order. Elliptic Functions of Second Order. There is an elliptic function with one second-order pole and two zeros per unit cell; it is called the Weierstrass elliptic function. Of greater use, however, are the Jacobi elliptic functions , which have two simple poles and two simple zeros in each cell. Therefore, by (4.5.65), each of these functions takes on all possible values just twice in each cell. We can set up such functions by means of series or infinite products expressions or by means of inverse integrals, by analogy with the trigonometric functions. For the first such function we choose to have a sequence of zeros along the real axis (like the sine function), each a distance irJ apart, where TJ (real) is the first period. We shall take the second period, T2, to be pure imaginary and have a simple pole a distance ihl above each zero. Above these again, along the top of the first row of cells, is another row of zeros. By analogy with the series of Eq. (4.5.59) we could set up a series which would account for all the zeros of this function. Suppose we assume that it starts out with unit slope at the origin, just as does the sine function; in fact let us assume that it behaves like a sine function at each of its zeros and that, when T2 ~ co and the function becomes singly periodic, it reduces directly to sin z. To emphasize this property we will call the function sn z. The poles of (l /sn z) (i.e., the zeros of sn) along the real axis are obtained by putting in a term (27r/TJ) csc (27rZ/TJ) in the series. This
430
Functions of a Complex Variable
[cH.4
will have the desired behavior near each of the zeros of sn along the real axis and will reduce to (l /sin z) if Tl is made to become 211" when T2 -> 00 . The next row of zeros of sn Z is at the points Z = T2 ± -!mTl, where T2 is pure imaginary. Here we should have another row of poles of (l /sn z). A term of the sort cos (211"TdTl) 1 sin (211"Z/Tl) - sin (211"T dTl) - sin (211"~/Tl) - 2 tan (211"T2/Tl) sin- (1I"~/Tl) where ~ = Z - T2, has poles which go to infinity at ~ = -!(nTl) just as does the function [l /sin (211"Z/Tl)] at Z = -!nTI. Combining this with a term giving poles at the points Z = -T2 ± mTl, we have 2 cos (211"TdTl) sin (211"Z /Tl) 4. sin (211"Z/Tl) cos (211"Tl/T2) sin- (211"Z/Tl ) - sin" (211"T dTl) = co~ (411"TdTl) - cos (411"Z/T2) Sin ce there are rows of zeros of sn Z at iy = ± nT2, we finally obtain a series .,
.J:....
sn Z
= 211" {esc (211"Z) Tl
Tl
+4 ~ \ ' sin(~211"Z/?\ cos (211"~~I/TY )} cos 1I"nT2 Tl - cos 1I"Z T2
(4.5.66)
n=l
which has the right distribution of zeros for sn z over the whole complex plane. It is not too difficult to show that the zeros of this series (the poles of sn) are midway between its poles (i .e., at z = imTI + (n + -!)T2), but it is much more difficult to calculate the residues at these poles of sn. These elliptic functions may also be approached from another direction. The series ., ., F(z) = e..in''Y+2inu = 1 + 2 qn. cos (2nu) (4.5.67)
l
l
n=-
n
00
ee
I
(where Irn l' > 0 and where q = eri'Y) is periodic in u with period 11". It is convergent as long as the imaginary part of l' is positive, so that Iql < 1. Interestingly enough the series is also pseudop eriodic/ iri u, with period 11"1', for ., ., F(u
l
+ 11"1')
n= -
l
e(..i'Y )(n'+2n)+2inu = e: ..i'Y- 2iu
n=-
aD
p(.. i'Y)( n + I) ' +2i(n + l )u 00
= q- 1e-2iuF(u) There are four such series, called the theta functions:
l
00
tJ 1(u,q)
= 2
(_l)nq(n+!)' sin [(2n
n=O
.,
tJ2(u,q) = 2
l
n=O
tJ1(u
+ l)u] ;
+ 1I"'Y ,q) =
q(n+!)' cos [(2n
tJ 1(u
+ 1I",q)
-NtJ 1(u,q)
+ l)u];
tJ2(u
+ 1I",q)
-tJ1(u,q) ; (4.5.68) -tJ 2(u,q) ;
§4.5]
Calculus of Residues; Gamma and Elliptic Functions t?-2(U
.,
t?-3(U,q)
Lr: cos
= 1+2
n -l
.,
t?-.(u,q) = 1
+ 'Tr'Y,q)
t?-3(U
= Nt?-2(U,q)
(2nu);
iJ3(u
+ 'Tr,q)
= t?-3(U,q) ;
+ 'Tr'Y,q) = Nt?-3(U,q)
L (_l)nqn. cos (2nu);
+2
431
(4.5.68)
t?-.(u
+ 'Tr,q)
=
t?-.(u,q);
n=l
t?-.(u + 'Tr'Y,q) = -Nt?-.(u,q) t?-2(U,q) = t?-I(U + j,r ,q) ; t?-3(U,q) = t?-.(u + j,r,q); etc . where q = e.-ioy and where N = q-I e-2iU: These are, of course, not true elliptic functions because of the presence of the factor N. By suitable manipulation of these series we can show that the quantities t?-Hu) + at?-~(u) d t?-Hu) + M~(u) t?-~(u) an t?-~(u)
are true elliptic functions and that, by suitable choice of a and b, we can arrange it so that they have at most only one simple pole in the unit cell, of side 'Tr and 'Tr'Y. But, from Eq. (4.5.62) , this means that these ratios, for these particular values of a and b, will be constants. Therefore it is possible to express t?-~ or t?-~ in terms of t?-f and t?-i. The constants can be determined by setting u = and jTr; and we finally obtain
°
t?-~(u)t?-~(O)
=
t?-~(u)t?-~(O)
t?-Hu)t?-~(O) = t?-~(u)t?-~(O)
- t?-Hu)t?-~(O) t?-i(u)t?-i(O) = - t?-~(u)t?-~(O) t?-Hu)t?-i(O) =
t?-~(u)t?-~(O)
-
t?-Hu)t?-~(O)
(4.5.69) t?-~(u)t?-i(O)
-
t?-~(u)t?-~(O)
where the second argument, q, has been omitted for brevity. From the theta functions, as we have seen, we can build true elliptic functions, by taking proper ratios so that the multiplicative factor drops out . For instance, the function t?-3(0,q) t?-1(U,q) z sn z = t?-2(0,q) t?-.(u,q); u = [t?-3(0,q)]2
(4.5.70)
has the same distribution of zeros and the same behavior at each zero that the function, defined in Eq. (4.5.66) has, provided 'Tr[t?-3(0,q)]2 is equal to TI and 'Tr'Y[t?-3(0,q)]2 is equal to T2, the second period . Moreover, treatises on elliptic functions show that from the series of Eq. (4.5.68), we can calculate, with a great deal of difficulty, a differential equation for the theta functions and finally for sn z. We find, for instance, that
!l- [t?-I(U,q)] = [t?-.(O q)]2 t?-2(U,q) t?-3(U,q) 't?-.(u,q) t?-.(u,q) du t?-.(u,q) and that if." = t?-l(U,q)/t?-.(u,q), we have, after further manipulation, (d.,,/du)2
=
[t?-~(O,q)
- .,,2t?-i(0,q)j[t?-HO,q) - .,,2t?-HO,q)]
432
Functions oj a Complex Variable
[cH.4
Finally, if we set y = [t1 a(0,q)!t12(0,q)]7] and z = u[t1 a(O,q)]2, we have for y (dyldz)2
= (1 - y2)(1 - k 2y2)
(4.5.71)
where Vk = t1 2(0,q)!t1a(0,q). All the properties of the elliptic function sn z = y may then be found in terms of the series (4.5.68), which converge rapidly. However, to discuss some of the properties it is useful to consider Eq. (4.5.71) as the basic definition. . Integral Representations for Elliptic Functions. This brings us to still a third way of defining elliptic fun ctions of second order, which is the most useful of all (except, perhaps, for numerical computation of tables) . We see that a formal solution of Eq. (4.5.71) is z
=
~v y(l
_
U~~1
_ k 2u 2 )
1
= sn- y
(4.5.72)
which defines the function sn- 1 y, inverse to the elliptic function sn z. Referring to the discussion of sin- 1 y, on page 426, we see that the present function y = sn z is periodic in z with period Tl = 4K, where
i'
K -
- )0 - 1 -
"2'
du _ y(1 - u 2)(l - k 2u 2) -
I"
)0
dtp y1 '- k 2 sin? tp
e yV(l _ dvv)(l _ vk2) -_ 2' F(1 .111Ik 1f
)0
(4 73) .5.
2)
"2' ,"2'
according to Eq. (5.3.16). As a matter of fact we can write sn (z + 2K) = - sn z. Equations (4.5.71) and (4.5.72) show that, for real values of z, sn z is never larger than unity. Examination of the series expansion for the integral near y = shows that sn z ---+ z - i(l + k2)za + . . . , so that sn is an odd function of z and the residue of the simple pole of (l/sn z), at z = 0, is 1. Suppose we next investigate the behavior of sn z near the point z = K, where it has the value unity. Going along the real axis away from z = K reduces the value of sn, so going at right angles to the real axis will increase the value of sn (why?) . Going upward from the real axis gives us
°
z = sn- 1 y = K
+ iv ;
v =
(v )1 y(u 2-
du ; 1)(1 - k 2u 2)
y = sn (K
+ iv)
Analysis of the equation for v shows that y is periodic in v with period 2K', where K ' --
1/k
1 1
y(u 2
_
d U _1f F (1.111Ik' 2) 1)(1 _ k2u2) - 2' "2' ,"2'
(4574) • •
where k' 2 = 1 - k 2. Consequently sn z is periodic in z with period = 2iK', the value of sn K being unity and the value of sn (K + iK') being 11k, and so on. Finally, if we make y imaginary, z is imaginary, and we can have y go to infinity for z finite. This means that there is a pole of sn z = y
T2
§4.5]
Calculus of Residues; Gamma and Elliptic Functions
433
on the imaginary axis, at the point ia, where
(00 a = )0
elk
dw
+ +
k 2w 2) = ) 1
w 2) (1
V( l
u =
v (u2
~
du 1) (1 _ k 2u 2) = K';
[/++k~;2r
As a result of t his type of exploration we find th at t he function sn z is an elliptic fun ction wit h one period 7 1 = 4K which is real and one period 7 2 = 2iK' which is pure imagin ary. The fun ction has simple zeros at z = 0 and z = 2K and at corresponding pla ces in t he other unit cells. It ha s simple poles at z = iK' and 2K + i K ' (and corresponding points), with the residue at t he pole at z = i K ' being 11k and consequent ly [from (4.5.62)] the residu e at z = 2K + i K ' is - (11k). The parameter k is called the modulus of the fun ction ; whenever its value is to be emphasized, t he elliptic fun ction is written sn (z,k) . Its value in terms of the quantity q is given in connection with Eq. (4.5.71) : usually, however, k is treated as the basic parameter, in which case K and K' are found from Eqs. (4.5.73) and (4.5.74) (K ' is the same fun cti on of k' = VI - k 2 as K is of k) and t hen q = e- 1r ( K 'I K ) . We can then work back and show that the sn (z,k) defined by Eq. (4.5.72) is the sa me function as t ha t defined by Eqs. (4.5.70) and (4.5.66) . Other Jacobi elliptic fun ctions are defined in similar ways: sn (z k)
=
,
tJ 3 (0,q) tJ1(u,q). tJ2(0,q) tJ 4 (u,q) '
z =
cn (z k)
=
,
( II )0
v(l - t2dt) (1 -
k 2t2)
= sn-I (y,k)
tJ 4 (0,q) tJ 2(u,q) . tJ2(0,q) tJ4(u,q )' z
(1 = )11
v (l _
dt t2)(1 _ k2
(4.5.75)
+ k 2t2) =
I
cn- (y,k)
dn (z k) = tJ 4 (0,q) tJ3(u,q). ,
tJ3 (0,q) tJ4 (u,q) ' z=
where
i' v(l _ t2) ([2dt + k2 _
)11
u = z[tJ3(0,q)]-2 = 7rz12K;
I
1) = dn- (y, k)
q = e- 1r ( K 'I K )
and the constants K , K' are given by Eqs. (4.5.73 ) and (4.5.74). Utilizing Eqs. (4.5.69) and our previous knowledge of the function sn, we can show that k 2sn 2 (z,k)
sn 2 (z,k) + cn 2 (z,k) = 1; en (O, k) = dn (O,k) = 1 + dn ! (z,k) = 1; (dl dz) sn (z,k) = cn (z,k) dn (z,k)
(4.5.76)
Many other properties of these functions ma y now be computed . Some of t hem are listed on pag e 486, and others are dealt wit h in t he problems. We shall find use for the fun ctions in a number of cases la ter in the book .
434
Functions oj a Complex Variable
[cH.4o
4.6 Asymptotic Series; Method of Steepest Descent To study the behavior of the various functions we shall encounter, for large values of [z], it will often be convenient to expand them in inverse powers of z :
z
A 2 + . .. ] + Z2
j(z) = r,o(z) [ A o + A I
(4.6.1)
where r,o(z) is a function whose behavior for large values of z is known . The expansion for r(z) given in the preceding section [Eq. (4.5.50)] is of this type. If j(z) j r,o(z) has an essential singularity at Izl-> co, the series in Eq. (4.6.1) will diverge. Nevertheless, the series may be useful, not only for a qualitative understanding of the function, but even in its computation for large values of z. The circumstance needed to make this possible is that the difference betweenj(z) jr,o(z) and the first (n + 1) terms of the series be of the order of Ijzn+l, so that for sufficiently large z, this difference may be made quite small . More precisely, the series is said to represent j(z) j r,o(z) asymptotically, i .e., .,
j(z) ~ r,o(z)
L [Apjzp]
(4.6.2)
p=o n
if
A:]} -> 0
lim {zn [j(Z) _ \ ' r,o(z) Lt z
Izl-- .,
(4.6.3)
p=o
This equation states that, for a given n, the first n terms of the series may be made as close as may be desired to the ratio j(z) jr,o(z) by making z large enough. For each value of z and n there will be an error of the order of Ijzn+l. Since the series actually diverges, there will be an optimal number of terms of the series to be used to represent [j(z)jr,o(z)] for a given z, Associated with this there will be an unavoidable error. As z increases, the optimal number of terms increases and the error decreases. An Example. Let us make these ideas more concrete by considering a simple example, the exponential integral -Ei( -x) =
Ix" [e-tjt] dt
the asymptotic series for the exponential integral is obtained by a series of partial integrations. For example, -Ei(-x) =
xe-:r: - J",r" [e-tJ t2 dt
Asymptotic Series; Method of Steepest Descent
§4.6]
435
Continuing this procedure yields . e-Et(-x) = x
Z [
1 1 - X
+ -x2! - -x3! + . . + .:. .-( ----'1x )_nn_! ] 2
3
n
+
(-l)n+l(n
The infinite series obtained by permitting n Cauchy convergence test yields
IUn+ll =
lim n-+
GO
Un
lim n--'
0:)
[~J ~ X
~
+
I)!
f" t~
n +2
dt
diverges, for the
00
00
Note that this indicates that two successive terms become equal in magnitude for n = x, indicating that the optimum number of terms for a given x is roughly the integer nearest x. To prove that the series is asymptotic, we must show that xn+Iez(n
+
1)!( -1 )n+l
i" ~:t2 dt ~O
}z t
x.....
00
This immediately follows, since
i.. L~:~2 ]
dt
< x:+21" e-
t
dt =
:::2
The error involved in using the first n terms is less than [(n + 1) !e- z /x n + 2] , which is exactly the next term in the series. We see that, as n increases, this estimate of the error first decreases and then increases without limit. Table 4.6.1 demonstrates how this works out for -4e 4Ei( -4) = 0.82533. Table 4.6.1 n
Value of nth term
Sum including nth term
Upper bound to error
0 1 2 3 4 5 6 7 8 9
1 .00000 -0 .25000 0.12500 -0 .09375 0 .09375 -0 .11719 0 .17579 -0.31013 0 .62026 -1 .39559
1.00000 0 .75000 0.87500 0 .78125 0 .87500 0 .75781 0 .93360 0.62347 1.24373 -0 .15186
0 .25000 0.. 12500 0 .09375 0.09375 0 .11719 0 .17579 0.31013 0 .62026 1.39559
-
The exact value is approached to within 5 per cent by the values 0.87500 and 0.78125. Averaging Successive Terms. The fact that these two values are on either side of the correct one suggests that the exact value might be more closely approached if the sequence defined by the average of two successive values is employed. In the present case, this average is 0.828125,
436
[cH.4
Functions of a Complex Variable
within t per cent of the correct value. follows:
L
We formulate this notion as
n
Let
s,
=
T n+1 =
Urn ;
+2 Sn+1
Sn
m=O n+l
then
T n+ 1 =
L
Urn-I;
Un
h Un = were
=
S
n
+2 Un+l ;
+ ~Un+l 1
U_ 1 =!uO
(4.6.4)
m=O
The sequence Un does not form an asymptotic series in the narrow sense, as defined by Eq. (4.6.1). We may, however, broaden our definition to include U« as follows:
..
L Up(z)
[f(z)/cp(z)] ~
p=O
if
lim {zn
Izl- ..
[~~~
n
-2: Up(Z)]} ~
0
(4.6.5)
p=o
It is clear that the statements following (4.6.3) which apply to the asymptotic series (4.6.1) apply to the more general case defined by (4.6.5) . For Up as defined by (4.6.4) n
L p= -1
n+l
Up(z) =
L
up(z) - tU n+l(Z)
p=O
If up(z) forms an asymptotic series, i.e., n
lim { zn[u/cp)- ~ up(z)J}~O
Izl-"
/::0
then Up(z) forms an asymptotic series. The method of averaging given by (4.6.4) is particularly useful when the series is an alternating one. We illustrate this by applying such a rearrangement to the case tabulated in Table 4.6.1 (see Table 4.6.2) . There are a number of questions we must answer with regard to asymptotic series before we will feel free to manipulate them. F irst let us note that an asymptotic series is not unique. For example, the two functions [f(z) /cp(z)] and 1/I(z) = [f(z) /cp(z)] + e» have the same asymptotic expansion when Re z > o. Moreover it is clear that the asymptotic expansion for 1/;(z) will change markedly in moving from Re z > 0 to Re z < o. Examination of the asymptotic form of 1/I(z) as the phase of z = Izlei " changes would thus show a discontinuity at "J = 71"/2, 311/2. These discontinuities are only apparent
Asymptotic Series; Method of Steepest Descent
§4.6]
437
and essentially are a result of the fact that asymptotic series are not unique. Therefore in our frequent dealings with asymptotic series we shall need to keep them under close scrutiny always to be aware of the range in the phase of z for which they are valid. The apparent discontinuity in the series will manifest itself often, indeed in the chapter on differential equations, it will be referred to as the Stokes' phenomenon (see Sec. 5.3). Table 4.6.2. Summing Asymptotic Series by Averaging Terms [see Eqs. (4.6.4) .and Table 4.6.1] I
Up
Tp
0 .50000 0.37500 -0 .06250 0 .015625 0 .00000 -0 .01720 0 .02930 -0 .06717 0 .15508 0 .38767
0 .50000 0 .87500 0 .81250 0 .82813 0 .82813 0 .810913 0.84023 0 .77306 0 .92813 0 .54046
n
-1 . 0 1 2 3 4 5 6 7 8
The following general properties of these series should be noted : Asymptotic series may be added term by term; asymptotic series may also be multiplied.
If x(z)
~
..
L
(Ap/zP) and 1/I(z)
p=O
L(~:); ..
x(z)1/I(z)
~
L(~:) ;
~
p=o
then
..
L n
where
en =
n=O
Asymptotic series may be integrated:
f
x(z) dz
=
ApB
n_ p
p=O
Aoz
+ Adn z -
.
L(~:~l) p=l
On the other hand, asymptotic series may be differentiated to obtain an asymptotic expansion for the derivative function only if it is known through some
other means that the derivative function has an asymptotic expansion. Integral Representations and Asymptotic Series. We shall often have occasion to compute the asymptotic behavior of an integral representation of some function-by a procedure known alternately as the method of steepest descent and as the saddle-point method. This technique will now be discussed and applied to the gamma function. First we should emphasize that the technique is successful only
[cH.4
Functions of a Complex Variable
438
when the function under examination can be represented by an integral of a rather particular form : J(z)
= Ie ez/(l) dt
(4.6.6)
where the contour C is such that the integrand goes to zero at the ends of the contour. This form is closely related to the Laplace transform (see Sec. 5.3). We may expect many of the integrals of interest in this book to be of form (4.6.6), inasmuch as solutions of the scalar Helmholtz equation (V 2 + k 2)1/I = 0 may be represented as a general superposition of plane waves, eik •r ; that is, 1/1 = f Cik.rf(k) dn n where dn n is the differential solid angle in the direction of vector k. In addition, it is often possible to transform an integral, not of this form, into one by some simple ruse . For example, the gamma function I'(z + 1) which for Re z > -1 is represented by substitution
r
= tz,
r(z
+
1) = zz+l
fa" c-Tr dr becomes, upon Z
I ., c- lzt dt = z
zz+l
fa"
CZ O D t-t)
dt
making the
(4.6.7)
The function I'(z + l) /zz+l is therefore of form (4.6.6). Let us now examine the behavior of J(z) as Izl--+ co, for a given phase cf> of z, Large values of Izl will usually result in some very rapid fluctuations in the value of the integrand. If z is complex, or if f(t) is complex along portions of the contour C, the imaginary part of [zf(t)] will generally increase as Izi increases. Consequently, the factor exp Ii Im [zf(t)Jl will oscillate rapidly, the oscillations increasing in frequency as Izl--+ co . The existence of such oscillations makes it difficult to evaluate the integral without some further transformations because of the resultant cancellations of the integrand ; in many cases large positive values are almost completely canceled by large negative values elsewhere on the contour. To obtain the relevant residue would require a forbiddingly accurate evaluation of the integral at every point. Under these conditions it is manifestly desirable to deform the contour .so as to minimize such effects. In the following it will be assumed that the requisite deformations are possible, that, if singularities intervene, it is possible in some manner to evaluate their effect. Choosing the Contour. In general the contour taken by the integral must run through regions where the real part of zf(t) is positive and other regions where it is negative. The former regions are more important, since here the integrand is larger, and in these regions, where Re [zf(t)] is largest, it is most important to reduce oscillations. What we do then is to search for a contour along which the imaginary part of [zf(t)] is constant in the region (or regions) where its real part is largest. Thus, in the region which contributes most to the final value, the integral may
Asymptotic Series; Method of Steepest Descent
§4.6]
439
be written J
(z)
=
fa
cz/{I)
dt =
CiIm[z/{I))
fa c~.[Z/(I)) dt
(4.6.8)
Then the path, in the regions where Re [zj(t)] is least , may be chosen so that Irn [zj(t)] varies if this turns out to be necessary to complete the contour. In this way we have ensured that the oscillations of the integrand make the least trouble.
Re(t)-
Fig. 4.29 Contour plot of real part of fun ction f 00, J(z) ~ ez!(lo) v27r/ze"ij"(to)
(4.6.13)
•
In the case of the gamma function [see Eq. (4.6.7)] f(t) = In t - t. Then I'(t) = (l/t) - 1, so that to = 1. We find that f(t o) = -1 and f"(to) = -1. The variable t runs from 0 to + 00. Transformation (4.6.12) changes to T = (t - l)ei~/2 so that the upper limit for T is + 00 . Equation (4.6.13) applies, and I'(s
+
1) ~ y'2; zZ+!e- z z-> co
in agreement with the first term of the asymptotic expansion for I'(s) given in Eq. (4.5.50) . See also the discussion of Eq. (5.3.77), where another example is discussed. The Rest of the Series. Equation (4.6.13) is the leading term in an asymptotic series . We shall now proceed to generalize the above discussion so as to obtain the next terms in the series. To simplify the discussion let the phase of z be included in f(t) so that in the following it will be possible to regard z as real. It is necessary to go back in our discussion to Eq. (4.6.11) and replace it by an exact substitution : f(t) = f(t o) - w 2
(4.6 .14)
Note that w is real in virtue of the choice Im [jet)] = Im [j(to)] . ducing (4.6.14) into J(z) yields J(z) = ez!(lo)
fa
e-zw'dt
or
J(z) = ez!(lo)
fa
e-
ZW
'
Intro-
(:~) dw
We once more assume that the original sense of the integration was such that the integration direction for w is from - 00 to + 00, so that J(z) = ez!(lo)
f-"'", e- (:~) dw ZW
'
(4.6.15)
To complete the story it is necessary to compute dt/dw, that is, to invert Eq. (4.6.14) and obtain dt/dw as a function of w. This should be done in the form of a power series, namely, (4.6.16)
442
[CH. 4
Functions of a Complex Variable
where, according to Eq. (4.6.14), only even powers of w will enter. Substituting form (4.6.16) into (4.6.15) will then yield as the asymptotic series J(z)
~ z-+
"
e./(Io)
00
t
/1f' a5 \' (a 2n) f(n t) (!)n '\J z ao f("2") z
Lt n=O
(4.6.17)
The coefficients an may be determined by the procedure described in the preceding section (see page 411). There it was shown that, if
L dn(t "
w =
to): = get)
n= 1
'.
[t - to]n+1 an = n! (if> (j(i) 1 dn
then
.(4.6.18)
In the present case get) = v'f(to) - f(t) . As an aid , we expand in a series : f (t o) - f(t)] = \' A (t - t)p [ (t - to)2 Lt p 0 P
Then Eq. (4.6.18) becomes
{ddx»
a - -1 n!
n
n
[2:
J-1"-I}
ApXp
%_0
P
or an is the coefficient of the nth power of x in a power-series expansion of (l;ApXp)-l n-l. We tabulate below the first three relevant coefficients in terms of A p : ao = l/VAo a2/aO = -lj-AIA o3 - jA 2Ao2 5 ·7·9'11 5 ·7·9 a./ao = 27 • 3 AtA o6 - -2-.- A o6AIA 2
·7 + 523 Ao·(A~ + 2A
1A
3) -
·M0 3A .
(4.6.19)
To illustrate the method let us again consider the gamma function. We havef(t) = In (t) - t, to = 1. The function [f(to) - j(t)]j(t - to) 2 is (t [
1) - In 1)2
(t -
t] = ! _ (t 2
so that An = [( -l)n/(n + 2)]. Eq. (4.6.19), are equal to ao
= 1/0;
3
1)
+
(t -
1)2 _ . . .
4
The values of an, by substitution into
Conformal Mapping
§4.7]
443
Therefore, by Eq. (4.6.17), the asymptotic expansion for I'(z becomes
r(z
+ 1)
=
+ 1)
.yz; z-+le-' [1 + _1 + 288z _1_ + ... J 12z 2
We shall use this saddle-point method several times in our investigations; in particular we shall study in detail the asymptotic behavior of Bessel functions and related functions by means of this technique in Sec. 5.3.
4.7 Conformal Mapping Most of the interesting and important geometrical applications of the theory of analytic functions of a complex variable to problems in physics may be grouped under the heading of conformal mapping. We employ the z Plane electrostatic field as an example. A common problem is that of a point cha rge q (line charge in three dimensions) enclosed by a metallic conductor forming a closed surface C and kept at zero potential (see Fig. 4.31). The electric field E will diverge from the source and strike the bounding surface orthogonally. We have Fig. 4.31 Equipotentials and lines of force for point source inside grounded sketched the equipotentials and lines enclosure. of force in the figure. Of course, lines of force and the equipotentials form an orthogonal grid. We may therefore expect that some function of a complex variable, w(z) = u iv, may be found for which the lines determined by the real part of w, u(x,y) = constant, yield the lines of force and the contour lines for the imaginary part of w, v(x,y) = constant, yield the equipotentials. The correct function must be singular, at the point z = Zq, where the charge is located. In fact, at Zq the function should have a logarithmic singularity, and as we shall see in Chap. 10, it should have the form w = - (2iq) In (z - Zq) plus some function analytic at Zq. The variables u and v may be used to define a conformal transformation in which u = constant lines are parallel to the imaginary axis, while the v = constant lines are parallel to the real axis. Since v = 0 on C, the bounding surface C transforms to the real axis in the w plane. The transform of Fig. 4.31 is illustrated in Fig. 4.32. The charge q is located at infinity (w = i 00) in the w plane. The constant u lines constitute just the parallel lines of force generated by the charge at infinity.
+
Functions of a Complex Variable
444
[cH.4
General Properties of the Transformation. The distortion of space resulting from the transformation can be given a more dramatic description, a description which is often useful in obtaining a " feeling " for the transformation. Essentially the transformation cuts C at some point and straightens C out, stretching it so that the point at which the cut is made goes to infinity. This procedure straightens out the potential lines; to straighten out the lines of force we must also move the charge q upward to infinity. One result of paramount importance should be noted. The function w(z) transforms the interior of the region bounded by C into the upper half plane. To perform this transformation it is only necessary to obtain
t
I I w Plane
± q
u-O ;;
- \~ v-o
c'/'
Fig. 4.32 Conformal transform of Fig. 4.31 to the w = u iv plane.
+
the point source function whose imaginary part satisfies the condition V = 0 on C and behaves as - (q/27r) In Iz - zql near the charge at Zq. It will be clear from the physics of the situation that such a point source function exists, with the consequence that a transformation taking the interior of C into the upper half plane exists. It is also clear that there are many such transformations, since the position of the charge can be anywhere in the interior of C. Assuming the existence of the transformation function fez) we may now go on to show its great value in the solution of various electrostatic problems which involve the interior of C. The essential point is that by means of the transformation it is possible to transform the problem from the complicated geometry of Fig. 4.31 to the simple geometry of Fig. 4.32 for which solutions may be much more easily obtained. For example, suppose that the potential varies along C, there being no charge in the interior of C. Then the transformed problem is one in which the potential V is specified along the real axis (v = 0) and it is desired to find V in the upper half plane. We know that V is the imaginary part of a function of a complex variable which is analytic in the upper half plane. It is therefore permissible to employ Eq. (4.2.13) : V(U,v)
f" [
=;;:v _..
V(u ' , 0) ] d I (u' _ U)2 v2 U
+
§4.7]
Conformal Mapping
445
It should be apparent that, once the transformation function is known, it becomes possible to solve any electrostatic problem associated with the interior of C. Schwarz-Christoffel Transformation. We now turn to the practical question of finding the transformation function. The most general contour C which may be discussed in great detail is that of the polygon . Special cases of importance which are also known are the ellipse and in y 0,
Positive Direction of 'PI
x
Fig. 4.33 Schwarz-Christoffel transformation of the inside of a polygon onto the upper half of the w plane.
particular the circle. Let us consider the polygon case; the transformation is then referred to as the Schwarz-Christoffel transformation. The polygon is illustrated in Fig . 4.33. The vertices of the polygon are labeled ao, aI, • • • ,the corresponding exterior angles ({'o, ({'I , • . . ,and the transforms of the vertices to the w plane bo, bl , Note that ({'o
+ ({'I + . ..
= 271"
(4.7.1)
The reader should pay particular attention to the manner in which the angles ({'i are defined and to the fact that the interior of the polygon is to the left as one goes around the perimeter in the direction of the angles. In this direction, ({"s which correspond to turning farther to the left are positive, those for reentrant angles, where the turn is to the right, are negative. In some "degenerate" cases, ({' may be 71" or -71"; which angle is the right one is usually determined by appeal to Eq. (4.7.1). Since the angles ({'i are not preserved by the transformation, it is evident that w(z) must be singuler at the points ai. As a consequence
446
[cH.4
Functions of a Complex Variable
we shall "round" these corners off as the contour C is described. The corresponding contour in the w plane is modified as indicated by means of the indicated semicircles. These are chosen so as to exclude the singular points b, from the upper half plane inasmuch as w(z) is to be analytic in this region. Consider the behavior of dz and dw as the polygon is traversed in a counterclockwise direction as indicated by the arrows in the figure. The phase of dw in the region to the left of bo is zero (dw is real), whereas the phase of dz is given by the direction of the line a4aO. At the point ao the phase of dz suffers a discontinuity of amount 1('0, whereas the phase of dw remains real. What must be the behavior of dz/dw in the neighborhood of the point bo? It must be real for w > bo and must suffer a change of phase of 1('0 while w - bo is changing phase by -7r, as illustrated in Fig. 4.33. The functional dependence thus described leads uniquely to dz/dw ~ A(w - bo)" w->b o
In order to obtain the correct phase change at bo, a = dz/dw ~ A(w - bo)-'Poh
1('0/7r,
so that (4.7.2)
w->b o
Performing this analysis at each point b, we obtain dz/dw = A(w - bo)-'Po/r(w - bl)-'P1/r(w - b2)-'P2I r
(4.7.3)
A must be a constant, for the only singularities and zeros of z in the upper half plane are at bo, bl , • • • • By Schwarz's principle of reflection (page 393) the function z may be continued across the real axis to the lower half plane, so that the only singularities of z in the entire plane occur at bi . Integrating (4.7.3) yields the Schwarz-Christoffel transformation. z = zo + Af(w - bo)-'Po/r(w - bl)-'P1/r(w - b2)-'P2I r . .. dw (4.7.4)
Recapitulating, Eq. (4.7.3) is a transformation of the interior of a polygon with exterior angles 1('0 • • • , into the upper half plane of w. The arbitrary constants zo, IAI, arg A must be adjusted so as to yield the correct origin, scale, and orientation of the polygon on the z plane. .The other constants bo, bl , . . . must be chosen so that they correspond to ao, al, . . . . Because of the three arbitrary constants available, it is possible to choose three of these points at will; i .e., three of the b's may be placed arbitrarily on the w plane . The remaining b's must be determined by first integrating (4.7.3), determining zo and A, and then solving for each of the b's by noting that, when w = bi , Z = ai. The transformation function z = z(w) will generally have branch points at the vertices bi . Usually one of the transformed points is placed at infinity. Suppose it is boo Then the transformation (4.7.4) becomes z = Zo + Af(w - bl)-'P1/r(w - b2)-'Pv r . .. dw (4.7.5)
§4.7]
Conformal Mapping
447
Transformations (4.7.4) and (4.7.5) transform the interior of the region bounded by the polygon into the upper half plane. It often occurs that the region of interest is t he exteri or of the polygon. In that case the polygon must be descri bed in the clockwise dire ction so the interior is to the right as one goes around the perimeter. The appropriate angles are now the negative of the angles 0) is integrable over (a,b). As a preliminary to proving the Fourier integral theorem, we shall first prove Parseval's formula. If f(x) belongs to Lebesgue class £2 in the interval (- 00,00), then, according to Parseval's formula, (4.8.11)
It will be seen later that the Fourier integral theorem is an almost immediate. consequence of the Parseval formula . To prove this formula, we consider the integral I = e- t& 2k2 IF(k ) 12 dk
J-.. .
Substitute the integral for F(k) in I :
f" = l.. f"
I =
1
1.. 21r
_
21r
-..
f f f" J(r) dr f"
e- t&'k' dk 00
00
-
f(x) dx
-..
f(x)e ikz dx
00
-
00
!(r)e- iki
dr
00
e- t& 'k'+ik(x- i) dk
-00
A function f(x) is said to have an ordinary discontinuity at x = t if lim [f (x ) ] r! x-+!+
lim [f(x)] and both limits exist.
In the first limit x approaches t.from the left, while
x-+!-
in the second it is from .t he right. .
Fourier Transforms
§4.8]
457
The int egration on k may now be performed :
= _~
I
V
f"
f(x) dx
21l" 0 -..
f"
1(ne-i(z-i)2/ 62 dr
(4.8.12)
- ..
We may now show that F(x) belongs to £2 in the interval (We write I = -1f(x)e- Hz- i)2/ 62 ](t) e-HZ-iJ 2/62 dx dr y'2;0 _ .. _ ..
~,oo).
f" f"
and by Schwarz's inequality! I
.
..
vk [ff I
ff If(r)l2e- Hz-j) '/6 2dx dr .. ~ rr If(x)l2e":'!
a
f(x) eikx dk
- a
Now consider H(k,a) = F(k,a) - F(k). The transform of this is a function which is zero in the range Ixl < a and equals f(x) in the range Ixl > a. Applying Parseval's formula, we have
f-.. .
IH(k,a)i2 dk
=
!-.. .
IF(k,a) - F(k)i2 dk =
t:
If(x) 12 dx
+
fa" If(x)i2 dx
Taking the limit a -+ 00 we see that flF(k,a) - F(k)i2 dk -+ 0, proving the theorem expressed in Eq. (4.8.16) . Properties of the Fourier Transform. Having once established the Fourier integral theorem, we may now turn to a consideration of the properties of the Fourier transform in the complex plane. The theorem we shall find of particular value is as follows : Let f(z) , (z = x + iy) be analytic in the strip y-
where y+ > 0 and y_ < O. If(z)l-+
+ iT)
y
<
y+
If, for any strip within this strip
{Ae
X
T -
;
Be T + X ;
Then F(k), (k = a
<
as x as x
-+ 00; -+ - 00;
T_
T+
0
will be analytic everywhere in the strip T_
< T < T+
and in any strip within this strip as u -+ as (f ~
+ 00 -
co
460
Functions of a Complex Variable
(cH.4
where A, B, C, D are real constants. The theorem is proved by noticing that the analyticity of F(k) is determined completely by the convergence of the defining integral :
f'"
F(k) = - 1-
V2; -'"
f(x) eikx dx = -1-
V2;
f'"
f(x)e-T:ceiv:r dx
-00
Uniform convergence at the upper limit of integration requires that eT-:c-r:c decays exponentially with x so that T_ < T, and vice versa at the lower limit of integration. Turning to the behavior of F(k) in the strip T_ < T < T+, consider the convergence of the integral for F(k). Since f(z) is analytic, we may take the path of integration along a line parallel to the x axis as long as y_ < y < y+. Then
f
F(k) = -1-
00
V2; -'"
e- TO, so that F+ is analytic in the region in the upper half plane of k = a + iT, above iTo. Similarly, the function F_ is analytic in the lower half plane T < TI. For example, consider the function f(x) = e1xl• Then TO = 1+, and Tl = -1- where 1+ = 1 + E, where E is small. Then F + should be analytic in the region T > 1+ while F_ should be analytic in the region T < -1-. We now evaluate F+ and F_ to check : F + = _1_
y'2;
r'" eX eikx dx =
Jo
F_ = _1_ fO
y'2; _ '"
_ _1_ (_1_); y'2; 1 + ik
e- Xeikx dx = _1_ (
y'2; -1
1
);
+ ik
T
> 1 T
<
-1
462
Functions of a Complex Variable
[cH.4
F+ has a singularity at k = i while F_ has a singularity at k = - i, which limits their range of analyticity. Function F+ is analytic above the singularity, for example. AsymptoticValues of the Transform. This is a convenient point at which to examine the asymptotic behavior of F+ and f+ and similarly for F_ and f- . Assume that it is possible to expand F+(k) in a series in inverse powers of k :
Introducing this expansion into the inversion formula (4.8.19)
we obtain We may employ the transform
1.. 211"
f..
+ iTO
_ .. +iTO
(e~ikr)
= { 1; x > 0
dk
~k
0; x
.
Tl > T~ . In the left-hand side of the equation Im r may take on all values between T~ and Tl so that Eq. (4.8.22) provides the analytic continuation for G(k) outside of its original region of analyticity. By the same procedure ", +i' O G(k) /",+M H(k) . - - dk + dk = 27rZ H(r) (4.8.23) - ee +in' Ie - r / - "'+i,o k - r where T~ < Irn r < TO . We see that representations (4.8.23) and (4.8.22) are both valid for Tl < Im r < TO . Pick r to be in this region, and subtract (4.8.22) from (4.8.23) : 21l'i[H(r)
+ G(r)]
= / "'+iTO G(k) dk _
f-
"'+iTO" G(Ie) dk "'+iTO" k - "'+i,o k "' +in' H(k) / ee + in H(k) + --dk--dk + in Ie / - ., +in' k = 1. G(k) dk + 1. H(k) dk 'fk-r t::
r
r
- "'
r
r
r
Since r is not within either contour, the right-hand side of the above equation is zero, thus proving the theorem. This is as far as we shall need to take the general theory. We have considered the Fourier integral theorem; we have discussed the analytic properties of Fourier transforms and obtained transforms and inversion formulas which apply when the functions do not satisfy the requirements of the Fourier integral theorem at both + 00 and - 00 • We now consider some applications. Faltung. The faltung of two functionsf and h is defined by the integral 10 _ / v21l'
f"'-", f(y)h(x .
y) dy
(4.8.24)
It is called faltung (German for folding) because the argument of h is "folded" (x - y) with respect to y. , We shall show, when both f and h
Fourier Transforms
§4.8]
465
are L 2 integrable in (- co, co ), that the Fourier transform of this integral is F (k )H (k ), that is, just the product of the two transforms. This relation is exceedingly useful in the solution of integral equations as discussed in Sec. 8.4. The theorem follows directly from Parseval's formula as given in Eq. (4.8.17). For if g(y ) = h(x - y) , then
- - y ) dy G = -1- /00 e+ikllh(x
and
'YI2; - 00
Now let x - y =
~
1 /00 e- ikllh(x - y ) dy G- = - = . y27r - 00
so that
G = e-ik:r / 00 eik~h W d~ V27r
= e-ik:r H(k )
- 00
Hence, by Eq. (4.8.17 ), 1 /00 j(y)h(x - y) dy = ~ ;n1 /00 F(k)H(k)e-ik:r dk (4.8.25) _ /v27r -00 v27r -00 which is just the theorem to be proved. In other words, the Fourier transform of FH is just the integral of Eq. (4.8.24). Because of the reciprocal relationship existing between functions and their Fourier transforms, we can give another form to Eq. (4.8.25) :
_ ~ /00 j(x)h(x)eik:r dx = _ ~ /00 F (l)H (k _ l) dl v27r -00 V27r -00 so that the faltung of the transforms FH is the transform of the product jh. This theorem may be generalized by regarding the integral on the right-hand side of Eq. (4.8.25) as a function of x. By mul tiplying by p(z - x) and integrating over x one obtains a function of z : 00
ff p(z -
x )h(x - y )j(y ) dy dz =
J-0000 P (k )H(k )F(k )e- ikz dk
To illustrate the value of Eq. (4.8.25), consider the simple integral equation 00 00 j(y)h(x - y) dy g(x) =
1-
where g and h are known and it is desired to find j . Take the Fourier transform of both sides of the equation. We have G(k) = yI2; F(k)H(k)
Hence
F(k) ., [ J27r
Z~~~ ]
2~ 1_ [Z~~n e-i k:r dk 00
or
j (x )
=
00
This is, of course, only a particular solution of the integral equation and, of course, is only valid when G/H is integrable £2 in the region (- co , co ). A fuller discussion of this type of problem will be considered in Sec. 8.4.
466
[cB.4
Functions of a Complex Variable
Poisson Sum Formula. The Fourier integral theorems also help us to evaluate sums. For example, the very general sort of series S =
I
00
n--
I(an) can be evaluated by the following process : Let I be
£(2)
in
00
the region (-
If we define
00 , 00 ) .
0;
Then
x
= { I;
f+(x)
I+(x)
=
x
0;
1 y/21r
f
-
I-(x)
OO
=
{IiO:
x
x
0
iTO +
00
+iTO
F+(k)e-ik:r:dx
Because of the integrability of I, constant
TO
(4.8.26)
can be less than zero. The
k =U+iT Plane Fig. 4.41 Contours for analytic continuation of functions G and H .
transform F+(k) is analytic for T > T~ , TO > T~ , as shown in Fig . 4.41. We also have oo+i" F_(k)e-ik:r: dx I-(x) = -1(4.8.27) ~ -OO+iTl
f
where Tl can be greater than zero and F_(k) is analytic for T < T~/, Tl < T~'. With these definitions in mind, it becomes convenient to break S up into two parts : ..
S+
=
I
-1
I+(an);
n=O
Consider S+ first.
S_
=
I
I-(an)
n=-oo
Introduce Eq. (4.8.26) into S+:
Because of the absolute convergence of F we can exchange sum and integral : S = -1+ - 10 ITo. In that event, integral (4.8.34) will define Fm(S) for only the region to the right of s = ITo, and the remainder of its dependence must be obtained by analytic continuation. An example of this procedure was discussed in Sec. 4.5 (the r function). For these circumstances inversion formula (4.8.37) is no longer correct. However, if we consider the (unction x"'f(x) (IT~ > ITo), then the above discussion again becomes valid. The transform of x"'f(x) is just Fm(s + IT~) . Applying Eq. (4.8.37) we find that Fm(s+IT~)d x'" 'f() x = -. • S
(l)fi'" 211't
Let s
+ IT~
=~.
- i'"
X
Then f(x) =
(1....) ji"'+
b
>0
Prove that
(2..
Jo
a
eqoo.,)
1 - 2p sin tJ
+ p2
{cos . sin sin (q tJ) + p cos (q sin
o + tJ) } ao
sin
cos (pq) sm
= 211" •
4.3
f.
Show that
_ .. (x2
cos (x) dx
+ a2) (x2 + b2)
[e- ae-"] ; b
11"
= a2 _ b2 b -
Re a
What is the value of the integral when a = b; Re a
>
>
O?
4.4 Show that
~
2"
o
4.6
Prove that { ..
Jo 4.6
2 n!
eoo. ,) cos (ntJ - sin tJ) dtJ = ...!!..
X 2"-1
1
+ x2dx
Calculate
f
.
_ .. (1
= (~) esc (1I"a);
X2
+ x 2) (1 -
dx 2x cos tJ
0
0
[cH.4
Functions of a Complex Variable
472
4.'1 Prove that
f
00
(1
x
a
+ X 2)2
dx
= ?r(1 -
- 1
a) .
4 cos (va) ,
<
a, p. > -1 , n is an integer greater than p., and
znF(z) approaches zero as Izi approaches infinity. The function F(z) is analytic over the whole complex plane except for a finite number of poles, none of which are on the real axis to the left of z = b. Show that the integral equals (1 /2i sin ?rp.) Je (z - a)n-I'-I(z -
b)I'F(z) dz
where the integral goes from z = b + E to - 00 just below the real axis, goes in a large circle counterclockwise to just above the real axis at - 00, and returns to z = b + E by a path just above the real axis. Hence show that the integral equals ?r csc (?rp.) ~[residues of (z - b)I'(z - a)n-I'-IF(z) at poles of F] Show that, for 0
G~ :}'
f~l 4.10
r
< p. < 1, k > 1,
x-a dx
00
4.11
+x
oo
?r 2
= sin?ra
at'J)
(sin sin t'J ;
-1
<
a
<
+
1)-11- 1
1, -?r
sin (kx) sin (k'x) dx = (~) {sin (k~) cos (k'~) ; x2 ~ cos (k~) sin (k'~) ;
e
-fJ. 4.17 Function y,.(x) is defined by the integral
+
+
1 {f ~+w (l z2) e- i zz f ~+iT (l z2) e- iu : dz} y,.(x) = 2-' ( 2 2)( + . ) dz ( 2 2)( . ) 7rt _ ~+i~ Z - Zo z ta - ~+ iT Z - Zo z-ta
where
< 1 and
T
y,.(x)
=
(J
> - 1. Show that
{(;2; zD
a1zl
e
+ ~o ~; ~ Z5) cos [zolx1 -
tan-
l
(~)]}
4.18 Show that z = 1 is a singularity of the function represented by the following series, within its radius of convergence ~
~
f(z) = '-' r(l
r(i)( -z)n n)r(j - n)
+
n =O
4.19
The generalized hypergeometric fun ction F(aO,al, by the power series
,a.lcl,c2
. . ,c.lz) is defined 1
+ aOal Cl
. . . a. z • C8
+ ao(ao + l)al(al + 1) Cl(Cl
Show that F is singular at z
=
[n• . [n
+
1) .. .
1 and that for ai
r(Cn)] r(p) .
n=l
r(a m )
m-O
. . . a.(a 8 C8(C 8 + 1)
,• ]
(1 - z)p
< c,
+ 1) (Z2) + 2!
[cH.4
Functions of a Complex Variable
474
L•
where p =
(an - cn)
+ ao.
n=l
4.20
n""" -
4.21
n=:&-
4.22
Prove that
GO
Prove that
QO
Show that an even integral function may be expressed as
...
fez)
= J(O)
Il [1 - (:;)] n=l
where J(a n ) = 0 and where we consider only one of the two roots an, -an in the product. Assume the restrictions on F(z) given on page 385. Show that, if J(O) = 1,
L(:~);
...
...
/,,(0) = -2
j(ivl(O) = 3[/,,(0»)2 - 12
n=l
L(:~) n=l
Employing product representation (4.3.9) for (liz) sin z, show that ... ...
L(~2)
n=l
4.23
=
(~2);
L(~4)
=
n=l
Prove that
n[1 +
(;~)
ee
eGZ - ebz = (a - b)ze1 (G+blz
~(a----:----::--,b)::-2_Z2] 2 4n2.n-
n=l
4.24
Prove that
4.26 Employing the generalized Euler transformation, show that F(ao,a1, •• •lb1,b2, .•• Iz) = F(a1 • • ·\b2 • • 'Iz)
CH.4]
Problems
475
4.26 By integrating e- k • zn-l around the sector enclosed by the real axis, the line '" = ex (z = rei'!'), the small arc r = E and the large arc r = R, prove that
J'" xa-1e-
pr
00'" C?S
sm
(px sin tJ) dx = p-mr(a)
C?S
sm
(atJ)
< j,r, a and p are real and greater than zero. 4.27 By integrating za::-le' about the contour starting from a point just to the right of the origin (z = E) to - OCJ just below the real axis, to OCJ i OCJ, to b - i OCJ to b + i OCJ , to - OCJ + i OCJ, to - OCJ, and thence to z = E by a path just above the real axis, prove that, for 0 < a < 1 and b > 0, eill(b + iy)a-l dy = 2e- b sin(n'a) rea) where tJ
f-"'",
and therefore that
hi.. 4.28
cos[tan tJ - (1 - a)tJ] seca+1(tJ) dtJ =
G)
sin(7I"a) rea)
Show that J = f f fxl-1ym-lzn-l dx dy dz
integrated over an octant of the volume bounded by the surface (x /a)p (y /b)q (z/cy = 1 is
+
+
a1bmcn) r(l/p)r(m/q)r(n /r) ( pqr r[(l/p) (m/q) (n/r)
+
+
+ 1]
4.29 Prove that r(z)r (z 4.30
+ ~) r
(z
+~) . . . r
(z
+n ~
1) =
(271")l(n-llni-n'r(nz)
Express the integral
Jo e- I't
n
-
X- 1 dt
occurring in Eq. (4.5.26), in terms of a gamma function . 4.31 Show that 1 _ cos (7I"ex)
cosh (7I"z)
271"r(i - iz)r (i
+ iz)
476
[cH.4
Functions of a Complex Variable
4.32
Show that
4.33
Show that
Ir(iy)12 = 1I"j(y sinh 1I"Y) Ir(z
+ 1)1 2~ 211"r 2
"+ 1
e- 2(1I'1'+ .1I) [1
+~ + 72r _1_ + . . .J 6r 2
2
where z = x + iy, r2 = x 2 + v', and I{J = tan- 1 (yjx). Show that the phase of I'(z) is given asymptotically by
h
In (x 2 + y2)
+ (x
-
-!)
tan- 1
(~) 1
- Y[ 1
4.34
+ 12(x2 + y2) -
Consider the integral 1 f(t) = - . 211"t
i C
3x 2 - y2 360(x2 y2)3
+
+
...J
ezt- >-...tz + vz) dz
Z(K
where the branch line for the integral is taken along the negative real axis and the contour starts at (- 00) below the branch line, encircles the branch point at the origin in a positive fashion, and goes out to ( - 00) above the branch line . Break up the integral into three parts, consisting of the two integrals extending from (- 00) to a small quantity EO and an integral over a circle of radius EO about the origin . Show that, in the limit of EO equal to zero, the third integral is 1j K while the first two may be combined into
...
- -1 11"
l:(
d [~ . . e:" sm(A Y u) duJ -
-n1 1) -
K
+
n
dAn
'
n=O
Show that
~ ... eo
ut
sin(A
.
0
d = .y; vU) .-!!: u
C
.
i:: 0
U
e-~2 d~
and hence that
4.36
Consider the integral g(z) =
f;.. G(t)r( -t)( -z)t dt
where G(t) is regular in the half plane Re t > 0, [G(t)ztjr(t + 1)] approaches zero for large values of Re t. The contour is along the imaginary axis except for a small semicircle going clockwise about the origin. Show by closing the contour with a semi circle of large radius in the half plane
Problems
CH.4]
Re t
> 0 that
..
g(z)
=
27ri
L
G(n)
(~n!)
n=O
..
Hence show that F(a b\clz) ,
477
r(c) \ ' rea + n)r(b + n) (zn) r(a)r(b) I'(c n) n!
=
Lt
n=O
=
r(c) 27rir(a)r(b)
+
fi .
+ t)r(b + t) + t)
rea
r( -t)( -z)' dt
'r(c
-i..
for Re a and Re b less than zero, a and b not integers. Show that the contour may also be closed by a circle in the negative real plane. Deduce that
~(~)(~)(b) F(a,b\clz)
r(~)(~(~~) b) (-Z)-"F(a,1
=
+ r(~(~(~ ~) a)
- c
(-z)-bF (b,1 - c
+ all
- b
+ b!1 -
a
+ aiD + bl ~)
[see Eq. (5.2.49)] . 4.36 Prove that
f
! .. (
_!..
cos
t?-)"-2
itt)
e
so -
7rr(a - 1) - 2"-2r[.Ha + t)]r[j(a _ t)] a > 1
by calculating f[z + (l jz)j"-2 Zl-l dz around a contour running along the imaginary axis from +i to -i and returning to +i via the right-hand half of the unit circle. 4.37 Using the method of steepest descents, show from the definition .
H~l)(Z)
that H~l)(V
sec ex)
~
i
= (e- .../
2 )
7r
f!.. +i .. ei(.eo. (10 while Fs. will be analytic for Re s < (11. Show that (10 < (11 if the Mellin transform exists . Show that
where (1 > (10 and T < (11 . 4.49 Find the Fourier transforms F + and F _ of cos (ax) for a complex, and determine the region in which each is analytic. 4.50 Find the Fourier transforms F+ and F_ of xne-", and determine the region in which each is analytic. 4.51 From the Poisson sum formula show that .,
~3(U
,
e-ia2)
= 07r \ '
~
a
e- (2/a2) (u+m1l")'
m=- co
(see page 431 for definition of
~3) .
Tabulation of Properties of Functions of Complex Variable A function j = u + iv of the complex variable z = x + i y is analytic in a given region R of the z plane when: a. Its derivative dj/ dz at a point z = a in the region R is independent of the direction of dz ; this derivative is also continuous in the region R. b. au /ax = av/ay; au/ay = -av/ax, these derivatives being continuous in R. c. .f j(z) dz = 0 for any closed contour within region R . Anyone of these three statements is a necessary and sufficient condition that j be analytic in R; the other two statements are then consequences of the first chosen . When any and all are true in R, then all derivatives of j with respect to z are analytic in R. In addition, for contour integrals entirely within a singly connected region of analyticity of j,
1..
r
j(z) dz = 27rij(a) ' (z - a) ,
1..
r
j(z) dz = 27ri j TO,
0)
o
)
Laplace Transforms.
>
y'2; (Re k > 0) (zo + Zl) e·,k (Re k < 0) (l/k o) V1r/2 sech(1rk/2k o) (i/k o) -v;j2 csch(-lrk/2k o) i y'2; ehiakia J a(2 0) e-!k2 J_ i (i k 2)
e-!·2
VZ J_i(iz
(Re k zok
< 0, if
If f(x) is zero for x
fo
00
If(xWe-2TZ dx is
and if
then Fl(p) is called the Laplace transform of f(x) and 1 f iOO+T j(x) = - . FI(p)e Pz d/p ; 21rt -iOO+T
T
> TO ; x>o
The faltung theorem becomes z
~o
f(y)h(x - y) dy
=
1 f iOO+T -2• FI(p)HI(p)e P z dp 1rt - iOO+T
For further discussion of the Laplace transform see tables at end of Chap. II. Mellin Transforms. If f(x) is defined for (0 ~ x ~ CL» , if
fo
00
If(x)12x-2cr- 1 dx
CR.
4]
Tabulation oj Properties of Functions of Complex Variable
is finite for a
> rTo,
485
and if F m(S)
=
fo 00 f(x)xB-1 dx
then F m(S) is called the Mellin transform of f(x) and
(d)
1 fiOO+O' f(x) = -2. Fm(s) ~ ; a'> rTo; 1I"t -ioo+O' Xo
«)
x> 0
The faltung theorem takes on the following forms: 00
or
fi 00 +0'
. V m(S)Wm(S) 1I"t -. 00 +0' 10o v(y)w ~Y ....J!..Y = -2' 00 1 fiOO+O' V m(P) W m(S 1I"t 00 +0' 10o v(x)w(x)xB-1 dx. = -2" ( )
1
-i
vex)
"AFm(S)
+ a) -sFm(s + 1)
In (x)f(x)
d ds Fm(S)
Fm(s
res) r(c) [r(s)r(a - s)r(b r(a)r(b) I'(c - s) [r(s)r(a - s)/r(a)] [11"/(1 - s) sin(1I"x)] (1I"/2s) tan(1I"s/2)
+ x)-a +
(l /x) In (1 x) tanh- I (x)
+ x)-mPm_
1
p) dp
a-oF m(S)
f(ax) xaf(x) d dxf(x)
F(a,blcl - x)
(1
X
Mellin transform F m(S)
Function f(x)
(1
(d~)
+ x)
1x (1
S)]
[r(s)/r(1 - s)][r(m - s)/r(m)]2 [I'(sj I'(e - s)r(c)/r(a)r(c - s)]
F(alel - x)
xjv(x)
[20-lr (v t s)/ r (v ; s+ 1)] [2s-ir (s + ; + 1)/r (1 + s)]
sin (x) cos(x)
res) sin(j-n-s) res) cos(j-n-s)
J vex)
Nv(x)
v ;
_
2~1 r
(s
t v) r
(s ;
v) cos[j-n-(s -
-n
where the functions J v, i.. F, P on, and N v are defined in the tables at the end of Chaps. 5, 10, and 11.
486
[cB.4
Functions of a Complex Variable
Tables of Special Functions of General Use (For other tables see the ends of Chaps. 5, 6, 10, 11, and 12)
The Gamma Function (see page 419)
fo" e-1tz-l dt(Re z > 0) i
fez) =
+ 1)
f(n
= n1i
f(z)f(1 - z)
I'(I) = 1; 11"
=
.
f~Z)
= ze'Y
(
SIn
.
Z
f(z
n(1
1I"Z
)
+ 1) .y;
= zf(z)
f({) = 2 2z- 1 f(2z) = _ r f(z)f(z V 11"
i
+ ~) e-z1ni
-y =
+ {)
0.577215 .
n -I
dId
= dz In [fez)] = fez) dz fez)
1J;(z)
..
--y
=
2: [n ~
+
1- z
n=O
1/!(n
+ 1) =
fez)
---t
--y
l
~n
1J;(z
+ 1)
+ 1 + ~ + ~ + ... +~;
1
= -
z
+ 1J;(z)
1J;({) = -'Y - 2 In 2
-yI2; zZ-ie- z ; for z» 1
= et 'Jo
B(x y)
Z- 1
=
2
(1 - t)Y-I dt
~i"" sin 2
z- 1
cP
= f(x)f(y) . Re z,
rex + y) '
Re y
>0
cP dcP
COS 2y-1
Elliptic Functions (see page 428)
(z )0 v(1
-
1v(1 1
x
1
dx
x 2) (1 - k 2x 2)
= sn-I (x k).
,
dx x 2) (1 - k 2 + k 2x 2)
X2)~:2 + k2 _ fax V (1 + X~~1 + k'2X2)
,
= crr'
(x ,k)
1
v(l _
K =
e v(1 -
)0
1)
=
drr? (x,k);
= tn-
dx . x 2)(1 - k 2x 2) ,
I
(x,k)
K'
=
e
Jo v(1 -
dx . x 2)(1 - k'2X2) '
k = sin
_r=::;===d~x:::;.:==~ - = Jo(x vx(1 - x)(1 - k 2x)
2 sn-1 .
(vx,k) ,.
+P ; d = e" = y'4ab;
eP =
VbfO,
so that, if d, a , and /3 are real quantities,
+
z = d cosh (w - /3) = e" cosh (h i~2 - /3) x = d cosh (h - /3) cos (~2) ; Y = d sinh (h - /3) sin (b) Idz/dwl = hI = h 2 = d ycosh 2 (h - /3) - cos" ~2
(5.1.15)
These are the elliptic coordinates, consisting of confocal ellipses and hyperbolas, with the foci at x = ± d and y = 0, as shown in Fig. 5.3. The constant /3 is usually set equal to zero for convenience. (However, it should be noted that, if we set a = /3 + In 2 and then allow /3 to go to negative infinity, the elliptic foci merge together into the origin and the coordinate system changes to polar coordinates in the limit.) The
504
Ordinary Differential Equations
[cH:5
forms of the partial differential equation in these coordinates and of the ordinary differential equations resulting from separation are
:~ + ~~ + d2k2[cosh 2 (tl 11'
=
F(~1)G(~2);
(d2F jdW
(d'l{}jdW - [d2k2 cos"
(~2)
- (3) - cos" hJlY = 0 + (d2k 2 cosh" (~l - (3) - a 2JF = 0 - a 2JG = 0
(5.1.16)
with a 2 again as the separation constant. Scale Factors and Coordinate Geometry. We shall see later in this cha pt er that equations of the form produced by separation in polar or elliptic coordinates, having exponential (or hyperbolic) functions in the coefficients Of F or G, are not in the most convenient form for analysis. The difficulty is primarily due to the geometry of these coordinate systems and to the fact that we have obtained them by use of the conformal transformation. Both of these systems have concentration points, where the scale factor Idzjdwl goes to zero. Near such points the coordinate lines are very closely spaced, corresponding to the smallness of the scale factor, and since the transformation is conformal, the scale factors for both coordinates vanish at these points. However, once we have determined the geometry of the separable system, we can modify the scale factors of each coordinate separately in such a manner as to retain the coordinate geometry but change the separated equations to forms which are more amenable to analysis. For instance, since algebraic functions for the coefficients of F and G are more desirable than exponential ones, in the separated equations, we could set eh to be a new coordinate r for the polar coordinates and cosh (~l - (3) to be J.l for elliptic coordinates. The transformation would not then be conformal, but the shape of the coordinate lines would be unchanged and the coordinates would still be separable. Referring back to Eq. (5.1.6), either for the wave equation or the Schroedinger equation, for separation we must have that k 21dzjdw l2 = k 2h 2 = !(h) + g(~2). In this case the separated equations are (d 2FjdW + h(h)F = 0; (d2GjdW + g(~2)G = 0; 11' = F(h)G(~2) (5.1.17) In line with the discussion of the previous paragraph, we choose some function J.l(h) of the variable h, such that the function f(~l), expressed as !(J.l) , is a simple algebraic function of u , Since! g is proportional to the scale factor Idz jdwl 2, the chances are that both! and g will go to zero at the concentration points of the coordinate system. We can choose our new coordinate J.l so the concentration point is for some standard value of J.l, such as 0 or 1 (or, perhaps, infinity) . The scale factor for p. is related to the scale factor h for ~1 and ~2 by the following relation:
+
h; = V(iJxjiJJ.l)2
+ (iJy jiJJ.l)2
= hiPl';
e,
= (dh jdp.)
(5.1.18)
Separable Coordinates
§5.1]
505
since ox/op. = (ox/oh)(d~ddp.). If b; does not go to zero at the concentration point of the coordinate system (where h is zero), then ~(p.) = h,./h must become infinite at these points. To transform the differential equation (5.1.17) for h to one in p., we also need the formulas d dh d. d _ 1 d. d 2 _ 1 d 2 ~~ d . ,_ d 2h ~=~~, dh-~~' da-~~2-~~' ~-~2
We can , if necessary, do the same for b, changing to a new function 71(~2) and obtaining a new scale factor h; = h~~, and so on. The new transformation and resulting separated equations then become w(z)
=
h(p.)
+ it2(71) ;
:;tF + cJ>'~2~dF=
d2 dp.2 -
cJ>:
dp.
h,. = V(OX/op.)2 + (oy/op.)2 = cJ>,.h; h; -h 2k 2if; .:: -[p(h) + q(~2)1if;; if; = F(P.)G(71)
+ cJ>~f(p.)F = 0;
d2G ~' dG d712 - cJ>: d71
+ cJ>~g(71)G = 0
=
cJ>~h
(5.1.19)
These last two ordinary differential equations look more complicated than Eqs. (5.1.17), but if cJ>~/cJ>,. and cJ>;f(p.) are algebraic functions of p. instead of transcendental or other functions , they are easier to analyze and solve. As we pointed out above, the function cJ>~ or the function ~;/cJ>,. or both will probably become infinite at the values of p. corresponding to the concentration points of the coordinate system, so that the singularities of the coefficients of dF/dp. and F in the equation for F are closely related to the geometry of the corresponding coordinate system. This interrelation will be referred to later in the chapter. To make specific these remarks we apply them to the polar coordinates of Eqs. (5.1.12) . We wish to change the scale of the radial coordinate so that the function e2h becomes an algebraic function. An obvious choice is p. = r = eh or tl = In p., where p. (or r) is the usual distance. The origin, the only concentration point for this coordinate system, is then at r = O. The scale factor and resulting equation for F are then I: ci
= In
r;
'" '¥r =
r;r ""
'¥r
=-
(T2; 1)
hr
= 1;
2F ddr2 + !r dF dr
+ 1. r2 f(r)F
=
0
(5.1.20) 2k2 2 where, for the wave equation, f(r) = r - a • We see, therefore, that the coefficients both of dF/dr and of F have singularities at the polar center, r = O. The scale of the b coordinate does not need to be changed, since the last of Eqs. (5.1.14) has no transcendental coefficients. Nevertheless ~2 is an angle, so that the solution G is a periodic function and w is a multivalued function of z. To remove the "multivaluedness" it is
[cH.5
Ordinary Differential Equations
506
sometimes useful to make the transformation 11 = cos ~2, where the useful range of 11 is between -1 and + 1. The related equations are
b = cos- 1 11 ;
cI>~ = -(1 -
11 2)- 1;
. _ _~ . d2G 11 dG h; - -r/v 1 - 11-, -d2 - -1--2-d 11
-11
11 2)- i
cI>~ = -':11(1 11
g(l1) G _ + -1--2 -11
0
(5.1.21)
where, for the wave equation, g(l1) = a 2, the separation constant. The singularities of the coefficients are here at 11 = ± 1, the two ends of the useful range for 11. It should be obvious by now that the simplest transformation for the elliptic coordinate case is cosh (h - (3) = p. and cos b = 11, where the interesting range of p. is from 1 to 00 and the useful range of 11 is from + 1 to -1. The transformation and modified equations are (for (3 = 0) x
e,
= dP.l1 ; Y = d 1
=~ ;
2F d dp.2
h" =
+ _p._ dF + p.2 -
11 2) ; ~l = cosh- 1 p.; ~2 1p.2 - 11 2 1p.2 - 11 2 d '\j p.2 _ 1; i; = d '\j 1 _ 112
Y(p.2 -
1 dp.
1)(1 -
2
f(p.) F = O. d G _ _ 11_ dG 1 ' d11 2 1 - 11 2 dl1
p.2 -
=
cos"
+ ~G 2
11
= 0
1 _ 11
(5.1.22) -d 2k 211 2
where, for the wave equation, f(p.) = d 2k 2p.2 - a 2 and g(l1) = + a 2• Here again, the coefficients in these equations have singularities only where the coordinates have concentration points (p. = 1, 11 = ± 1). 11 = (l Id) y(x d)2 y 2 = rtld and We should also note that p. J!. - 11 = (l Id) y(x - d)2 + y2 = r2ld, or
+
p.
= (rl
+ r2) /2d ;
+
11
= (r2 - r2) /2d
+
(5.1.23)
where rl, r2 are the distances from the point (x,y) to the two foci of the coordinates (x = +d, y = 0) . Therefore the line p. = constant corresponds to the locus of points such that the sum of the distances of the point from the two foci are constant, a standard definition of an ellipse. Similarly the line 11 = constant involves the difference between the distances, a standard definition of an hyperbola. Separation Constants and Boundary Conditions. If the coordinates suitable for the boundary surface of a given problem are separable coordinates, then one can, in principle, satisfy reasonable boundary conditions by the right combination of the factored solutions (what boundary conditions are "reasonable" and how to find the "right combination" will be discussed in Chaps. 6 and 7) . For instance, the boundary may be the line h = constant in one of the separable twodimensional coordinates we have been discussing. This boundary may be finite in length (as for a closed boundary) , in which case the factors X2(~2) must be continuous as we vary ~2, going along the line ~1 = constant from a given point clear around to the same point.
Separable Coordinates
§5.1]
507
For instance, for the polar coordinates r, Ip, the line r = constant is a circle of finite size which is completely traversed by letting the angle Ip go from 0 to 211". The equation for the Ip factor in the separated solution is (d 2if! /dIp2)
+ a 2if!
=
0;
where a is the separation constant
with solutions either cos (alp) or sin (alp) or a linear combination thereof. In order to have if! continuous along the boundary r = constant, the solution if! must have the same value at Ip = 211" as it does at Ip = 0 ; in other words if! must be periodic in Ip with period 211". Such a periodicity requirement imposes restrictions on the allowable values of the separation constant a. In our example, in order that cos (alp) or sin (alp) be periodic in Ip with period 211", the separation constant a must be an integer m = 0, 1, 2, 3, . .. . In other' cases where ~2 is a periodic coordinate, there are similar restrictions on the values of the separation constant for solutions to be continuous around the circuit of the boundary ~l = constant. We can always order these allowed values, letting the lowest value be al and so on, so that an+l > an , and we can label the corresponding factors in the solutions XH~2), X~(~2), . .. . The factor for h also depends on a, so that the complete solution corresponding to the allowed value an of the separation constant is Xr(h)X~(~2) . It will be shown in Chap. 6 that any "reasonable" function of the periodic coordinate ~2 can always be expressed in terms of a series of these allowed functions ., !(b) =
L AnX~(~2)
n = 1
The rules for computing the coefficients An will also be given later. Therefore if the solution of interest if;(h,~2) is to be one which satisfies the boundary condition if;(c,b) = !(~ 2) along the boundary h = c, then the solution may be expressed in terms of the separated solutions for the allowed values of the separation constant : (5.1.24)
Other, more complicated boundary conditions may be similarly satisfied. We notice that the final solution, if;(h,~2), is not separable but that it can be expressed in terms of a series of separable solutions. In each case the conditions of periodicity pick out a sequence of aliowed values of the separation constant, and the general solutions are given in terms of a series over these allowed values. Even for open boundaries, generalizations of the periodicity requirements allowed us again to express a solution satisfying specified boundary conditions in terms of a series
508
Ordinary Differential Equations
[cH.5
(or an integral) of separated solutions over the allowed values of the separation constant. But this discussion has carried us as far afield as is needful here; we must next investigate the separability of three-dimensional partial differential equations. Separation in Three Dimensions. Separation of solutions in two dimensions is particularly simple for several reasons. In the first place there is only one separation constant, so that the factored solutions form a one-parameter family, which makes the fitting of boundary conditions by series relatively straightforward. In the second place, the conditions for separation are simple : in the equation 2 2 a "" + a "" +k21dZ12 2 aU aV 2 dw "" = 0 the term k 2ldz /dwl 2 must consist of a simple sum of a function of u alone and a function of v alone, or else the equation would not separate. And third, the only cases giving rise to families of nodal surfaces which coincide with the coordinate surfaces are cases where the solutions have the simple factored form Xl(h)X2(~2) . In all three of these respects the three-dimensional separation problem is more complex. Since there are three separated equations, there are two separation constants instead of one. Each of the three equations may contain both constants, in which case each of the three factors in the separated solution depend on both separation constants in a complicated manner and the satisfying of boundary conditions even by a series of separated solutions is a tedious and difficult task. For some coordinate systems, however, the results are such that one (or two) of the separated equations contains only one separation constant; in these cases the general series solution takes on a simpler and more amenable form . Taking up the third item next, it turns out that in the three-dimensional Laplace equation V 2"" = 0, there are some coordinate systems in which the solution takes on the more complex form R(h,~2,h)Xl(h) ' ·X2 ( ~2) X 3( ~ 3) , where the additional factor R (which might be called a modulation factor) is independent of the separation constants. For these systems there is a measure of separation, and boundary conditions can be satisfied, for the common factor R can be taken outside the summation over allowed values of the separation constant and the sum has the same general form as for the cases without a modulation factor. Returning to the second item enumerated above, the term in the threedimensional partial differential equation corresponding to the term k 2Idz /dw\2 for two dimensions does not need to be simply a sum of functions each of only one coordinate ; separation may be attained for more complicated cases than this.
Separable Coordinates
§5.1]
509
The Stickel Determinant. The general technique for separating our standard three-dimensional partial differential equation
depends on the properties of three-row determinants. The relation between such a determinant S and its elements ~mn is given by the equation S
=
I~mnl
=
~ll
~12
~13
~21
~ 22
~23
~ 31
~ 32
~3 3
=
~1l~22~ 33 -
+ ~12~23~ 31 + ~1 3~21~32
~1 3~22~31 -
~1l~23~ 32 -
(5.1.25)
~12~21~ 3 3
The first minor of the determinant for the element ~mn is the factor which multiplies the element ~mn in the expression for the determinant. For instance, the first minors for the elements in the first column, ~ll, ~21, ~31, are M 1 = as/a~l1 = ~22~ 33 - ~ 23~ 32 (5.1.26) M 2 = as/a~21 = ~1 3~ 32 - 12~33
M 3 = as/ a31 =
12~ 23
-
~1322
Since we need here only minors for the first column, we need not give the M 's two subscripts. The important property of determinants, which we shall use in separating three-dimensional equations, is the orthogonality property relating elements and first minors. For instance, for the minors defined in Eq. (5.1.26) we have 3
I
n=l
3
Mn~nl = s,
I
n=l
Mn~nm = 0; m
= 2,3
(5.1.27)
as though there were vectors with components n2 or ~n3 which were perpendicular to a vector with components M n . Therefore if the separated equations for the three-dimensional case were (5.1.28) we could combine the three equations in such a manner as to eliminate the separation constants k« and k 3• For by multiplying the equation for Xl by (M d S)X 2X3, and so on, and adding, we obtain (5.1.29) This equation would correspond to our standard equation v 21J;
+ ki1J; = 0
510
Ordinary Differential Equations
if the expression for the Laplacian in the
~
[cH.5
coordinates, (5.1.30)
were equal to the first expression of Eq . (5.1.29). In order to have this happen, several restrictions must be placed on the scale factors h and on the elements 4>nm of the determinant S . In the first place , since Eq. (5.1.28) is supposed to be a separated equation, the fun ctions i-; 4>nl, 4>n2, and 4>n 3 must all be functions of ~n alone . A determinant with elements 4>lm in the top row which are fun ctions of h, with elements 4>2m in the second row functions of ~2 and with elements 4>3m in the bottom row functions of ~ 3 only, is called a Stock el determinant. It is basi c for the study of separation in three dimensions. We also note that, if 4>lm is a function of ~l , etc., the first minor M 1 is a function of ~2 and ~3, but not of h, etc . Next, we must have the quantity hlh2h3/h~ be equal to the product of the function fn of ~n alone times some function gn of the other es. Then, for instance, the term in the Laplacian for h becomes
which has the general form of the term (MdSh)[a(ha,y/ah)/a~d in Eq. (5.1.29) . In fact in order to have the terms identical, we must have l/h~
= Mn/S
and this, together with the original restriction on the Robertson condition
(5.1.31) hlh2h 3/h~,
leads to (5.1.32)
which at the same time determines the Stackel determinant and limits the kinds of coordinate systems which will allow separation. When it holds, the quantity h 1h 2h3/hf is equal to h, a function of h, times M J?i'3, a function of ~2 and ~3 but not of h, thus fulfilling the requirement stated earlier in this paragraph. These interrelated requirements on the scale factors severely limit the number of independent coordinate systems which answer the specifications. The detailed analysis necessary to determine which systems do satisfy the requirements is much more tedious than was the corresponding analysis, given on page 500, for the two-dimensional case. There we showed that the separable coordinate systems (for the Wave equation) consisted of confocal conic sections (ellipses and hyperbolas) or their
Separable Coordinates
§5.1]
511
degenerate forms (circles and radii, confocal parabolas, or parallel lines) . The detailed analysis for the three-dimensional case reveals a corresponding limitation ; for Euclidean space, the separable coordinates for the wave equation are confocal quadric surfaces or their degenerate forms . Confocal Quadric Surfaces. The equation ~2
x2 _
a2 + ~2
y2 _
b2 +
Z2
e_
c2
=
1; a ~ b ~
c
~
°
(5.1.33)
for different values of the parameter ~, represents three families of confocal quadric surfaces. For ~ > a we have a complete family of confocal ellipsoids, with traces on the y, z plane which are ellipses with foci at y = 0, Z = ± Vb2 - c2, traces on the x, z plane which are 2 ellipses with foci at x = 0; z = ± c2 and traces on the x, y plane 2 which are ellipses with foci at x = 0, y = ± b2 • The limiting surface of this family is for ~ ~ a, which is the part of the y, z plane 2 inside an ellipse having major axis 2 c2 along the z axis and minor 2 2 b along the y axis. axis 2 For a > ~ > b we have a complete set of confocal hyperboloids of one sheet, with traces on the y, z plane which are ellipses with foci at y = 0, Z = ± yb 2 - c2, with traces on the x, z plane which are hyperbolas with foci at x = 0, z = ± ya 2 - c2 and with traces on the x, y plane which are hyperbolas with foci at x = 0, y = ± ya 2 - b2 • The limiting surfaces are for ~ ~ a, which is the part of the y, z plane outside the ellipse having major axis 2 ya 2 - c2 along the z axis and minor axis 2 b2 along the y axis, and for ~ ~ b, which is the part of the 2 x, z plane outside the hyperbolas having transverse axis 2 Vb 2 - c2 2 along z and conjugate axis 2 b2 along x. Finally for b > ~ > c we have a complete family of confocal hyperboloids of two sheets, with traces on the y, z plane which are hyperbolas with foci at y = 0, z = ± Vb 2 - c2 , with traces on the x, z plane which are hyperbolas with foci at x = 0, z = ± ya 2 - c2 and with no traces on the x, y plane. The limiting surfaces here are for ~ ~ b, which is the part of the x , z plane inside the hyperbolas having transverse axis 2 2 Vb2 - c2 along z and conjugate axis 2 b2 along the x axis, and for ~ ~ c, which is the x, y plane. Incidentally, we can set c = without any loss of generality. Since the three families of surfaces are mutually orthogonal, we can consider the three ranges for the parameter ~ to correspond to three families of coordinate surfaces, with h(~l > a) corresponding to the ellipsoids, b(a > b > b) to the hyperboloids of one sheet, and ~3(b > ~3 > c) to the hyperboloids of two sheets. It can easily be verified that the relation between the coordinates x, y, z and the ellipsoidal coordinates h , h , ~3 (for c = 0) with their scale factors are
va
va
va
va
va
va
va
°
512
Ordinary Differential Equations
h > Z
=
a
>
~2
[cH.5
>b>
~a
>0
hh~a ---ab
(5.1.34)
_ Im-mm-m.
h 2 - '\j (~~ _ a2)(~~ _ b2)'
etc .
Following our earlier discussion, we find that h1h 2h a/hi equals
Vm - a 2 )m- b2 ) Vm - ~i)2/m - a2)(~~
times the function which does not contain h.
- b2)(~i - a2)(~i - b2) , Consequently the function
(5.1.35) is the one defined in Eq . (5.1.32). minant : S
= h1h 2h a = fdd..
m-
a2)(~~
This in turn fixes the Staekel deter-
m-2 wm-2 m(~i - W - a )m- a )m- b2)(~~ -
b2)(~i
- b2)
and this, together with Eqs. (5.1.31), _ S _ M1
-
hi -
m- a 2)(l;i -
-
l;i -
.
~i
a2)(~i - b2)(~i _ b2)' etc .
enables us to compute the elements of the Stackel determinant. They are q,nl(~n)
1
= 1; q,n2(~n) = ~~ _ a 2; q,na(~n)
=
1 (l;~ _ b2)(a2 _ b2)
(5.1.36)
The Helmholtz equation and the resulting separated ordinary differential equations for these coordinates are therefore
2: m- m(l;~:!nmm - ma~n [fn :tJ + n
G1 = an
d
V(~~
1 _ a2)(l;~ _
+
m- m;
d b2) d~n
[lei + (l;~ :
[V(
a2)
+
kif/;
=
0;
m-
G2 = W; Ga = 2 2)(e b2) dXnJ ~n - a n d~n
(~~ _ b2~~a2 _
b2)
Jx, = 0
m-
tV
(5.1.37)
This is the sort of ordinary differential equation we shall encounter in our future studies. We note that in the three-dimensional Schroedinger equation for a particle the term k 2 (which we have been calling kD is not a constant but is the difference between a constant ki (the total energy of the particle) and the potential energy of the particle as a function of the t's. To have separability the potential energy of the particle must be such as
§5.1]
513
Separable Coordinates
to subtract some function J.ln(~n) of ~n alone from the coefficient of X n in Eq. (5.1.37). This means that the allowed form for the potential energy is (5.1.38) We note also that for ellipsoidal coordinates each of the three separated ordinary differential equations contains k1 and both separation constants k 2 and k a• Recalling our discussion of quantum mechanics in Chap. 2, we can consider the process of separation a process of rotation, in abstract vector space, from a vector set specified by the coordinates (x, Y, z or h , h, ~a) to one specified by the parameters k1, k 2, ka. The factored solutions are the transformation functions (direction cosines) going from the eigenvalues for coordinate position to the eigenvalues for the k's. What we have found is that this transformation, for ellipsoidal coordinates, yields transformation functions which are separated for the coordinates (into factors) but which are not separated for the parameters k. For some of the degenerate forms of ellipsoidal coordinates the factored solutions are also separated with regard to the parameters, which adds considerably to the ease of manipulating the solutions. It is these degenerate forms of the ellipsoidal coordinates, obtained by setting a, b, c equal or zero or infinity, which are of more use and interest than the general form . There are 10 such forms which are recognized as "different" coordinate systems and given special names. These 11 systems (the general ellipsoidal system plus the 10 degenerate forms) are the only systems which allow separation of the wave equation or the Schroedinger equation in three dimensions [even with these the Schroedinger equation separates only if the potential energy has a certain functional form, see Eq. (5.1.38)]. These forms, with the related forms for scale factors h, the determinant S , and so on, are given in the table at the end of this chapter. Degenerate Forms of Ellipsoidal Coordinates. Starting from the transformation (xi - a 2 ) (xi - a 2 ) (xi - a 2 ) a 2(a 2 y'
=
-
(32)
(xi - (32) (xi - (32) (xi - (32) . (32«(32 _ a2)
,
,
XIX2Xa
z = ~
for the generalized ellipsoidal coordinates, we can obtain all the ten degenerate forms listed at the end of the chapter by stretching, compressing, and translating. For instance, stretching all focal distances indefinitely gives, at the center of the ellipsoids, I. Rectangular Coordinates. We set xi in the equations above equal to a 2 + ~, xi = (32 + ~i, and Xa = ~a; we let (3 ::;: a sin I(), where I() can be arbitrary, and then allow a to become infinite. This will give the
Ordinary Differential Equations
514
[CH. 5
coordinates defined under I at the end of the chapter. On the other hand, letting {3 go to zero symmetrizes the ellipsoids into flattened figures of revolution: IX. Oblate Spheroidal Coordinates. Setting a = a, xi = a 2 ~L xi = a 2 - a2~i , X 3 = {3~3 and letting {3 go to zero, we change from ellipsoids to oblate (flattened) spheroids, from hyperboloids of one sheet to hyperboloids of revolution (still of one sheet) and from hyperboloids of two sheets to pairs of planes going through the axis of revolution. In order to have the form given in the table at the end of the chapter we must set x' = z, y' = y, z' = x , which makes z the axis of revolution. Making the ellipsoid into a figure of revolution about the longer axis gives us VIII. Prolate Spheroidal Coordinates, This is done by letting {3 ~ a, according to the following formulas, = a, {32 = a 2 - E, Xl = ~l, xi = a 2 - E~i, xi = a2~i, E~ O. If then we stretch the long axis out indefinitely, we arrive at II . Circular Cylinder- Coordinates. We set a = a, {32 = a 2 - E, xi = a 2 + ~i, xi = a 2 - E~i, Xa = ~3, let E~ 0 and then a ~ 00, giving the simplest rotational coordinates. If we had stretched the long axis out indefinitely before symmetrizing, we would have obtained III. Elliptic Cylinder Coordinates. Setting {32 = a 2 + a2, xi = a 2 + ~i , xi = a 2 a2~i, X3 = ~3 and letting a ~ 00 but keeping a finite, results in this system. On the other hand if we shorten the long axis of the prolate spheroidal coordinates, instead of stretching it, we arrive at the completely symmetric V. Spherical Coordinates. Setting a = a, {32 = a 2 - E, Xl = ~l, xi = a2 - E~i , Xa = a~3, we first let E~ 0, then let a ~ 0, which produces complete symmetry. Finally, if we let {3 be proportional to a and have both go to zero at the same time, we obtain VI. Conical Coordinates. These have spheres and cones of elliptic cross section as coordinate surfaces. They are obtained by setting a = ka, (3 = k'a, k 2 k'2 = 1, xi = ~U(k2 - k'2), xi = a 2[2k2k'2 + (k 2 - k'2) ~i], x~ = a 2[2k2k'2 - (k 2 - k'2Hil and then letting a ~ O. The parabolic systems are obtained by changing the position of their origin to the "edge" of the ellipsoid before stretching. The most general case is Paraboloidal Coordinates. Here we set a 2 = d2 a 2d, {32 = d 2 b2d and set the new origin at z' = d, so that x = x', y = y', and z = z' - 'd. We let xi = d 2 + TJid, xi = d 2 + TJid, and xi = d 2 + TJid and then finally allow d ~ 00 . The new coordinates are
-r.
a
+
+
+
x=
(TJi - a 2)(TJi - a 2)(TJi - a 2) .
a2 - b2 ' z = ·HTJr + TJi +
y =
+
(TJi - b2)(TJi -
TJi - a2
b2)(TJ~
b2 - a2
-
b2)
- b2)
§5.1]
Separable Coordinates
515
which correspond to the surfaces x2
y2
-;;---,; + --7]2 a 7]2 b 2
2
=
7]2 -
2z
For 7] = 7]1 > a they are a family of elliptic paraboloids with traces which are parabolas on the x, z and y, z planes but which are ellipses on the z, y plane. For 7] = 7]2 (where a > 7]2 > b) the surfaces are hyperbolic paraboloids, with traces which are parabolas in the x, z and y, z planes but which are hyperbolas in the x, y plane. Finally for 7] = 7]3 < b (we must let 7]i become negative to cover all of this family) the surfaces are again elliptic paraboloids, pointed in the opposite direction with respect to the z axis. The limiting surfaces are for 7]1 ~ a, which is that part of the y, z plane which is inside the parabola with vertex at z = ja 2 and focus at z = jb 2 ; for 7]2 ~ a, which is the rest of the y, z plane; and for 7]2 ~ b, which is the part of the x, z plane outside the parabola with vertex at z = jb 2 and focus at ja 2 ; while the limiting surface for 7]3 ~ b is the rest of the z, z plane. The scale factors and related functions for these coordinates are given on page 664. As before, other coordinate systems can be obtained by modification of interfocal distances. For instance, if we set a = b, we obtain the rotational VII. Parabolic Coordinates. We set b2 = a 2 - E, 7]~ = ~~ + a2, 7]i = a 2 - E~~, 7]~ = b2 - ~i and then let E ~ 0, giving the simpler system. On the other hand, if we pull out the longer elliptic axis, we eventually obtain IV. Parabolic Cylinder Coordinates. Here we set 7]i = a 2 + ~~, 2 ~i and x = Z' - jb 2, Y = y', z = x' fa and then 2 7]i = b + ~~, 7]~ = b let a ~ 00. This completes all the different degenerate systems which can be obtained from ellipsoidal coordinates. It is of interest to examine the Stackel determinants and the finally separated equations for these cases to see whether there are systematic trends. Confluence of Singularities. We have arranged the scale of all these coordinates so that the functions in and nm are algebraic functions of the coordinate ~n and so that, when the separated equation is put in the form
(5.1.39)
the functions p and q have singularities at the concentration points for the corresponding coordinates. For instance, for the ellipsoidal coordinates,
516
Ordinary Differential Equations
[CH. 5
p and q have poles at ~ = ± a, ± b, and at infinity (i .e., if we change to the variable u = 1/~ the corresponding fun ctions p and q have poles at u = 0, or ~ = 00). We obtain the degenerate forms of the coordinate systems by merging two or more of these singularities. A point which is a singularity for p or q is called a singular point of the corresponding equation, and a merging of singular points is called a confluence of singular points. In the case of prolate spheroidal coordinates, for instance, there is confluence between the singular points at a and b and at -a and -b which, together with a change of scale, puts the singular points for the h equation at ± a and infinity, for the ~2 and ~ a equations at ± 1 and 00. In the spherical coordinates we let a go to zero, so that the singular points for the h equation are at 0 and 00 , and so on. Wherever there is a singular point of the differential equation, there is a singularity (pole, branch point, or essential singularity) of the general solution of the equation. Consequently we ca n say that the factored solution if; = X IX2X a usually has a singularity at all the concentration points of the corresponding coordinate system. We can also say that all the ordinary differential equations obtained by separating the equation V 2if; + k2if; = 0 (which includes a large part of the equations we shall study) are obtained from the general equation with five singular points by confluence of the five singular points to form four, three, or two . Just as the specification of zeros and singularities specifies a function of a complex variable, so a specification of the position and nature of the singular points of a differential equation specifies the equation and its solutions, as we shall see later in this chapter. This is, of course, just another way of saying that the geometry of the coordinate system determines the nature of the solutions of the separated equations, a not surprising result. Separation Constants. Examination of the Stackel determinants of the 11 coordinate systems tabulated at the end of the chapter shows that there are quite a number with some of their elements equal to zero. For instance all the rotational coordinates have .pal = .pa2 = O. This means that the factors X a(~a), for rotational coordinates involve only the separation constant ka• Since ~ a for rotational coordinates corresponds to the angle about the rotational axis, it is not surprising that this factor is particularly simple . We also see that all the cylindrical coordinates have two of the three elements .pnl(~n) equal to zero, which means that only one of the factors X n depends on k l . Therefore for some of the degenerate forms of the ellipsoidal coordinates, the factored solutions achieve a certain amount of separation of the parameters k. Another way of stating this is in terms of the actual process of separating the equation, as it is usually done . One takes the equation
517
Separable Coordinates
§5.1]
and divides by y; = X IX2X3. If the coordinates are separable, so that Eqs. (5.1.29) and (5.1.31) hold, we have 3
~ 1 d [I dXnJ k - 0 Lt h~lnXn d~n n d~n + 12
(5.1.40)
n=!
In some cases it will be possible to multiply this equation by some function of the es so that at least one of the four terms in the resulting equation depends only on one coordinate and the other terms do not depend on this coordinate. We can ' then set this term equal to a constant IX (for a fun ction of ~n alone to be equal to a function of the other es can only be if the function is independent of all es constant), and in such a case the corresponding factor X depends only on the one constant IX (which is a separation constant, either ki or ki). The possibility of separating the equation in this simple manner depends on the nature of the scale factors h n , as Eq. (5.1.40) clearly indicates (all other fa ctors in the nth term are functions of ~n alone, so that, if Ii; were a constant or a fun ction of ~n alone, this term would be all ready for separation without further manipulation) . Three cases may be distinguished : A. Solution Completely Separable for Separation Constants. In this case one can find a multiplier ,",(h,~2,~3) such that two of the terms are functions of one coordinate each (suppose that they are for ~2 and ~3)' Then the term for ~2, (,",/h'if2X2)[d(f~Xdd~2) /d~2] may be set equal to the constant ki and the corresponding term for X 3 may be set equal to the constant k~. Consequently the remaining equ ation for X I is
hihX
I
d~1 VI ~11) + ki + k~ + ,",k~ = 0
and the factored solution takes on the form (A)
(5.1.41)
where two of the factors depend only on one parameter k as well as on just one ooordinate s. Comparison with the Stackel determinant method of separation indicates that the solution can have the form of Eq. (5.1.41) only if two rows of the Stackel determinant each have two zeros, and reference to the table at the end of this chapter indicates that only solutions for rectangular and circular cylinder coordinates have this simple behavior. [Spherical coordinates give a solution of the form XI(h;kl,k2)X2(~2,k 2,k3)X3(~3,k3), which is as simple as Eq. (5.1.41) for
518
Ordinary Differential Equations
[CH. 5
the Laplace equation, when k l = 0.] This type of separation requires a high degree of symmetry in the coordinate system. B. Solution Partially Separable for Separation Constants. In this case only one term (the one for ~3, for example) can be separated the first time round ; the remaining equation
must be multiplied by still another factor, V(h,~2) , in order that a second term can be split off. The factored solutions, therefore, have one of the following forms: (B I) (B 2)
y;
= XI(h ;k2,k3)X2(~2 ;k2k3)X3(~3;kl,k3)
Y; =
Xl(~I ;kl,k2,k3)X2(~2 ;kl ,k2,k3)X3a3 ;k3)
.(5.1.42)
Reference to the table shows that parabolic cylinder coordinates correspond to (B I ) and elliptic cylinder, parabolic, oblate, and prolate spherical coordinates (all the rest of the cylindrical and rotational coordinates) correspond to (B 2) . Here only the last row of the Stackel determinant has two zeros. C. Solution Nonseparable for Separation Constants. In this case no Stackel element in the second or third column is equal to zero and the full machinery of the Stackel determinant must be used to achieve separation. The possible forms are (C I ) (C 2)
if; = Xl(~I ;kl,k2,k3)X2(~2 ;k2,k3)X3(b;k2,k3) if; = Xl al ;k 1,k 2,k 3)X 2(~2 ;k l ,k 2, k 3)X 3(~3 ;k1,k 2,k 3)
(5.1.43)
Only conical coordinates give form (CI ) . Ellipsoidal and paraboloidal coordinates result in form (CI ) , where no Stackel element is zero. It should be obvious that forms (A) are comparatively simple to apply to a given problem, forms (B) are more difficult, and forms (C) are very much more difficult to use. Laplace Equation in Three Dimensions, Modulation Factor. It is obvious that the Laplace equation, v21f; = 0, to which our standard equation reduces when kl = 0, is separable in all the 11 coordinate systems enumerated in the table. But since the Laplace equation in two dimensions separates in more systems than did the two-dimensional wave equation, we may well inquire whether this is also true for three dimensions. Investigations show that there are no more coordinate systems for which solutions of the Laplace equation can take on the form Xl(~I)X2(~2)X3(~3) of type (A) , (B), or (C). However, it turns out that other systems can be found for which a series of solutions of the Laplace equation can be found having the more general form (5.1.44)
§5.1]
Separable Coordinates
519
where R is independent of the separation constants k 2 and k s (see page 509). The same investigation shows that the wave equation is not amenable to this same generalization, so that the additional coordinate systems are separable only for the Laplace equation. The factor R can be called a modulation factor; it modifies all the family of factored solutions in the same way . Its presence modifies the formalism of the Stackel determinant to some extent. For instance, we now set [instead of Eq . (5.1.32)] h 1h2h s/S
where u is a function of h, (5.1.31)]
= ft(h)h(b)fs(~ s)R2U
~2,
h
(5.1.45)
We also require that [instead of Eq.
1 /h~ =
(5.1.46)
Mn/Su
where, in these two equations, the Stackel determinant S and its first minors M n satisfy the same requirements as before (the elements nm of S are functions of ~n only and therefore the minors M n do not depend on ~n). Inserting all this into Laplace equation we first obtain the equation
This separation of terms for X and terms for R is the reason for the insertion of R in both Eqs. (5.1.44) and (5.1.45). If now we can find a form for R which satisfies the equation '\'
1
e [
fJRJ
~ h~fn fJ~n fn fJ~n
k¥R
+U
= 0
(5.1.47)
n
then, using Eq . (5.1.46), we finally obtain (5.1.48) which separates, like the wave equation, into the ordinary differential equations (5.1.49) d~n fn ~~nn + [k¥nl + k~n2 + k~ns]Xn = 0
*
from which the factors X n can be determined. Confocal Cyclides. The fourth-order surfaces most nearly analogous to the quadric surfaces (ellipsoids, hyperboloids, paraboloids) are the cyclides. One interesting property of such a surface is that its inversion in a sphere is also a cyclide . The equation for the surface can most simply be expressed in terms of homogeneous coordinates X, p., v, p
x
= X/P ;
y
= p./P :
z
= v/p
(5.1.50)
520
[cH.5
Ordinary Differential Equations
or in terms of " pent aspherical coordinates" Xl = i(X2 + p.2 + v2 + p2) = i p2(X 2 + y2 + Z2 + 1) X2 = (X2 + p.2 + v2 - p2) = p2(X 2 + y2 + Z2 - 1) (5.1.51 ) Xa = 2pX = 2p2X; x. = 2pp. = 2p2y; X5 = 2pv = 2p2Z
The surface defined by the equation
Lf ~~a: 5
(5.1.52)
an+l 2: an
= 0;
n=1
where ~ and the a's are constant, is a cyclide. The surfaces obtained by taking different values of ~, keeping the a's constant, is a family of surfaces which may be called a family of confocal cyclides. One complete family is generated by taking all values of ~ between a2 and aa, another by taking values between aa and a4, and a third corresponds to the range between a4 and a5. Furthermore these families are mutually orthogonal, so they can be used for coordinate surfaces. We take for h the range between a2 and aa, for b the range aa to a4, and for h the range a4 to aD. The Laplace equation V 2if; = 0 is equivalent to the equation iJ2if; iJx~
+ iJ~ + iJ2if; iJx~
=0
iJx~
+
since X = Xa/2 p2, y = x4/2 p2, Z = XS/2 p2, P = - (X2 iXI)/2p when if; depends on Xa , X4, X5 alone. But also, since Xl and X2 enter in the combination X2 + iXI, we can say that, when if; also depends on XI and X2 (or on p), iJ~ + iJ~ = 0 iJx~
.
z>"
n=O
n=O
""
= (z -
- l)n
n=O
Lb~(z ""
(z - 1)'"
n=O
=0
- l)n
y; in
540
Ordinary Differential Equations
[CR. 5
or else in terms of a fundamental set of solutions about any ordinary point (see page 531). What we have to do next is to determine the relationship between the series coefficients an and b« for different n's so we can compute an in terms of ao and b; in terms of b« and therefore obtain the series for the solution. Insertion of anyone of the six possible series into the equation and equating coefficients of powers of z, (z - 1), or liz to zero will give us a whole series of equations of the general form Dn(a n) =
r nlan-k+l + r n2an-k+2 +
... + I'nkan =
0
(5.2.40)
which are to be solved to obtain an in terms of ao (or b; in terms of bo) . These equations are called k-term recursion formulas. The coefficients r are functions of the constants in the equation, are different for the differ ent singular point and index chosen, and also depend on their subscripts nand j (j = 1, 2, .. . , k). Since the series (5.2.39) are the correct forms, we shall find that r Ok = 0, which ensures that there need not be any a's with negative subscripts. For Eq. (5.2.38), for instance, for z = 0 and for the index X. the recursion formula is Dn(an)
= [n(n
+ X-
+ + + + /I -
XI - Jl - JlI 1) - (/I X)(XI Jl JlI - [en 1)(2n 2X - 2XI - (/I X)(X I Jl JlI
+
+
+
l)]a n
+ 3) - JlJlI + + + + /I - l)]an+l + (n + 2)(n + X - XI + 2)an+2 = 0 Jl -
JlI
(5.2.41)
With such a three-term recursion formula, an explicit formula for a.Ja; is an extremely complicated thing, however. At any rate we have reduced the problem of finding the solution of the differential equation to one of solving an infinite sequence of algebraic equations determining the successive coefficients of the series solution . In other words, we have made yet another transformation, from the continuous variable z to the sequence of integer values of the subscripts o(the coefficients of the power series, the n of an or bn; correspondingly the equation has changed from the differential equation L.(Y;) = 0 to the difference equation Dn(a n) = 0. The coefficients in the difference equation correspond to and are determined by the coefficients p and q in the differential equation. The interrelation between differential equation and recursion formula is given its most suggestive form when we express D n in terms of the difference operators: /jean) = an - an+l ; /j(nan) = nan - (n + l)a n+l ; etc. /j2(an) = o(oan) = an - 2a n+l + a n+2; etc .
For instance, the recursion formula for the series solution of Eq. (5.2.38), given in Eq. (5.2.41), can be written
General Properties, Series Solutions
§5.2]
Dn(a n) = (n
541
+ X - X' + 2)o2(nan ) - (J.l + J.l' + 3)6(nan ) + (X + v)(X + v')o(a n ) - J.lJ.l'o(a n) + (J.lJ.l' + 2n)an
=0
(v' = 1 - X - X' - J.l - J.l' - v)
As long as we are limiting ourselves to the use of power series for the solutions, it behooves us to transform the independent and dependent variables of the differential equation into the form which will give the simplest recursion formula possible. The three-term recursion formula of Eq. (5.2.41), for instance, corresponds to a second-order difference equation; perhaps a transformation of dependent vari able will change this to a two-term recursion formula, which would correspond to a firstorder difference equation and which would be very much simpler to solve. The Hypergeometric Equation. Having transformed the independent variable to shift the positions of the singular points to their standard position, we next transform the dependent variable so that the indices at the singular points in the finite part of the plane are as simple as possible. Such a change of Yt can change only the sum of the indices at the singular point ; it cannot change their difference. We can, however, divide out by ZA(Z - l )«, so that one solution at each point is analytic, the other having the index X' - X or J.l' - J.l. In other words, our solution is Yt = ZA(Z - l)~y, where the Riemann symbol for y is .
y=P
{
0 0 X' - X
1 0 J.l' -
00
J.l
}
v+X+J.l z 1 - X' - J.l' - v
The corresponding equation for y [obtained by substituting Yt = 1)~ into Eq. (5.2.38) or by modifying Eq . (5.2.38) to correspond to the new P symbol] is
ZA(Z -
" + [X
Y
- X'z + 1+ J.l - z-1 J.l' + 1] , Y + (v
+ X + J.l)(1
- X' - J.l' z(z - 1)
=
v)
0
Y
But we have now too many constants for the quantities we have to fix. Only three constants are needed to fix the indices; there are four indices which have not been set zero (one at 0, .one at 1, and two at 00), but the sum of these four must equal unity, so three constants are enough . It will be found most convenient to set the two indices at 00 equal to a and b, the one at 0 equal to 1 - c, so that the one at 1 equals c - a-b. The Riemann P symbol is then F = P {
~ 1- c
1
00
o
a
c-a-b
b
[cH.5
Ordinary Differential Equations
542
and the corresponding differential equation z(z - 1)F"
+ [(a + b +
l)z - c]F' + abF = 0
(5.2.42)
called the hypergeometric equation, is the standard equation for three regular singular points. The analytic solution of this equation about z = 0 is called the hypergeometric function . To obtain its series expansion we set F = ~anzn and insert in Eq. (5.2.42). The coefficient of z" gives the recursion formul a for the series Dn(an) = (n = (n
+ a)(n + b)an - (n + 1)(n + c)an+l + c)(n + l)o(a n) + [n(a + b - 1) -
(n
+
l)c
+ ab]an = 0 . (5.2.43)
This is a two-term recursion formula, a first-order difference equation, with the simple solution ao = l' an = ,
a(a
+
1) .. . (a + n - l)b(b + 1) . .. (b + n - 1) [1 ·2 . .. n] c(c 1) . . . (c n - 1)
--'--~':--=-------'---.---;---;-,-:;-;--,---,-;---,---'----:;--,--~
+
+
The corresponding series F( a,bl c 1z ) = 1 + ab c z
+ a(a +
l)b(b + 1) 2 2!c(c + 1) z
+
(5.2.44)
is called the hypergeometric series (see page 388). It is the analytic solution of Eq. (5.2.42) at z = 0. , It converges as long as Izt < 1, for the next singular point is at z = 1. All the solutions of Eq. (5.2.42) about all its singularities can be expressed in terms of such series. For instance, if we insert the function zl-cF2 (where F 2 can be analytic, for the second index at z = 0 is 1 - c) into Eq. (5.2.42) , we obtain z(z - 1)F~
-
+ [(a + b -
2c
+ 3)z
- 2
+ c]F~ + (a - c + l)(b
- c
+ I)F
2
=
0
which is another hypergeometric equation with analytic solution the hypergeometric series F 2 = F(b - c + 1, a - c + 112 - c] z) . The general solution of Eq. (5.2.42) is therefore . AF(a,blclz)
+ Bzl-cF(b
- c + 1, a - c
+
112 - c] z)
(5.2.45)
Working backward now, the general solution of Eq. (5.2.38) at
z = 0 is
+ +
AZA(Z - 1)I'F(X /1. II , 1 II + BZA'(Z - 1)I'F(X'
-
X' - /1.'1 X - X' + 11 z) II, 1 II X - J.l'LX' - X + 11 z)
+ J.l +
General Properties, Series Solutions
§5.2]
543
and the general solution of Papperitz equation (5.2.36) .at w = a is A
(~)~ w-c (~)~ w-c F 1-
v-
(x + p+. , v
c)
X' - p.'1 X - X' + 11 w - a b w-cb-a
+ B (~)~' (~)~ w-c w-c F (XI + p+. , V
1 - v - X - p.'1 X' - X + 1w-ab-c) 1---w-cb-a
(5.2.46)
Solutions of Papperitz equation at w = b or w = c can be obtained by interchanging X, X' with p., p.', etc., since the equation is symmetrical for interchange of singular points and corresponding indices (if we remember that Vi = 1 - X - X' - p. - p.' - v). The formula (5.2.45) expresses the general solution unless 11 - c] is zero or an integer, in which case the series for one or the other of the solutions will have a zero factor in the denominator of all terms above a certain term in the series. For instance, if c = 3, the series for the second solution will have its first two terms finite but all the higher terms will be infinite because of the term (2 - c + 1) in the denominator. This is an example of the special case, mentioned on page 529, where the indices for a given singular point differ by an integer. As mentioned on page 532, we must then find the second solution by means of Eq. (5.2.6). Here Yl = F(a,blclz) and p = (c/z) + [(a + b + 1 - c)/(z - 1)] so that e-!pdz = z-c(1 - z)c-a-b-l. The second solution is Y2 = F(a,blclz)f[F(a,blclz)]-2z-C(1 - z)c-a-b-l dz As long as Izl < 1, we can expand (1 - z)c-a-b-l[F(a,blclz)]-2 in a power series in z, which we can write Yo + Y1Z + . . ' . The series for the integrand is then ~ ZC
+~ + . . . + Yc-l + Yc + Yc+1Z + . . . zC-l Z
where we have assumed that c is a positive integer, so that Yc is multiplied by the zero power of z and Yc-l is divided by the first power of z, and so on, with no fractional powers entering. In this case Yc-t/z will integrate to a logarithmic term, and the second solution will be Y2 = F(a,blclz)Yc-l In z + (l /z C-1)(h o + h1z
+ .. . + hc_2zc- 2 + hczc + . . .)
where the coefficients h n are functions of a, b, c, and n. Functions Expressible by Hypergeometric Series. The series of Eq. (5.2.44) can express a great variety of functions. For instance, some
544
[CR. 5
Ordinary Differential Equations
of the simpler cases are (1
+ z)n =
F( -n,blbl
-
In(1
z);
+ z)
=
zF (I ,1121 - z)
The separated equation for ~2 for circular cylinder coordinates and for ~a for spherical, parabolic, prolate, and oblate spheroidal coordinates is d [_ ~dX]
1.
~d~ V.1-~-df d
2X
d~2
or
+
[i ~+
1
i ] d~
+~_
1
dX
This has the form of Eq. (5.2.36) with a - 1
= -i =
Jl
+ Jl'
1, XX'
-
=0 =
the usual Eq. (5.2.35), X + X' + Jl symbol for the solution is therefore
=
JlJl',
k~
+
1-~2
- ~2
k~ _
X
=0
1X = 0
+ 1, b = 1'1"
=
+ J.l.' + + I'
-1, c = 00, X + X' [and, of course, = 1]. The Riemann
-k~
1"
X=P{~o -~0 -k~~} a
and the general solution about
X=
A
~
=
+ 1 is (from
Eq. 5.2.46)
~ F (1 + k a, 1- kal ill ; ~) + B VI + ~F (i + k a, i - kal ill ; ~)
These solutions are called Tschebyscheff functions . They turn out [see Eq. (5.2.54)] to be proportional to sin (kip) and cos (klp),respectively, where ~ = cos Ip, as a transformation of the differential equation above will show. These functions are discussed in Sec. 5.3. The separated equation for ~2 for spherical coordinates and also for spheroidal coordinates when k, = 0 is d
2X
d~2
+
[1 ~-
1
1 ] d~
+~+
dX
1
k2
- [ e --=. 1 + (~2 -
k 2/ 2
la)(~ + 1) - (~2
k 2/2
] 1\(~ _ 1) X = 0
-
which is called the L egendre equation. It corresponds to Eq. (5.2.36) if a = + 1 b = -1 c = 00 X = - X' = m/2 = " = - II' I' = - n 1" = n + 1, where we have set k a = m and k 2 = n(n + 1) in order to make the results easier to write out . Therefore the solution corresponds to the Riemann symbol ,
"
X = p {
I"'"
m~2 m/~ ':n ~}
-m/2
-m/2
n
+1
f"""
,
General Properties, Series Solutions
§5.2]
and the general solution about X
= A(1 -
~2)m/2F (
~
545
"'" 1 is [see also Eq. (5.2.52)]
m - n, m + n
1 - - ~) + 111 + ml2
~)m/2 F (-n, n + 111 + B ( 11 + _ ~
1 - ~) - ml--
2
(5.2.47)
The first solution is known as the Legendre function of the first kind, and the second solution is called the L egendrefunction of the second kind. If m is a positive integer, this second solution, as written, will "blow up" and a second solution will have to be constructed by use of Eq. (5.2.6) (see page 532). These Legendre functions and the related Gegenbauer functions will be discussed again in a few pages and in very considerable detail in Sees. 5.3 and 10.3; for they are of very great importance in our future discussions. Analytic Continuation of Hypergeometric Series. As an exercise in dealing with these solutions we can derive an expression for the behavior of the hypergeometric series as z approaches unity. By referring to the Riemann symbol above Eq. (5.2.42), we see that the indices for the singular point z = 1 are and c - a-b. Using Eq. (5.2.46) with a = 1, b = 0, c = 00, >. = 0, 'A' = c - a - b, Il = 0, Il' = 1 - c, p = a, v' = b (in other words, exchanging the two singular points and 1), we have that the general solution of the hypergeometric equation (5.2.42) about z = 1 is
°
°
AF(a,bla
+ b - c + 111 + B(1 -
z) z)c-a-bF(c - b, c -
al c -
a - b
+ 111 -
z)
Now the hypergeometric series F(a,blclz) is a solution of Eq. (5.2.42), and by analytic continuation, it must be some combination of the two solutions about z = 1. In other words we can be sure that F(a ,blclz) = aF(a,bla + b - c + 111 - z) + ,8(1 - z)c-a-:bF(c - b, c - al c - a - b + 111 - z)
(5.2.48)
If we could somehow determine the values of the coefficients a and ,8, we should have a means of computing the exact behavior of F(a,blclz) at z = 1 and even beyond, almost out to z = 2. On page 388 we showed that, when b < c < a + b,
+ b - c) (1 _ )c--b F( a,bl cIz) --;:; r(c)r(a r(a)r(b) z where we have substituted gamma functions for the factorials, so that
a, b, and c can have nonintegral values. All that this equation shows
is that, when c < a + b, the highest order infinity at z = 1 is of the form (1 - Z)e-a-b, and it gives the coefficient of this term. There are,
546
,Ordinary Differential Equations
[tH .5
of course , other terms of the form (1 - z)c-a-b+l " (1 - z)e-a-b+ 2 etc 0'' some of which may also go to infinity at z = 1, but very close to z = 1 they are "drowned out" by the (1 - z) c-a-b term. The rest of the terms can be supplied, however, by use of Eq. (5.2.48). Since F(c - b, c - al c - a - b + 11 1 - z) ~ 1, we have that z-+1
the principal term on the right-hand side of Eq. (5.2.48), as z ~ 1, is . ~(1 - z)c-a-b as long as c < a + b. Comparing this with the results referred to above we see that (3
= [r(c)r(a
+b-
c)]j[r(a)r(b)]
which fixes the coefficients of all the terms in negative powers of (1 - z). But we can go further than this in the case that c < 1. We can -use the limiting formula above in reverse by seeing what happens to Eq. (5.2.48) 'when z is allowed to go to zero. In this case the left-handside goes to unity, of course. Using the limiting formula of page 388 on the right-hand side, we have eventually 1 ~ a rea
+b-
c + l)r(c - 1) Zl-c r(a)r(b)
z-+O
+ (J r(c -
a - b + l)r(c - 1) Zl-c r(c - a)r(c - b)
If c < 1, the right-hand side will become infinite at z = 0 unless the two terms just cancel each other. This gives a relation between a and (3 which enables us to solve the problem. Inserting the value of ~ already obtained and utilizing the property of the gamma function, I'(u + 1) = ur(u), we obtain
1----+ (a
+b_
c) rea
21-+0
+b-
c)r(c - 1) r(a)r(b)
[a _r(c r(c -
a - b)r(c)] Zl-c a)r(c - b) + finite terms
In order that this be true for c < 1, we must have the quantity in square brackets equal to zero, which gives an equation for a . Consequently, at least for c < 1, c < a + b, we have the useful formula
r(c)r(c - a - b) F(a,blcjz) = r(c _ a)r(c _ b) F(a,
+ r(c)r(a +bI'(cj I'(b)
bl a + b -
c) (1 _ z)c-a-bF(c _ a c ,
+ 111 bl c - a -
c
z) b + 111 - z) (5.2.49)
which enables us to extend the solution through the singular point at
z = 1. In the next section we shall show that the equation is valid over a much wider range of values of c than our present derivation would lead us to believe (for c larger than a + b, for instance). In fact, except
547
General Properties , 'Series Solutions
§5.2]
for the places where the gamma functions in the numerators go to infinity, this formula is universally valid . Another set of useful formulas relating hypergeometric series may be obtained by use of this" joining equation " plus a certain amount of manipulation of the hypergeometric equation (5.2.42). In this equation we set a = 2a, b = 2f3, C = a + f3 + i and change the independent variable from z to u = 4z(1 - z), giving the equation dP u(u - 1) du 2
+ [(a + f3 + l)u -
(a
+ (3 + i)] dF du + a{3F
= 0
which is the hypergeometric equation again, with new parameters a, f3 and (a + f3 + i) instead of a,. b, and c. Therefore we have shown that the function F(a, f31 a +.f3 + il 4z - 4z 2) is a solution of Eq. (5.2.42) for a = 2a, b = 2f3, C = ~ + f3 + i and must be expressible in terms of the two solutions aboute = 0 [see Eq. (5.2.45)] :
+ f3 + il z) . + Bzi F(a, if31 a + f3 + il 4z
AF(2a, 2f31 a
a-PF({3
- a
+ i, a -
f3
+ il i-a -
f31 z)
2 However, 4z 2) is an analytic function of z at z = OJ therefore the second solution, having a branch point, cannot enter and B is zero. Also, since F = 1 for z = 0, A must equal unity, We therefore have
F(2a, 2f31 a
+ f3 + il z)
= F(a, f31 a
+ (3 + il 4z
- 4z 2 )
(5.2.50)
This formula, relating an F for 2a, 2f3, and z to an F for a, f3, and Z2 might be called a duplication formula for the hypergeometric function. Gegenbauer Functions. We mentioned earlier that it is sometimes desirable to have the canonical form for three regular singular points so that two of them are at ± 1 rather than 0 and 1. This is particularly true when the indices at +.1 are the same as the indices for - 1. This suggests the Riemann symbol I/;
=P
- I 0
{ -{3
o1 -f3
oc
-a
a
Z
}
+ 2f3 + 1
The corresponding equation (Z2 -
1)1/;"
+ 2(f3 + l)zl/;'
- a(a
+ 2{3 + 1)1/; =
0
(5.2.51)
is called Gegenbauer's equation [compared with the Legendre equation on page 544]. It is a satisfactory form for the equations for ~2 for circular cylinder and spherical coordinates. The solutions can, of course, be expressed in terms of hypergeometric functions:
.
548 AF ( -a, a
+ 2{j + 111 + (j11 ~ z)
+ B(l + z)-fJF ( " " or
[CR. 5
Ordinary Differential Equations
aF ( -a, a
+ 2{j + 111 + (j11 ;
+ b(l -
{j, a
+ {j + 111 _
(j11
~ z)
z)
z)-fJF ( -a - {j, a
+ {j + 111
_ (j11 ; z)
The solution which will be most useful to us has the various forms 2{j + 1) F( + 2 + 1 _ 11 + I ++l)r({j + 1) a {j , a (j"2""2"Z (5.2.52) r(a + 2{j + 1) sin [1r(a + (j)] 2fJr(a + l)r({j + 1) sin (1r{j) -Ft« + 2{j + 1, -all + (jl i + ·iz) (1 + z)-fJ sin (1ra) I - r(l _ (j) sin (1r{j) F( - a - {j, a + {j + 1 1 - (jl "2" + "2"z)
fJ ( ) r(a T CIt Z - 2fJr(a TfJ( ) = CIt z
1 _
1
)
1
1
where we have used Eq. (5.2.49) and the relation sin(1ru)r(u)r(l - u) = 1r in order to obtain the second form . The function T~(z) is called a Gegenbauer function. If a is not an integer, T has a branch point at z = -1 (unless (j is an integer). When a is zero or a positive integer, T is a finite polynomial in z and, of course, is analytic at z = ± 1. The second solution about z = 1 is also sometimes used . As a matter of fact (l - z2)lfJ times this second solution,
P~+fJ(z) =
r(l
= (a
~ (j)
e :Y ~
fJ
+ {j + 111 Z2)jfJ {sin {1r(a + (j)] .
(jl i - iz)
F(-a - {j, a
+ 2{j + 1)(1 + l)r({j + 1) sin (1ra) . F(a + 2{j + 1, -all + (jl i - iz) - s~n t{j~ F(a + 2{j + 1, -all + (jl i + iz)} SIn 1ra
2fJr(a
is often called the generalized Legendre function of z. When a is an integer n = 0, 1, 2, . . . , it can be shown by expansion of the polynomial and term-by-term comparison that F(n
+ 2{j + 1, -nil + (jl i + iz) = (-l)nF(n
2nr(n
+ 2{j + 1, -nil + (jl i + 1) d (1
r({3 {3
n
+ +
1)(1 - Z2)11 dzn
- z
- iz) 2) +11 n
and, consequently, T (z) n
=
(-l)nr(n 2n+fJnlr(n
+ 2{3 +
+ (3 + 1)1) (1 -
Z2)-1l
~ (1 -
dzn
z2)n+1l
(5253) ..
§5.2]
General Properties, Series Solutions
549
which can be called a Gegenbauer polynomial (it differs, by a numerical factor, from the polynomial C~+l often called a Gegenbauer polynomial). When 13 is an integer m = 0, 1, 2, . .. but a is not, it will be shown in the next section that T>;; has logarithmic branch points at z = ± 1 but that (1 - z2)-lmP::+m is analytic over the whole range -1 ~ z ~ 1. Finally, when both a and 13 are positive integers (or zero), Tr;:(z) = (1 - z2)-lmp,;:(z) and both functions are analytic over this range of z, Since 1 dn n! d n+ 2m (1 _ z2)m dzn (l - z2)n:-m = (-I)n (n
+ 2m)! dz n+2 m (Z2
- l)n+m
we have .
Tr;:(z)
d n+2m
1
+ m)! dz n+2m (Z2
.,,---,---0------,--,--
2.1l+ m(n
- l)n+on
which is sometimes called a tesseral polynomial. The case of m = 0 is of particular importance, enough importance to merit giving the polynomial a special symbol and name. The function
is called a Legendre polynomial [see Eq. (5.2.47)]. in the rest of the book. We see that
We shall meet it often
We shall have much more to say about these functions in the next section. In the special case 13 = ±i, a = n = 1, 2, 3, . . . , the polynomials T can be shown (by direct expansion) to have the following forms: T;i(Z)
=!n \j;~ cosh [n cosh"? (z)]
THz) =
~z22~ 1 sinh [(n + 1) cosh-
(5.2.54) 1
(z)]
called Tschebyscheff polynomials. Incidentally, the general solution of the separated equation for ~2 in circular cylinder coordinates and for ~3 in parabolic, prolate, and oblate spheroidal coordinates, (Z2 - l)f'
+ zy;' -
a 2y; = 0
has for general solution
y; ftl!l
= AT;I(z)
+ B ~ T1_1(Z)
a comparison with the equation for T will show.
550
Ordinary Differential Equations
[cH.5
One Regular and One Irregular Singular Point. For this equationit is usual to put the regular singular point at z = 0 and the irregular one at z = 00. Equations of this sort occur for the following separable coordinates: . 1. For the wave equation in circular cylinder (z = kl~l, if; = Xl), spherical (z = klh, if; = v'z Xl), and conical (z = kl~l, if; = v'z Xl) coordinates the equation is the Bessel equation d2if1 dz2
+ ~Z dz dif; + (1
_
2) n if; = Z2
0
2. For the wave equation in parabolic cylinder coordinates (z = "H i, i~~, if; = Xl , X 2) 2if; d 2 dz
+ Z~ #dz + (k 2 + 2a) if; = 0 z
3. For the wave equation in parabolic coordinat es (z Xl, X 2)
=
~~, ~~ ,
if;
=
which includes the parabolic cylinder function as a special case. 4. For the Schroedinger equation for one particle in a coulomb 1/r field, in spherical coordinates, for the radial factor (z = h, if; = Xl) 2if; d dz2
+ ~ dif; + [-E + 2a Z dz
_n(n Z2+ l)J if; = 0
Z
This suggests that we study the general equation d2.I.
dZ~
+ p(z)
d; +
d'"
q=
q(z)if;
-t»
+
= 0;
ez
a )
p =
(1 - A - A')
+ C~')
z
;
(5.2.55)
This is not the most general equation with a regular singular point at z = 0 and an irregular one at infinity. The function p could have a constant, and both p and q could have terms like az or bz, etc ., added to the expressions shown in Eq. (5.2.55). But these additional terms would make the essential singularity in the solution at z = 00 more "singular," and since none of the equations coming from the separation of the wave equation exhibit such additional terms, we shall not include them here . A few more complicated cases will be included in the problems , however! Returning to Eq. (5.2.55), we see that the indicial equation indicates that the expansions about the regular singular point at z = 0 are Z>'Ul(Z) and Z>.IU 2(Z) , where Ul and U2 are analytic functions at z = O. As before,
§5.2]
General Properties, Series Solutions
551
we arrange it so that one solution is analytic, by setting'" that f(z) = Ui(Z) + ZA'-AU2(Z). The equation for f is then
f" +
+ X-
[(1
X')/z]f'
+ [(2a/z)
- k 2]f
=
0
= zAf(z), so (5.2.56)
Setting z = 1/w, we next examine the singular point at infinity. The equation, in terms of w, becomes 2
df dw 2
+ [' 1 + X'
- X] df
dw
W
+ [2a
w3
2
_ k ] f = 0 w4
which indicates an irregular singular point at w = 0, due to the terms 2a/w 3 and k 2/ W 4• This equation has no indicial equation, so both solutions have essential singularities at w = O. However, setting f = e-k/wF, we can arrive at an equation F"
+ [2k + 1 -
X + X'] F' _ [k(l
w2
+ X-
w3
W
X') - 2a] F = 0
which does have an indicial equation, with one root, 2ks - [k(1
+X-
X') - 2a]
=0
Therefore a solution for F is WtlVi(W), where Vi is an analytic function at w = 0 and {j = ~(1 + X - X') - (a/k) . Another solution for f. is ek/ ww lJ'V2(W ) , where {j' = ~(1 + X - X') + (a/k) and V2 is analytic at w = O. Thus the essential singularity at w = 0 (z - ? 00) is of the form e+k/w = e+kz. ' We can now return to Eq. (5.2.56) in z and set f = e-kzF(z) (or'" = zAe-kzF) and be -sure that one solution for F is analytic at z = 0 and has a branch point only at z -? 00 . The equation for F in terms of z is F"
+ [1 + ~ -
X' _ 2k] F' _ [k(l
+ X~
which reduces to its simplest form when we set z a = ~(1 + X - X') - (a /le), d 2F x dx 2
+ (c -
X') - 2a ] F = 0
= x/2k, c = I' + A - X',
dF x) dx - aF = 0
(5.2.57)
which is called the confluent hypergeometric equation. This name is chosen because Eq. (5.2.57) comes from the hypergeometric equation (5.2.42) by suitable confluence of the singular points at z = 1 and z = 00 . The confluence may be more easily seen by starting with the Papperitz equation (5.2.36) for the case a = 0, c = 00,
v/' + [1 - X - X'
+1-
+[
+
z
-XX'b Z2(Z - b)
p. -
z- b
p.'] v/
p.p.'b z(z - b)2
+
11(1 - X - X' - p. - p.' z(z - b)
II)] '" = 0
Ordinary Differential Equations
552
[cH.5
We now let X = 0, 1 - X' = c (using c now for another constant, not , the position of the third singular point which wentto infinity), p.' = -b" and p. = a - v (not the a which went to zero). We finally arrive at the confluent hypergeometric equation (5.2.57) by allowing the second singular point to merge with the third at infinity, that is, we let b ~ 00. At the same time that the singular points merge, one of the indices at each of these points (that is, p.' and v') also tends to infinity. ' The one solution of Eq. (5.2.57) which is analytic at x = 0 may be found by substituting the general series form F = 1:anx n into the equation. The coefficient of the power x n gives the recursion formula (n
+ 1)(n + c)an+l -
(n
+ a)a n =
(5.2.58)
0
which is again a two-term recursion formula, with a simple solution. Therefore the solution which is analytic at z = 0 is given by the series _ F(alclx) - 1
~ a(a + 1) a(a + l)(a + 2) + c x + 2!c(c + 1) x + 3!c(c + l)(c + 2) x + 2
3
• • •
(5.2.59) which is called the confluent hypergeometric series. This series converges in the range - 00 < x < 00 . Using the methods discussed on page 388 we can obtain an indication of the asymptotic behavior of this series. For large values of z the terms in the higher powers of z preponderate. But to the first approximation in z/n and when a and c are integers, the term in the series with zn (n large) becomes (c - 1) !(a + n)! (c - I)! nr:» (c - I)! zn zn '"'-' z" '"'-' 7---:+-; -,------.,---.,.-, n!(a - 1)!(c + n)! - (a - I)! n! - (a - I)! (n - a + c)!
----;':;------':;-;-';;-,------;--'--c-;
Therefore when z is large enough, the series approaches the series (c - I)! \ ' zn (c - I)! \ ' zm (c - I)! Z (a - I)! (n - a + c)! ~ (a - I)! za-c m! = (a _ I)! za-Ce
Lt
Lt
n
m
.
Inserting gamma functions instead of factorials we finally arrive at the indication that F(alclz) ~ r(c) za-CeZ (5.2.60) z-->
00
r(a)
In the next section we shall show that this asymptotic formula is valid over a wider range of values of a and c and z than the present derivation would concede. By reversing the procedure by which we obtained F from if;, we see that the general solution of Eq. (5.2.55) about z = 0 is if; = Ae-kzzxF
C+ ~ -
X' -
+ Be-k'zX'F
III + X -
C- ~ +
X' -
X'I 2kz)
~ 11 ~ X+ X'I 2kZ)
, (5.2.61)
General Properties, Series Solutions
§5.2]
553
I
unless (A - A') is an integer, in which case the second solution contains a logarithmic term and must be obtained by use of Eq. (5.2.6) . From this general solution, we can also see that a second solution of the confluent hypergeometric equation is xl-cF(a - e
+ 112 -
elx)
However, we can also see that, if we set z = - ~ in Eq. (5.2.57) and set F = e-W 2 , the equation for F 2 is also that for a confluent hypergeometric function . In other words, another solution of Eq. (5.2.57) is e"'F(e - aiel - z) . This is not a third independent solution, however, for series expansion and series multiplication show that e"'F(e -
aiel - z)
(5.2.62)
= F(alelx)
Incidentally, comparison with Eq. (5.2.60) indicates that an asymptotic formula which might be satisfactory for z large and positive might not be satisfactory at all for z large and negative. In fact if Eq. (5.2.62) is correct and Eq. (5.2.60) is valid for Re z ~ 00, then F(alelz)
~ r( r~) z-> -
e
00
a
) (-z)-a
(5.2.63)
However, we should postpone further discussion of this matter until next section, when we shall be much better equipped for the discussion. Comparison of the equations on page 550 with Eq. (5.2.61) shows that the general solution of the Bessel equation is 1/; = e-iz[Azn F(i
+ nil + 2nl2iz) + Bz:» F(i -
nI1-2nI2iz)]
or, using Eq. (5.2.62), 1/;
= Ae-izznF(i
+ nil + 2nl2iz) + Be izz-nF(i -
nil - 2nl-2iz)
The general solution for the wave equation in parabolic coordinates, which includes that for the parabolic cylinder, is 1/; = Ae- ikz zmF
G+ + i; 11 + m
2m l2ikz)
+ Be ikz z-m F (!2 - m. -
ia
k
11 -
2m l-2ikz)
and the general solution of the Schroedinger equation for a particle in a coulomb potential field is, for E = -k 2 , 1/;
=
Ae- ikz zn F (n
+ 1 + i; 12n + 2
12ikZ)
+ Be iko z-n-l F ( -n _
i; 1-2n
l-2ikz)
554
Ordinary Differential Equations
[CR. 5
Many other functions (such as the error function and the incomplete gamma function) can be expressed in terms of the confluent hyper': geometric function. Asymptotic Series. Although we shall postpone a complete discussion of the behavior of the confluent hypergeometric fun ction about z = 00, we should investigate the series expansion of the solutions suitable about the singular point at z = 00 . Making the transformation w = l/x in Eq. (5.2.57), we see that the confluent hypergeometric equation has the form (5.2.64) about the point at infinity. Although this is an irregular singular point, there is one solution of the indicial equation, s = a. Inserting F = :ZanWa+n into Eq. (5.2.64) we find again a two-term recursion formula for the a's (n + l)an+l + (n + a)(n + a - c + l)an = 0 Therefore a series expansion for F about w = 0 (z = F
1
=
Az-a [1 _ a(a - c 1!
+ 1) ~ + a(a + l)(a z
c
00)
is
+ l)(a -
c + 2) .
2!
.~ - ...J
(5.2.65)
where we have inserted l/z for w again. This should be compared with Eq. (5.2.63). Comparison with Eq. (5.2.60) suggests that a second solution should be of the form we-a el/w:Zbnw n. Solving for the b's results in a two-term recursion formula similar to the one above. The second series solution about z = . 00 is then F 2 = BzG-ce z
[1 + (1 - a)(c 1! + (1 -
a) ~ z a)(2 - a)(c - a)(c - a 2!
+ 1) 1. + .. .J Z2
The chief trouble with these two series solutions .is that they do not converge, except at z = 00 . The divergence is of a peculiar type, however, for when z is large but finite, the series first converges and then, as we take more and more terms, eventually diverges. To be specific, we find that the difference ~n(z) between F 1 and the first n terms of the series in Eq. (5.2.65) first diminishes as n is increased and then increases without limit. For small values of z the minimum value for ~n comes at a relatively small value of n, and this minimum value is relatively large . As z is increased, the minimum for ~n comes for larger and larger n and the value of the minimum gets smaller and smaller. For any finite
General Properties, Series Solutions
§5.2]
555
value of z, therefore, it is possible to obtain a fairly accurate value of F by taking a finite number of terms, whereas many more terms would result in a less accurate answer. As long as z is finite, there is a certain irreducible error in the computed result even if the optimum number of terms is used, but this error rapidly diminishes as z increases. In many cases of interest, this irreducible error is already smaller than 0.1 by the time z is as large as 10. In many such cases the first term only in the expansion is a satisfactory approximation for z > 20. Such series, which are divergent but which can be used to compute a value that never exactly equals the "correct" value but rapidly approaches this "correct" value as z is increased, are called asymptotic series. They were discussed in some detail in Sec. 4.6. In some respects they turn out to be more useful than convergent series as long as they are handled with tact and understanding. Some of the requisite understanding will be supplied in next section; the tact must be left to the user. Two Regular, One Irregular Singular Point. The equation for hand b for elliptic cylinder coordinates has the form (I/; = Xl, X 2, Z = hid, ~2) (Z2 - 1)1/;"
+ zl/;' +
(h 2z2 - b)1/; = 0
and the equations for hand b for prolate and oblate spheroidal coordinates have the form [X = (Z2 - l)i"l/;] (Z2 - 1)1/;"
+ 2(a + l)zl/;' + (h 2z2 -
b)1/;
= 0
(5.2.66)
The first equation is but a special case of the second equation (a = To show the singular points we rewrite the second equation as 21/; d 2 dz
+ [aZ -+ 11+ az + 1] dl/; + [h2 + Z2h2 -+ 1 dz
-t).
bJ I/; = 0
1
It has regular singular points at z = ± 1, with indices 0 and - a at both and an irregular singular point at z = 00 . The equation is not the most general one having two regular and one irregular singular points, but it is the equation which is encountered in our work. The singular points are at standard positions (we could change the regular ones to 0 and 1, but ± 1 is more convenient), and one solution at each regular point is analytic, as we have previously required for a canonical form; therefore we shall consider Eq. (5.2.66) as the canonical form for this type of equation. If we insert a power series for z in Eq. (5.2.66), we obtain the threeterm recursion formula (n
+ 1)(n + 2)a + 2 + [b n
n(n
+ 2a + 1)]a + ha _ 2 = n
n
0
from which we can obtain the two fundamental solutions about the regular points z = O. The series expansions about the singular points,
556
Ordinary Differential Equations
[cH.5
in powers of 1 - z or 1 + z, produce four-term recursion formulas, which are even more difficult to compute and analyze. In such cases we try expansions in series of appropriate functions rather than in series of powers of (1 ± z) . For instance, by transforming the independent variable in the elliptic cylinder case (a = -i) we can obtain d 2'1/; Z = cos 1/>; dl/>2 + (b - h2 cos" 1/»'1/; = 0
z = ~ (x +~); x = e i
2'1/; 2d x dx2
d'l/; + x dx +
(h"4
2 x
2
+ v"2 -
(5.2.67)
';
v
1)
b + "4 x2 'I/; = 0
The first of these equations is called Mathieu's equation. The second is quite interesting, for it is algebraic in form but the transformation from z to x has changed the singular points so it now has two irregular singular points, one at 0 and the other at eo , The first form of the equation can be used to bring out an interesting property of the solutions. Since cos- I/> is periodic in I/> with period 7r, if '1/;(1/» is a solution of the equation, then '1/;(1/> + 7r) is also a solution. For instance, if '1/;1 and '1/;2 are two independent solutions, we shall have, in general, '1/;1 (I/> + 7r) = au'l/;l (I/» + a 12'1/;2(1/» and '1/;2(1/> + 7r) = a21'1/;l(l/» + a22'1/;2(1/» where the a's are constants determined by the parameters band hand by the particular set of solutions '1/;1 and '1/;2. From this fact it can be shown that it is possible to find a solution of Eq. (5.2.67) which has the form ei • .; times a function which is periodic in 1/>. This solution (call it '1') would be, of course, some combination of '1/;1 and '1/;2, '1' = A'I/;l(l/» + B'I/;2(1/» = ei'W(I/» , where F is periodic in I/> with period 7r ; that is, F(I/> + 7r) = F(I/» . Using the properties of the 'I/;'s we have that eu(.p+r)F(1/>
+ 7r)
+ + +
+ +
= A'I/;l(1/> 7r) B'I/;2(1/> 7r) = (Aal1 Ba21) 'I/; 1 (I/» (A a12 = erueuW(I/» = eru[A'I/;l(l/» + B'I/;2(1/»]
+ Ba22)'I/;2(1/»
Equating the coefficients of '1/;1(1/» and '1/;2(1/» we have two simultaneous equations for A and Band eru : A(all - eru) + Ba2l = 0; Aa12 + B(a22 - eri.) = 0 For this to be soluble, the determinant of the coefficients must be zero: ru (a u - e ) a 21 . = 0 al2 (a22 - er,,)
l
I
General Properties, Series Solutions
§5.2]
557
This is a quadratic equation in e"i. with two roots, corresponding to two independent solutions of Eq. (5.2.66) , The possibility of such solutions of Eq. (5.2.67) is called Floquet's theorem. Referring back to the second form of Eq. (5.2.67), Floquet's theorem states that it should be possible to set up two independent solutions of the form x'(x = ei. Such a series is periodic in 4>, with period 7l', as we have just shown it must be, and it can represent the function around both irregular singular points z = 0 and z = 00. We therefore choose as a solution
L ..,
if;
=
L ane2in~ ee
anxM-2n = ei'
n=-·oo
(5.2.68)
n=-oo
Setting this into the second Eq. (5.2.67) we arrive at the basic recursion formula h2a n+1 [2h 2 - 4b 16(n tsF]a n h 2a n_1 = 0
+
+
~2(an)
+ :2 (h2 -
+ b
+ 4(n
+ + i-s)2]an = 0
(5.2.69)
This is a three-term recursion formula for the unknowns an and s. It would be much more desirable to make other transformations of dependent or independent variable to obtain a two-term recursion formula. Unfortunately, as will be shown later in this section, such a pleasant outcome is not possible for equations of this complexity, so we are forced to undertake the analysis of three-term recursion formulas. If we start with arbitrarily chosen values of ao, aI, and s, we can compute all the other a's for positive and negative n's. But in such a case the a's will not necessarily become smaller as n increases, so the series will not, in general, converge. Only for certain values of s and of al/aO will the series converge, and we must find a way to compute these certain values. Continued Fractions. First, of course, we must make sure that the series can converge for some values of a I/ao and of s. To do this we compute the value of an/an_l for large positive n and of an/an+l for large negative n. If these approach zero as n ~ ± 00 , we can be sure the series converges over the range 0 < z < 00 .
an an-l an an+l
+ tsF + 16(n + i-S)2 +
16(n
-h2 2h 2 - 4b -h2 2h 2 - 4b
+ h2(an+I/an) + h2(an_l/an)
(5.2.70)
The first of these equations shows that, when n is large and positive, an/an-l is small and approaches zero proportional to 1/n 2 if the next ratio an+l/an also approaches zero for large n, for then for large enough n the
558
[CR. 5
Ordinary Differential Equations
term 16(n + tS)2 is much larger than all the other terms in the denominator and an/an_l -+ - (h 2/16n 2) . Use of the methods of page 388 indicates that for large values of x (t/J large and along the negative imaginary axis) l/I is approximately proportional to x' cos (hx/4) . Likewise the second of Eqs. (5.2.70) shows that, when an-dan is small for large negative values of n, then an/an+l -+ - (h2/ 16n 2) . Therefore for very small values of x (t/J large and along the positive imaginary axis) l/I is approximately proportional to x' cos (h/4x). However, we still must find out how to compute the ratios an/an-l for small values of n . and also how to determine the correct value of s, Equations (5.2.70) can give us the hint which sets us on the right track, however. If we cannot start from ao, al and work outward, perhaps we can start from very large values of n and work inward. Suppose that we start from a value of n that is so large that an+da n is very nearly -h 2/ 16 (n + tS)2. Substituting in the first equation gives a fairly ' accurate expression for an/an-I; then substituting this in again into the equation for a n-dan- 2 gives us a still more accurate expression for a n-dan- 2; and so on, until we arrive at a value for at/ao al ao
= 16(1 +
-h2 2 tS)2 2h - 4b - 16(2
+
+ tS)2 ~ 2h 2 _
4b _
h4
16(3
+ ts)2 + 2h
2
-
4b - - -
Similar use of the second equation for negative n's gives us -h 2 h4 2 16(1 - !S)2 + 2h - 4b - 16(2 _ tS)2 + 2h 2 _ 4b _
h4
16(3 - !S)2
+ 2h 2 -
4b - - -
These expressions are called continued fractions . Questions of convergence can be answered by adaptation of rules for series. Using Eq. (5.2.69) for n = 0 gives us a formula relating at/ao and a_I/ao:
(5.2.71) This, together with the continued fraction equations above, results in an equation from which s can be determined in terms of band h. We compute the continued fractions for assumed values of s and then check it by means of Eq. (5.2.71). If it does not check exactly, we use the square root of the right-hand side of the equation for a new value of s to insert in the continued fraction. Unless we have made too bad a guess at first, this process of reinsertion converges rapidly and a value correct to five or six places can usually be obtained in less than a dozen runs.
§5.2]
General Properties, Series Solutions
559
Of course, if h is small, we can perform this iterative process analytically . To the first orderin h2 , we have s = Vb - (h2 /4 Vb). Inserting this into the two continued fractions (and omitting the h' terms in them), we have
or
..
L
S(b,h,eicf» = eiocf>
n=-
ane2incf>
(5.2.72)
00
using these values of s and the a's is one solution of Eq. (5.2.66) . The other solution is for the opposite sign of s, where an and a_ n change places : S(b,h,e-icf»
..
I
= e- i8cf>
n=-
ane- 2incf> 00
which is the complex conjugate of the first series . The corresponding real functions are obtained by addition or subtraction :
..
Se(b,h;z) So(b,h;z)
=
I .. I
n=-
IlO
n= -
00
an cos [(s
+ 2n)cfl];
z
= cos q, ; (5.2.73)
an sin [(s
+ 2n)q,]
These functions are even or odd about q, = 0 hut are not periodic in q, with period 7r or 27r unless s is an integer or zero.
560
Ordinary Differential Equations
[cH.5
For certain ranges of values of b (for negative .values of b, for instance, or for b near 1 or near 4 or near 9, etc.) s turns out to be a complex number. In this case the real solutions Se and So have somewhat more complicated forms. The larger h is, the larger the ranges of b where s is complex. These ranges are called the ranges of instability of the solutions, so named because the real exponential factor, which then is present in the solutions, becomes extremely large for large values of z, positive or negative. The Hill Determinant. Before we continue with our discussion of the solution of Mathieu's equation, we shall indicate a quite different method of computing s and the coefficients an, which is successful because of the particular symmetry of the recursion formulas (5.2.69). These equations are, of course, a set of homogeneous simultaneous differential equations for the a's (an infinite number because the a's are infinite) . In order · that they be solved to find the an's in terms of ao, for instance, the determinant of the coefficients must be zero. This is an infinite determinant, so that we must watch its convergence. However, we can divide the nth recursion formula by 2h 2 - 4b + 16n2 before we form the determinant, which will improve convergence. The resulting determinant (0'
+ 2) -
a2
~(8
{32
0
4 - a2
4 - a2 {32 1 - a2
(0'
+ 1)2 -
a2
1 - a2 {32 -a 2
0
{32 1 - a2 0'2 - a 2 -a 2 {32 - 1 - a2
0
0
0 0 {32 -a 2 (0' - 1)2 - a 2 1 - a2
(5.2.74) where 0' = j-s, a 2 = ib - j-h 2 , and {3 = h/4, is called Hill's determinant . All we have to do is to solve it for e ! Astonishingly enough such solution is possible because of the periodic nature of the dependence on a, which relates ~ to the trigonometric functions. First of all we simplify our determinant by multiplying the nth row (measured from the row n = 0) by [n2 - a 2J1[(O' + n)2 - a 2], . and so on, obtaining a new determinant D(O'), where
~(s)
= D(O')
n no::::l -
(0'
+ n)2 n
2 -
a
2
a
2
GO
"
{(O'
+ a)/nI
2
][1 - {(O' - a)/nI 2]
[1 - (a/n)2J2
§5.2]
General Properties, Series Solutions
561
The determinant D(O') has a sequence of unities along the main diagonal, flanked by a sequence of the sort ,82/[(U + n)2 - a 2] on both neighboring diagonals, with all other terms zero. Referring to Eq. (4.3.9) we see that .1(s) = -D(O') sin 1r(O' + .a ) sin 1r(O' - a) = D(O') sin2(1r~) - sin 2(1rO') sm 2(1ra) sm 2(1ra) (5.2.75) which displays some of the periodi c dependence on a and O'. But determinant D(O') also has definite periodicity in 0', having simple poles at 0' = ±n ± a, due to the elements ,82/[(0' + n)2 - a 2] . In fact the only poles of D(O') are at these points, and the function also is bounded at infinity (D - / 1 as 0' ---7 00), so it is an obvious function on which to practice the legerdemain of pages 381 to 385. We first subtract off all the poles :
..
K(O')
=
D(O') - C
I
n=-
(0'
+ n~2 _
a2
GO
= D(O')
_!!- ~ [ 2a
'-'
n=-
0'
1
+a+n
-
1 0' - a
+n
]
00
where C equals the residue at each of the poles of D. However analysis of the sort resulting in Eq . (4.3.7) (see Prob. 4.21, page 474) gives 1
1r cot(1rx) = X
2x 2x +-1 + -4 + x x 2 -
2 -
n=-
10
so that the function K(O') can be rewritten. This function 1rC K(O') = D(O') - 2a [cot 1r(O'
+ a)
- cot 1r(O' - a)]
has no poles for any value of s and is bounded at s ---7 00. By Liouville's theorem (page 381) it must be a constant, and by letting 0' ---7 00, we see the constant K is unity. Therefore, amazingly enough , D(O') = 1
+ 1rC 2a [cot 1r(O' + a)
- cot 1r(O' - a)]
Returning to Eq. (5.2.75) we see that A( S )
L.l
2(1rO') = 1 _ sin + 1rC cot(-scc) • 2()
sm
1ra
a
where the only constant as yet undetermined is the constant C, the residue at the poles of D(O') . This can be computed by setting (1 = OJ
562
Ordinary Differential Equations
[CH. 5
(7re/ (1) cot(7rOl) = ~(O) - 1. Therefore, returning to the original notation, we have that the original determinant has the relatively simple functional dependence on s: sin 2(7rs /2) ~(s) = ~(O) (5.2.76) 2(j,r sin Vb - -!h2) where
h2 144+2h 2-4b
0
0
h2 64+2h 2-4b
1
h2 64+2h 2-4b
0
0
h2 16+2h 2-4b
1
h2 16+2h 2-4b
0
0
h2 2 2h - 4b
1
0
0
0
h2 16+2h 2-4b
1
~(O)
=
is a convergent determinant which is independent of s. Since Ll(s) is to be zero, the equivalent of Eq. (5.2.71) for determining s is sin 2(7rs/2) = Ll(O) sin 2(-!7r
Vb -
-!h2)
Mathieu Functions. We are now ready to return to our previous discussion of the allowed values of s and the solutions S. The quantity Ll(O) sin- (j,r Vb- - -!h2) is a periodic function of 01 = -! Vb - -!h2 with period 1. For h = 0, Ll = 1 and s = ± 201 = ± Vb; this is the limiting by; = O. When b - -!h2 case where Eq. (5.2.66) reduces to (d¥/dcf>2) 2(7r0l) is large enough negative, Ll(O) sin is negative and s is pure imaginary. The whole range of values of hand b shown shaded, to the left of line 0 in Fig. 5.4, corresponds to unstable solutions, with a real exponential factor. For some value of b, depending on the value of h and plotted as curve 0 in Fig. 5.4, Ll(O) sin 2(7r0l) is zero, so that s is zero. In this case the solution S(b,h;ei4» is real, symmetrical in tP, and periodic in tP with period 7r [for an = a_ n and S is a Fourier series in cos(2ncf»). This function is called the Mathieu function of zero order_and is given the special symbol
+
..
Seo(h,z) =
l n=
B 2n cos(2ntP); z = cos cf> 00
where the B's are proportional to the a's, but adjusted so that Seo(h,l) = 1. When s = 0, the two solutions S(b,h;ei~) and S(b,h;e-i~) are equal and the second solution, independent of Se«, must be obtained by means
§5.2]
General Properties, Series Solutions
563
of Eq. (5.2.6). It has a logarithmic term in it and hence is not periodic in q,. Over the range of band h indicated in Fig . 5.4 by the unshaded area between the line marked 0 and the line 10 expression .1(0) sin 2(ll'a) is less than unity and thus there is a solution for s which is real and less than unity. Se and So as given in Eq. (5.2.73) are independent solutions, and the best way to compute s and the coefficients an is by means of the continued fraction of Eq. (5.2.71).
h
2
4
6
Separation Constant
b
Fig. 6.4 Values of separation constants for periodic solutions of the Mathieu equation.
For the set of values of band h given by the curve marked as 10, .1(0) sin 2(ll'a) and s are unity. It turns out that an = -a-n-l so that both S(b,h,e i~) and S(b,h,e-i~) are proportional to SOl(h,z)
=
LB
2n
+1 sin(2n
+ 1)q,
n=O
which is called the odd Mathieu function of the first order. ' Above this, in the next shaded region, .1(0) sin 2(ll'a) is larger than unity, s is complex, having the value 1 + ei, and consequently the solution is unstable. At the upper edge of this region of instability s is again unity, but this time an = a-n-l, so that both S solutions are proportional to
..
Sel(h,z)
=
LB
2n
+ 1 cos(2n
+ 1)q,
n=O
the even Mathieu function of the first order. The second solution again has a logarithmic term. This behavior continues for increasing values of b: alternating regions of stability and instability, the boundary lines between, corresponding
564
Ordinary Differential Equations
[cH.5
to the special case where s is an integer and where one solution is periodic, either an even function (using cosines in the Fourier series) or an odd function (using sines), and the other solution is nonperiodic, containing a logarithm. For the rest of the range of band h, exclusive of the boundary lines, the solution is nonperiodic but oscillating, with solutions of the form given in Eq. (5.2.73), or else nonperiodic and unstable, with solutions containing a complex exponential factor times a Fourier series. In many cases of physical interest the coordinate corresponding to q, is a periodic one, returning on itself as q, increases by 211". This being the case the only useful solutions are the periodic ones, which we have called Mathieu functions, for integral values of s (one odd function SOm and one even function Se-; for each integral value m of s). When h goes to zero Se; reduces to cos(mq,) and SOm reduces to sin(mq,). To compute the allowed values of the separation constant b corresponding to these periodic functions we can use, instead of the Hill determinant, the continued fraction equation (5.2.71) in reverse to find b when s is given . We let s = m, an integer, and find, by successive approximations, a consistent solution of
where the ratios al/aO, a_l/ao are given in terms of continued fractions on page 559. Except for m = 0, there are two different solutions for each value of m, one giving rise to a sine series and one giving a cosine series (that is, an = ± a2m-n). When s = 0, aI/ao = a_I/ao, and we solve the equation 2b
=
h
h'
2 -
16 + 2h2 _ 4b _
L
h'
16 . 4
+ 2h
2
-
4b - - - ;
~
then
Seo(h,z) =
n=O
etc .
L ~
B 2n cos (2nq,) ; B 2n = an/[
an]
n-O
where the coefficients D are normalized so that Se« = 1 for q, = O. The allowed value of b for this case ca n be labeled beo(h) . However if we are interested only in the Mathieu functions , the periodic solutions, we can simplify the calculations considerably by making use of the fact that the solutions are Fourier series. We transform the first of Eqs . (5.2.67) into (d~/dq,2) + (b - jh 2 - ih 2 cos 2q,)1/I = 0 As we have been showing, the periodic solutions of this equation are of four different types : 1. Even solutions of period 11", s = even integer = 2m, allowed values
General Properties, Series Solutions
§5.2]
of b = be2m
565
..
l
Se2m(h, cos tjJ) =
B 2n cos(2ntjJ)
n=O
II. Even solutions of period 211", s = odd integer = 2m b = be2m+l
..
Se2m+l(h, cos tjJ)
=
l
B 2n+ 1 cos(2n
+
1, for
+ l)tjJ
n=O
III. Odd solutions of period 11",
S
S02m(b, cos tjJ) =
= even integer = 2m, for b = b02m
..
l
B 2n sin(2ntjJ)
n=l
IV . Odd solutions of period 211", b = b02m+ 1
S
= odd integer = 2m
+ 1,
for
.
S02m+l(h, cos tjJ)
=
l
B 2n+1 sin(2n
+
l)tjJ
n=O
where the coefficients B depend on h, on m (i.e., on the value of s and the related value of b) and, of course, on n. Inserting the Fourier series of type I into the differential equation and utilizing the identity cos a cos b = i- cos(a + b) + i- cos(a - b), we have where
B 2 = koBo; B 4 = k 2B2 - 2B o; k 2nB2n = B 2n+2 + B 2n- 2 k« = h- 2(4b - 2h 2 - 4m 2)
From these equations, by rearranging into continued fractions , we can compute the ratio of coefficients and also the allowed value of b, which is be2m. Letting the ratio be represented by
we have two alternative sets of equations for the G's : 1
, (5.2.77) or
G2 = ko; G4 = k 2 - (2/k o) 1 G2n = k 2n- 2 - k 1 2n- 4 - k 6 _ 2n-
1 k 2 - (2/k o);
n
>
2
(5.2.78)
either of which can be used, depending on the relative ease of calculation and speed of convergence.
566
Ordinary Differential Equations
[cu. 5
Equating the two expressions for G2 gives a continued fraction equation for determination of the allowed value of b. Setting a = b - th 2 , o = ih2 , we have 2 a = 20 82 a - 4 82 a - 16 - - - - - ; : 0 - ; : ; - - a - 36which is equivalent to the equation of page 564. An infinite sequence of solutions of a as functions of 0 can be found, from which values of be2m can be determined. Some of the values are given in the tables. Values are also shown graphically in Fig. 5.4. For solutions of type II, by the same methods, we can arrive at the following equations for the coefficient ratios G and the separation constant be2m+l = a + ih2 : B 1 G3 = - 3 = k 1 - 1; G2n +1 = k 2n - 1 - k 1 B1 1 2n-3 k - _ 2n
6
- k1
n G2n -
1
1
= k 2n-l
1
k
-
2n+ l -
a
=
1
+0+a
>1
1 k
9-
n>O
2n+ 3 -
82 -
1;
-
2
a -
8 82 25 - ---;-:::--a- 49-
In both of these cases it is convenient to normalize the function so that Se« = 1 at r/J = O. This means that ~Bn = 1. For type III solutions we have B o = 0, and the equations for the G's and bo's are 1 G = 0 ; G4 = k 2 ; G2n = k 2n _ 2
-
1
k:-----
2
G2n
a
=
=
k
1 2n -
4
+a
n>2
2n - 4 -
1 k 2n + 2 -
a - m2
km = 8-
82 02 - 16 - a _ 36 _
Finally for type IV solutions we have G2n + 1 = k 2n G2n -
1
1
= k 2n- l
-
1 k 2n + 1 -
1 -
1 -,;-k--- 2n+l -
82 a = 1 - 002 a - 9 - ----:::-::---a - 25-
1
k1
+ 1;
n> 1 n>O
I
General Properties, Series Solutions
§5.2]
567
For both sine series it is convenient to normalize so that the slope of the function, dSom/drP, is unity at rP = O. This means that ~nBn = 1. When computing values of a (and thus of b) we can juggle the continued fractions to ease the work. For instance, for the type I solutions for the value of a near 16, we can invert Eq. (5.2.77) twice, to obtain a= 16
+
2 0 a - 4 - (202/ a )
_-----,0=-2:--_ _----: - 36 02
+a
a-------:6,....,4-----
It turns out that (unless h is quite large) the values of B; are largest for n ~ m. Consequently On = B n/Bn-2 is small for n > m and is large for n < m. Experience will show that the finite continued fractions, like Eq. (5.2.77), are best to use when computing G; for n < m and the infinite fractions, like Eq . (3.2.78), are best for the values of n larger than m. Mathieu Functions of the Second Kind. As we have mentioned on page 559, when s is an integer both S(b,h,ei~) and S(b,h,e-i~) are proportional to either Se« or SOm, the Mathieu function of the first kind, corresponding to b = be; or bo- ; respectively. For these particular values of b the second solution has a logarithmic singularity in z = ei~ (in other words, it is not periodic in rP) , and we must set up special solutions for these special cases. We indicate here the method for obtaining the second solutions corresponding to the even functions Se2m(h, cos rP). Since rP = 0 is an ordinary point for the Mathieu equation
1/1"
+ (b -
ih 2
-
ih 2 cos 2rP)1/I = 0
one should be able to set up a fundamental set of solutions, one with unit value and zero slope and the other with zero value and unit slope at rP = O. For b = be2m, the first solution is Se2m(h, cos rP), which has unit value and zero slope. The second solution should be of the general form Fe2m(h, cos ep) = 'Y2m [ rP Se2m(h, cos ep)
+
..
L D2n sin (2nrP) ]
(5.2.79)
n=1
which has a logarithmic singularity in z at ep = cos- 1 (z) = O. It is also not periodic in rP. Inserting this into the Mathieu equation and remembering that Se2m = ~B2n cos(2nrP) is a solution of the same equation for the same value of b, we finally obtain
..
'Y2m
L [-4nB 2n -
(2n) 2D 2n + -ih2D 2n_2
n=l
with the term in D 2n -
+ (be2n 2
- jh 2)D 2n + -ih2D 2n+2] sin(2nrP) = 0
being absent when n = 1. This gives rise to a
568
Ordinary Differential Equations
[cH.5
set of simultaneous equations [be2 - jh 2 - 4]D2 + iND 4 = 4B 2 ih 2D 2 + [be2n - jh 2 - 16]D4 + ih 3D 6 = 8B 4 ; etc . from which the D's can be obtained in terms of the B's (this is not simple , but we can find a solution for which the series converges) . We fix the value of the constant 'Y2m by requiring that the slope of Fe at q, = 0, which is 'Y 2m[1 + ~2nD2n] , be unity,
r
00
'Y2m
=
[1 + l
2nD 2n
n=1
Therefore we have the fundamental set for
q"
s«: = 1; (d/dq,)Se 2m = 0 } q, Fe2m = 0 ; (d/dq,)Fe2m = 1
,
The Wronskian A(Se ,Fe), with respect to
q"
=
0
is a constant, and therefore
d
d
Se2m(h, cos q,) dq, Fe2m(h, cos q,) - Fe2m(h, cos q,) dq, Se2m(h, cos q,) = 1
for all values of q,. See also Eq. (5.3.91). The second solutions for the other Mathieu functions are obtained in a similar manner. For instance, for b = bo2m+1, the second solution is
l
00
Fo2m+1(h, cos q,)
= 02m+1 [ q,S02m+1(h, cos q,) +
D2n+ 1 cos(2n
+
1)q,]
n-O
with equations for the D's similar to those written above. In this case we normalize so that F02m+1 = 1 at q, = 0 (it has a zero slope there), which results in an equation for
l
00
o2m+1 = [
D2n+ 1]-1
n=O
and the Wronskian for this pair of solutions equals -1. Therefore we have (at least) indicated the form of the second solutions for those values of b for which S(b,h,e-~) is not independent of S(b,h,ei . ) . For all other values of b the two S functions are independent and constitute a satisfactory pair. More on Recursion Formulas. We are now in a position to be a little more sophisticated about series solutions of differential equations and their related recursion formulas. Suppose that we have a differential equation £(1/;) == 1/;" + p1/;' + q1/; = 0, for which we wish to obtain a series solution about one of its singular points. For ease in computation we shall make the origin coincide with the singular point in question,
I
General Properties, Series Solutions
§5.2]
569
which can be done without disturbing the other singular points. Then p or q or both have poles at z = O. If p has no more than a simple pole and q no more than a pole of second order at z = 0, then the singular point there is regular, and we can, if we wish, expand the solution directly in a double power series in z , Each series is of the form z'~anzn, where s is one of the two roots of the indicial equation S2
+ (P
- l)s
+Q= 0
where P = lim zp(z) and Q = lim Z2q(Z). %-0
The coefficient of zn+. in the
.-0
series resulting when £(y;) is applied to the series ~anzn+. is the recursion formula Dn(a n) for the power series about the singularity at z = O. It, with its fellows, for other values of n , constitutes an infinite sequence of linear equations for the unknown coefficients an. If either p or q, or both, is a function of z which requires an infinite series to represent it, then each recursion formula D; involves all the a's from ao to an (or perhaps even higher) . In principle these simultaneous equations can always be solved , to obtain the ratios between an and ao. But if the recursion formulas have more than two terms apiece (i.e ., involve more than a pair of adjacent a's), then the task of computing the series and testing it for convergence, for asymptotic behavior, and so on, becomes very much more difficult. Let us see what we ca n say about the possibility of obtaining a set of two-term formulas. Short recursion formulas can be obtained only when p and q are rational functions of z, i .e., are ratios of polynomials in Z (see page 382). If they are not rational functions, we can try to transform the independent variable so that the new p and q are rational functions : if this can be done, we can then proceed ; if not , we are left to struggle with infinite recursion formulas. The denominators of p and q, if they are rational functions, are determined by the position of the singular points of the equation. At least one of the two denominators has Z as a factor, for one of the two functions has a pole at z = O. If there are other singular points for finite values of z (say at z = Zi , i = 1, 2, .. . , N) , then there must be fa ctors of the form (z - Zi) in the denominators of either p or q or both. At any rate, when we clear the equation £(y;) = 0 of fractions, it will have the form N
n i-O
N
(z - Zi)n.y;" + F(z)y;'
+ G(z)y; = 0;
M =
L ni;
Zo
=0
i-O
where F and G are polynomials in z: Parenthetically, if there is to be no singular point at infinity, G(z) must be a polynomial in z of order M - 4 or less and F must be a polynomial of order M - 1, with its highest order term equal to 2zM - 1(why?) .
570
Ordinary Differential Equations
[CR. 5
It is not difficult to see that, in general, such, an equation will have an M-term recursion formula. If the point at infinity is a regular point, this can be reduced by one or two terms by transforming the independent variable [by the transformation w = zj(z - Zj)] so that the jth singular point is placed at infinity. When this is done (if it can be done) and there is a singular point at zero and at infinity, the equation will still have the form given above but the order of the polynomials multiplying v/', vI', and if; will be the smallest possible for the particular equation chosen. We now can see that usually it is possible to have a two-term recursion formula only if there is but one other singular point beside the ones at zero and infinity, for the coefficient of if;" must have the form zno(z - Zl), F must have the form as az no + bzno-1, and G the form az no-1 + {3z no-2 in order that the powers of z arrange themselves in the series for £(if;) = 0 so that two-term recursion formulas will result. One other, rather unlikely, case is for there to be two other singular points, symmetrically placed (that is, Z2 = -Zl) so that the coefficient of if;" is zno(z2 - 4). If then F = azno+ l + bzno-1 and G has the unlikely form az no + {3z no-2, a two-term recursion formula results, relating an and a n+2 (rather than an and an+l). Even if there is only one other singular point, the functions F and G may not have the requisite simple forms . In this case we can sometimes help by transforming the dependent variable, letting if; = u(z)f(z) , where u is some power of z times some power of (z - Zl). Usually the proper powers are one of the indices s at each of the singular points, so that the new dependent variable f has one analytic solution both at z = 0 and z = ZI. This often reduces the order of the polynomial G and produces a two-term recurs ion formula. It is the trick which was used successfully on the Papperitz equation, to result in the hypergeometric equation. When more than one singular point is irregular, F or G does not have the right form for a two-term recursion formula to result. As we have seen, the best we can do for the case of two irregular as well as the case of two regular and one irregular points is to obtain a three-term formula. Larger numbers of singular points or a higher species of irregularity will result in still more complex formulas. Luckily such cases have not as yet turned out to be of great practical importance, so we shall dismiss them without further ado, beyond remarking that, if they become important, further research will be needed to establish techniques for handling these more complicated recursion formulas. Functional Series. But after all, it is not necessary to confine ourselves to a series of powers of z; a series of terms containing a set of functions in:
§5.2)
General Properties, Series Solutions
571
can also be used . To see how this generalization may ' be ca rried out, we return to the power series method and ask why the set of functions
was so useful. One obvious reply is that eminently simple recurrence relations:
In,
for this case, satisfies
Z!n = !n+l; !~ = n!n-l Using these relations it is possible to redu ce a differential operator .,c to a form in which only different powers of zn are involved. In order to use another set of In's to represent the solution, the new set of functions must also satisfy recurrence relations. Another important and useful property of power series is its property of completeness. By completeness we mean that, subject to certain conditions, a linear combination of the powers of z may be used to represent any function. This statement is a consequence of Laurent's theorem [Eq. (4.3.4)) and is subject to the conditions involved in Laurent's theorem. Before we can use other sets of functions, we must also show what functions may be represented by their means and what others cannot be so represented. Later on, in the chapter on eigenfunctions, we shall devote considerable space to a discussion of this question. However, it is worth while discussing whatever can be done at this point with the techniques we have already developed. We shall give a few examples and then return to the original question of the solving of equations by the functional series. The method generally employed involves relating the functional series to be used to the power series. Then, by using the known properties of the power series, it is possible to obtain information about the set In. As a first example , we can establish the completeness of the Fourier series in ei n 8 directly from the Laurent series. From Eq. (4.3.4)
Consider now the values of if;(z) on the unit circle, z
..
Then
i8
if; (e
)
=
La
ne
= ei 8•
in 8
From the completeness property of the power series, we may now conclude that the functional set eM will represent any reasonable periodic function of 0 of period 211'. The necessity for the periodicity is a consequence of the fact that we are representing by the Fourier series the values of if; on a circle and if; repeats itself as we complete the circuit.
Ordinary Differential Equations
572
[CR. 5
By considering even or odd functions of B, we immediately generate the ' sine or cosine Fourier series. As a second example, let us examine the first solution of Legendre's equation (5.2.49) for m = 0 and integer values of n.
X = F( -n, n
+ 111[ j
- jz)
For integer values of n, X is a polynomial in n, the Legendre polynomials P n to be discussed in more detail later on page 595. Let us tabulate the first few of these functions: Po = 1 PI = Z P 2 = (3z 2
-
P 3 = (5z 3 - 3z)/2 P 4 = (35z 4 - 30z2 + 3)/8 P« = (63x S - 70z 3 + 15z)/8 ; etc.
1)/2
From this sequence, we may prove that any zn may be expressed as a linear combination of P n ' For 1 and z it is obvious. We write out the answer for the next few powers: 1
= Po
z = PI Z2 = !-(2P 2 + Po) Z3 = t(2P 3 + 3P I ) Z4 = :fiJ(8P 4 + 20P 2 ZS = -h(8P s + 28P 3
+ 7P o) + 27P I ) ;
etc.
From the fact that the power series in positive powers of z is complete for functions which are not singular, we may conclude that these fun ctions may equally well be expressed in terms of the Legendre polynomials. In order to include singular functions, it would be necessary to include the second solution of the Legendre equation in our discussion (corresponding to the negative powers of z). . For present purposes it is not necessary to determine coefficients in the above expressions explicitly. It is enough to demonstrate the possibility of such a representation. For example, in the case of the solution of Bessel's equation for integer n (5.2.62) we have one set of solutions
In(z) = e- i·znF(j
+ nil + 2nJ2iz) ----+ zn(l + . ..) z--+O
[see Eqs. (5.2.63) et seq.]. We may expect that by suitable combinations of these functions it would be possible to represent zn. Similarly for z-n, the second solutions, the Neumann fun ctions (see page 625) would be useful. Again one sees that these fun ctions are as complete as the power series. This statement is somewhat more difficult to verify than the similar one for the Legendre functions because the Bessel functions are not polynomials but rather infinite series. However, it is relatively easy to demonstrate that in principle a Bessel function representation is
General Properties, Series Solutions
§5.2]
573
,
possible. We now are able to turn to some examples of the use of functional series, from which we shall be able to deduce the sort of maneuvers which must be usually employed. As a first example, consider the Mathieu equation discussed earlier (page 556) : 1/;"
+
[b -
~2
~2 cos 2,p ] if; =
_
0
This equation resembles to some extent the equation satisfied by the exponential functions fn
= ei2(n+ . ) ;
.f~
+ [4(n + s)2]fn
= 0
L 00
(the cos 2,p term is not present).
We substitute
Anfn
into the
Mathieu equation:
L: 00
An {[ -4(n
+ S)2 + (b
- ~)]
e2i(n+.)8 _
~2 e 2i(n+.+l)8 2
- "4 h e2i(n+.-1) 8} =0 Rearranging terms so that e2i (n+ . ) 8 is a common factor, we obtain 00
2 2 2 ~ {h Lt -"4 A n- 1 - h "4 A n+ 1 + An [h b - "2 -
4(n
+ s)2 ]}
e
2i(n+.)8
=0
-co
From the completeness of ei (n+. )8, the coefficient of each term must be zero. (Where we are using the result that, if a power series is identically zero, the coefficient of each power must be zero.) We thus obtain the three-term recursion formula :
~2 A n+ + An [4(n + sF _ b + ~2] + ~2 A n1
1
=
0
This is identical with (5.2.69) (of course) . As a second example, consider a special case of the equation resulting from the separation of spheroidal coordinates: (Z2 - 1)"/' + 2z1/;' + (h 2z 2 - b)if; = 0 Compare this with the equation satisfied by the Legendre polynomials Pn : . (Z2 - l)P~ + 2zP~ - n(n + l)pn = 0 If w.e insist that 1/; be nonsingular at ± 1, the singular points of its differential equation, the use of P n = [« is suggested; i.e., let
574
[cH.5
Ordinary Differential Equations
The following recurrence relation, which will be derived later, will be useful here: n(n - 1) Z2P n = 4n 2 _ 1 Pn-2
+
[n2 4n 2 _ 1
(n
+ 1)2
]
+ (2n + 1)(2n + 3) P; (n + 1)(n + 2) + (2n + 1)(2n + 3) P n+
2
Substituting in the differential equation yields 00
~ Lt P;
{
2 (n A n+2h (2n
+ 2)(n + 1) + 3)(2n + 5) + An [ (n)(n + 1)
- b
+h
2 (
2 n 4n 2 - 1
o
(n
+ 1)2)J
+ (2n + 1)(2n + 3)
+
2 h (n - 1)(n) } 2 nA (2n - 1)(2n _ 3)
=
0
This is a three-term recursion formula which must now be solved for An, subject to the conditions A_ 1 = A_ 2 = O. What we have done here is to find some complet e set of functions fn(z), in terms of which we choose to expand our solution. In practice we choose f's such that their differential equation '.mn(fn) = 0 is not very different from the equation we wish to solve, £(1/1) = O. We then use the relations between successive fn's in order to express the difference between £(fn) and '.mn(fn) in terms of a series of fn's. [£ - '.mmlfm =
I 'Ymnfn
(5.2.80)
n
For instance, for the Legendre polynomial series discussed above, [£ '.mmlP n = (h2z2 - b + n(n + 1)]Pn, which can be set into the series form of Eq. (5.2.80), with only three terms in the series (for P n - 2, P n , and Pn+2). If we have chosen well, the series in fn will be finite, with a few terms. We call these equations, relating simple operators operating on [« to simple series of fm's, recurrence formulas in order to distinguish them from the recursion formulas of Eq. (5.2.40). Setting our series into the operator £ , we have £(J:.amfm) =
[I (£ -
'.mm)amfm
m
+ 2: am'.mm(fm) ] m
=
I am [2: 'Ymnfn] = 2: [I am'Ymn]fn m
n
n
= 0
m
where '.mm(fm) = 0 by definition. If the set fn is complete, we can equate the coefficients of fn in the last series to zero separately:
I am'Ymn m
= 0
§5.2]
575
General Properties, Series Solutions
which are the recursion formulas for the coefficients am. If these can be solved, we have obtained a solution for the equation £(if;) = O. The general usefulness of the expansion in question depends, of course on its rate of convergence, which in turn depends on the behavior of an as n ~ 00. To obtain this, consider the above equation in the limit n ~ 00: A n +2 ~2
+ An
[n2 - b + ~J
+ h2 A~-2
= 0; for
n~
00
This is just the recursion relation (5.2.69) derived for Mathieu functions if the substitution 8
= 0; A n +2 = j-Cn +2 ; n = 2{3
is made. It may be recalled that the Mathieu recursion relation has a convergent result for an as n ~ 00 if, for a given 8, b takes on only certain ' special values. These values must be determined from the recursion formulas for an by the methods outlined in the section on continued fractions. Many other cases of series expansion of a function in terms of other functions will be found later in this book. Series of Legendre functions (hypergeometric functions, see page 593) and of Bessel functions (confluent hypergeometric functions, see page 619) will be particularly useful. In general, what we can try to do with such series is to use solutions of equations with a given set of singular points to express solutions of equations with one more singular point (or with more complicated singular points). For instance, according to Eq. (5.2.30) a power of z is a solution of a differential equation with two regular singular points, at o and 00 . Therefore solutions of an equation with three regular points (hypergeometric function) or of one with one regular and one irregular point (confluent hypergeometric function) can be expressed in terms of a power series in z with comparative ease. On the other hand solutions of an equation with two regular points and one irregular point (spheroidal functions) can most easily be expressed in terms of a series of hypergeometric functions (Gegenbauer functions) or of confluent hypergeometric functions (Bessel functions). We defer the Bessel function series and any discussion of series of more complex functions until later in the book [see Eqs. (5.3.83) and (11.3.87)]. As a final remark, note that the series '2,anf n(z) may be generalized into an integral as the Fourier series was generalized into a Fourier integral.
For example, instead of if;(z) =
Lanf,,(z) we could write "
if;(z) = fK(z,t)v(t) dt
It is clear that the variable integer n has been replaced by the con-
576
Ordinary Differential Equations
[CR. 5
tinuous variable t, that the functions fn(z) have gone over into K(z,t), and that the coefficients an have become v(t). We may, by comparison with the procedure for the series form, outline the manner in which we obtain the integral representation of y; (as the above integral is called). First we apply the operator z, where now, to indicate that .c operates on the z variable only, we write .cz for .c. Then
.c.y; =
f.c.[K(z,t)]v(t) dt = 0
In our treatment of .cz[fn(z)], we made use of the recurrence relations of I; to change the differential operator into a set of difference operators, by the recurrence relations
where 'Y np are numerical coefficients. This amounted to replacing the operation on z by an operation on the subscript n. In the case of the integral representation, this means that we could arrange to have .cz[K(z,t)] = mtt[K(z,t)], where mtt is a differential operator in t, so that
o=
fmtt[K(z,t)]v(t) dt
The next step in the series representation in terms of I; was to rearrange the series so that fn was a common factor; the coefficient of t-; involving several an's, was then put equal to zero, yielding a recursion relation for an. Thus the operation on [« was transformed to an operation on an. Similarly here the operation mtt must now be transformed over to an operation on v. This may be accomplished by integrating by parts or equivalently by means of the adjoint to mtt defined earlier (page 528). We recall that d vmtt[u] - umti[v] = dt P(u,v) Therefore
0
=
fK(z,t)mti(v) dt
+
[P(u,v)]
with the second term evaluated at the limits of integration. We now choose the limits or contour of integration so that the P(u,v) term is zero; then the original differential equation is satisfied if the "amplitude " v(t) in the integral representation satisfies the equation mti(v) = 0 This equation is the analogue of the recursion formulas for an. If we can solve the differential equation for v(t), we shall then have a solution of the original differential equation for y;(z), which has some points of advantage over a series representation. But this subject is a voluminous one, and we had better devote a separate section to it.
Integral Representations
§5.3]
577
5.3 Integral Representations We have now pro ceeded far enough to see the way series solutions go. Expansion about an ordinary point is straightforward. The unusual cases are about the singular points of the differential equation, where the general solution has a singularity. We have indicated how we can obtain series solutions for two independent solutions about each singular point, which will converge out to the next singular point (or else are asymptotic series, from which reasonably accurate values of the solution can be computed over a more restricted range) . In other words, we have worked out a means of calculating the behavior of any solution of the second-order, linear, differential equation in the immediate neighborhood of any point in the complex plane. In particular we can set up the series expansion for the particular solution which satisfies any reasonable boundary condition at any given point (what constitutes "reasonableness" in boundary conditions will be discussed in Chap. 6). Quite often these boundary conditions are applied at the singular points of the differential equation. We have seen that such singular points correspond to the geometrical "concentration points" of the corresponding coordinate system. Often the shape of the physical boundary can be idealized to fit the simple shape corresponding to the singular point of one of the dimensions (for instance, the origin corresponding to r = in spherical coordinates, the disk corresponding to J.L = for oblate spheroidal coordinates, the slit corresponding to ep = 0, 1r for elliptic cylinder coordinates, etc .). Often only one of the solutions, for only one of the indices (if the singular point is regular) , can fit the boundary conditions, so one of the solutions discussed in the previous section will be satisfactory. If we wish values of the allowed solution nearby, the series expansion is satisfactory and, indeed, is the only way whereby these intermediate values can be computed. But quite often we wish to calculat e values and slopes near the next singular point, where the series which is satisfactory for the first singular point either converges extremely slowly or else diverges. For instance, we often must satisfy boundary conditions at both ends of the range for a coordinate corresponding to two consecutive singular points. What is needed is a joining factor, relating one of the series about one singular point with the two solutions about the other point, for then we should not need to stretch the convergence of the series. Suppose Ur, Vr are the two independent series expanded about the singular point z = ar and U2, V2 are the series about z = a2. If we can find relations of the sort, Ur = 'YllU2 + 'Y12V2, etc. , we can then use Ur, Vr when we wish to put boundary conditions at ar and U2, V2 for putting conditions at a2. We shall have no convergence problems if we
°
°
578
Ordinary Differential Equations
[cH.5
can express each solution at one end in terms of the solutions at the other. For the simplest sorts of differential equation this joining of the behavior about one singular point to the behavior about another is easy. As long as the solutions are either rational functions (Eq. 5.2.29) or elementary transcendental functions [Eq . (5.2.33)], we know the behavior of the solutions at both ends of the range ; the" joining" has already been done for us. For the next more complex equations this joining is not so simple. Equations (5.2.51) and (5.2.60) are examples of such joining equations, but our derivation of these formulas from the series solutions was not rigorous, nor was it even valid for much of the ranges of the parameters. Series solutions, as Stokes put it , "have the advantage of being generally applicable, but are wholly devoid of elegance." What we should prefer would be to express the solutions in terms of the rational fun ctions or the elementary transcendentals in some finite way , which would be just as convergent at one singular point as at another. This can be done for some equations by the use of integrals, instead of series . The expression y;(z) = fK(z,t)v(t) dt (5.3.1) is general enough to express nearly any solution. If the functions K and v turn out to be rational functions or elementary transcendentals, then we have a" closed " form which can be used to calculate the solution anywhere on the complex plane. All we need is to determine how we are to find the correct forms for K and v for a given equation. Some Simple Examples. Our acquaintance with the techniques of contour integration enables us to work out a few simple cases to illustrate the relationship between integral representation and series solution. For instance, since 1
t - n n= -
or 7r coth(7rt)
1
=
QO
t - in n= -
GO
(see page 561), we can utilize the residues of this function to produce a Fourier series . For instance, the integral
I(z) = ¢c cot h (7rt) F(t)e zt dt can be developed into a series if F is a rational function of t (see page 413) with all its poles to the right of the imaginary axis, which behaves at infinity such that tF(t) stays finite or goes to zero as t -t 00 . The contour C includes a line just to the right of the imaginary axis, going from E i 00 to E + i 00 , and then returns to - i 00 by going along the semicircle of infinite radius inscribed about the negative half of the t plane. Since tF(t) does not become infinite at Itl-t 00, the contour integral
§5.3]
Integral Representations
579
about the infinite semicircle is zero [since coth(1Tt) -t 1 when Itl-t 00 if Re t > 0] as long as Re z > O. The poles inside the contour are therefore those of coth (1Tt), which are ± i n. Inside the contour we have required that F be analytic everywher e. Along t he imaginary axis F has a symmetric and un symmetric part. F (t ) = set) + u (t) ; set) = !F(t ) + M" (t) = Re F; u(t) = !F(t) - !F(t ) = i Irn (F) The cont our integral is 21Ti times the sum of the residues for all the poles of cot h(1Tz) :
f
..
i
~ +. cot h(1Tt) F(t) ezl dt = 4i ~ [Re F n cos(nz) - Im F n si n(nz)] ~
-tOO+1l!
n=O
(5.3 .2) where E; = F(in) . A more immediately useful integral representation can be obtained from the properties of the gamma function I'( - t). The fun ction -1T
r ( - t)
=
r et
+ 1) sin (1Tt ) -;:;:
-
1 ( - I)n (t - n)r (t
+ 1) ; n = 0,1,2, .
has simple poles at t = 0, 1, 2, . . . . If we include in the integrand a factor z', and if the whole integrand converges, then the sum of the residu es will turn out to be a series in integral powers of z. As was shown on page 486, the asymptotic expression for r et + 1) is r et
+
1) ~ yI2; e-I- 1t l +! t->
00
Therefore if G(t) is su ch a fun ction that [G(t)zljr(t + 1)]-t 0 when 00 for Re t > 1, and if all the singularities of G are to the left of the imaginary axis, then the integral
t -t
t:..
..
G(t )r ( -t)( -Z)l dt = 21Ti
~
G(n)z njn!
(5.3.3)
n=O
where the cont our passes to the left of the pole at t = 0 and is completed along the semicircle of infinite radius surrounding the positive half of the t plane, going from +i 00 back around to - i 00. Thus we have a fairly direct means of going from a series to an integral. If G(n) is a "closed function" of n (i. e., if the successive coefficients are related by a simple formula such as a two-term recursion formula), then the integrand is a closed form. The uses of such a formula and the precautions necessary to ensure convergence are well illustrated by applying it to the hypergeometric
580
Ordinary Differential Equations
series [see Eq. (5.2.44)]
..
[CR. 5
+
fCc) \ ' f(a n)f(b + n) F(a,blclz) = f(a)f(b) f(c + n)n ! zn
4
n=O
. f(c) /2ri I'(c + t)f(b + t) It appears that the function G should be f(a)f(b) f(c + t)
if this function should have its poles to the left of the imaginary axis and if Gzl/f(t + 1) sin(1I"t) converges. Using the asymptotic formula for the gamma function, we find that, if t = Rei 8 = R cos 0 + iR sin 0, for R large enough f(a I'(c
Also if z = (-Z)l
d an
=
re i';
+ t)f(b + t) '"'-' eHc-a- b Ra+b-c-l
+ t)f(t + 1) -
and (- z) =
exp{R[(ln r) cos 0
re i-i7<
+ (11" -
ei(a+b-
e- J) 8
= eln r+ i(-7 0>
(11"/2) -(11"/2)
Therefore as long as the magnitude of z is less than unity (i.e., In r < 0) and the argument of z larger than zero and less than 211"(0 < q, < 211") , the integrand will vanish along the semicircle part of the contour [R ~ ~, - (11"/2) < 0 < (11"/2)] . As long as these conditions hold, an integral representation for the hypergeometric series is F(a,blclz)
fCc)
= f(a)f(b)
fi"
-i..
+
+
f(a t)f(b t) I 211"if(c t) f( -t)( -z) dt
+
(5.3.4)
if we can draw the contour between -i~ and +i~ so that all the poles of I'( -t) are to the right and the poles of f(a + t)f(b + t) are to the left. Figure (5.5) shows that this may be done, even if Re a and Re b are negative, as long as neither a nor b are negative integers. If either a or b is a negative integer, Eq. (5.2.44) shows that F is a finite polynomial, not an infinite series (it also indicates that c should not be a negative integer) . This integral representation does not seem to be a useful or elegant result ; in fact at first sight it appears less elegant than the expression for
§5.3]
581
Integral Representations
the series. It is not difficult to show, however, that the possibilities for rearranging the contour without changing the value of the integral make for considerable flexibility in expanding the result, just the flexibility needed to relate the expansion about one singularity to expansions about another singularity. For instance, further examination of the asymptotic behavior of the integrand of Eq. (5.3.4) shows that it goes to zero when R --t 00 for the range (71"/2) < (J < (371"/2) , (t = Rei 9) , i .e., for the infinite semicircle enclosing the left-hand part of the t plane. Therefore the contour enclosing all the poles of r( -t) (which results in the hypergeometric , Plane
-4; b -3;b
2
3
. . .
-4-0
-3- 0 - 2- 0
Fig. 6.6 Contour for integral representation of hypergeometric function by Mellin transform.
series above) is equal to the contour enclosing all the poles of rea + t) reb t) . Using the equations rea + t) = [7I"/r(1 - a - t) sin 7I"(a + t)], etc., and evaluating residues at the pairs of poles, we have
+
00
r(a)r(b) F(
r(c)
bl I )
a, c Z
= \ ' rea I'(I
4 n=O
+ n)r(1 - C + a + n) . + n)r(1 + a - b + n) sin 7I"(c - a - n) cos(n7l") sin 7I"(b -
-z
a - n) (
-G-n
)
00
+ \'
reb
+ n)r(1
=
r(a)r(b -
4 r(1 + n)r(1 n=O
- c - a
+ b + n)
+ b + n)
sin 7I"(c - b - n) -z -1>-n cos(n7l") sin 7I"(a - b - n ) ( )
a) ( (-z)-GF a I I'(c - a ) '
+ r(b)r(a ~) b) (-z)-bF (b rc-
1
c + all -
1- c
+ bll -
b a
1)
+ al-z
+ bl~) z
(5.3.5)
If la - bl is an integer or zero, one of these series "blows up," for the second solution should contain a logarithmic term.
582
Ordinary Differential Equations
[CR.
5
The series on the left-hand side converges when Izi < 1, whereas the series on the right converges when Izl > 1. Strictly speaking, the two series should not be equated ; what we should say is that the integral representation given in Eq. (5.3.4) has a series expansion in terms of F(a,blclz) which is valid for Izi < 1; it also has another series expansion, given by the right-hand side of Eq. (5.3.5), which .is valid for Izi > 1. The integral representation has validity for all (or nearly all) values of z and can be considered as the "real solution" of Eq. (5.2.42). The series expansions can be considered as partial representations of this "real solution," valid over limited ranges of z, By means of the integral representation we can perform the analytic continuation of the solution from one region of convergence around one singularity to another region of convergence about another. Therefore we might call the integral on the right-hand side of Eq. (5.3.4) the hypergeometricfunction, F(a,blclz), one solution of Eq. (5.2.42). When Izl < 1, we can compute this function by using the series representation given in Eq. (5.2.44), the hypergeometric series in z. For other ranges of z the hypergeometric function can be computed by using hypergeometric series in l iz, as given in Eq. (5.3.5), or in (1 - z), as given later, and so on. The function itself, exhibiting different facets from different points of view, is the integral representation. General Equations for the Integrand. The characteristics exhibited by the representation for the hypergeometric function are typical of integral representations in general. They make analytic continuation of a solution almost a tautology. When such a representation can be found, it is usually not difficult to fit boundary conditions at both ends of a range of e. When a representation cannot be found, such fitting is difficult and" untidy." The methods used to obtain the integral representations of the previous subsection were far from straightforward ; what is needed is a technique of proceeding directly from the differential equation to the form for the integrand. We shall first use the form given in Eq. (5.3.1), with a kernel K(z,t) and a modulation factor v(t). We choose a form for K which seems appropriate and then find out what differential equation v satisfies . If this equation is simple enough (enough simpler than the equation for if;), then a closed form for v can be obtained and then the integral representation is manageable. The differential equation to be solved is a second-order equation, with the independent variable transformed so that the coefficients are algebraic. Rather than using the form given in Eq. (5.2.1), it is better to clear of fractions, using the form £z(if;)
= fez)
d 2if; dz 2
+ g(z) dif; dz + h(z)if; = 0
(5.3.6)
§5.3]
Inlegral Representations
583
where I, g, and h are then polynomials in z. When the differential operator £ is applied to the integral form of Eq. (5.3.1), the operator can be taken inside the integral, to operate on the z part of the kernel K, £(fKv dt) = f£/K(z ,t))v(t) dt
if the integral is a reasonably convergent one. Operation by £ z on K(z,t) produces another function of z and t. If we have picked a satisfactory form for K , this new function of z and t is also equal to some operator function of t, operating on K(z,t),
[In some cases it is sufficient to have £z(K) equal to an operator mtt , operating on some other kernel K'(z ,t).] It is this equivalence between an operator in z and an operator in t, when operating on the kernel K, which at the same time makes possible the calculation of vet) and also severely limits the choice of forms for K(z,t) . Not many functions of z and t possess a simple reciprocal relation such as that exhibited by the exponential kernel ezt [used in Eq. (5.3.2)]
where a differential operator involving derivatives in z and powers of z is transformed into an operator involving corresponding powers of t and derivatives in t. Nevertheless a number of other kernels have been found useful in various cases: zt(used in Eq. 5.3.3), (z - ty, various functions of the product (zt) , etc. In many cases several different kernels can be used, giving rise to several alternative integral representations of the same solution. Which kernel is likely to produce the most useful representation depends on the relation between the singularities of the kernel and the singular points of the differential equation. For instance, for the hypergeometric equation, with three regular singular points, one might expect a kernel of the form (z - t)1' to be better than ezt , which has an essential singularity at infinity. On the other hand, the confluent hypergeometric equation, having an irregular singular point at infinity, would appear to be suitable for the kernel ezt • But let us return to the immediate subject, which is the setting up of an equation for vet). We have reached the point where the integrand consists of a function v of t multiplied by a differential operator mtt operating on the t part of the kernel K(z,t). Referring to Eq. (5.2.10), we see that we can next transform this to K times the adjoint (see page 527) differential operator mt operating on v, plus the derivative of a bilinear concomitant P. Symbolically it goes as follows:
584
Ordinary Differential Equations
£ z(y) = f(z) £z(y) =
=
~~ + g(z) ~~ + h(z)y = 0 ;
f f[
£z(K) v dt = Kmtt(v)
where, if operator is
~(K)
f ~t(K)
_
~(v)
2K
jdt 2)
f
K(z,t)v(t) dt
v dt
+ :t P(v,K) ] dt
= a(t)(d
y =
[cH.5
=
(5.3.7)
f
K(z,t)mtt(v) dt
+ [P(v,K)]
+ (3(t)(dK jdt) + 'Y(t)K ,
d2
the adjoint
d
= dt 2 (av) - dt ((3v ) + 'YV
and the corresponding bilinear concomitant is P(v,K)
d
d
= av dt K - K dt (av) + (3vK
If now the limits of integration and the cont our along which the integral is taken are such that P returns to i ts initial value at the end of the contour, the integral of dP j dt is zero and
Therefore if v(t) is a solution of the differential equation mtt(v) = 0, the integral y = I Kv dt is a solution of the differential equation £ z(y) = 0, which is the equation we wish to solve. If we have been lucky in our choice of kernel K, the equation mtt(v) = 0 will be simpler than £ z(y) = 0 and v will be a simple function of t. . Usually there are several different possible integration paths and limits, anyone of which makes J(dP j dt) dt = O. These different integrals correspond to different independent solutions of £z(y) = O. From another point of view, what we are doing here is investigating integral transforms, of the sort of the Fourier transform f(lI) =
f'"_... F(J.L) e
i" .
dJ.L;
F(J.L)
=~ 211"
f'"_ ... f(lI)e-
i" .
dll
discussed in Sec. 4.8. We transform from function 1/;(z) to function v(t) by means of the kernel K(z,t) , 1/;(z) = Iv(t)K(z,t) dt
and try to find a type of transformation for which v, the transform, is a simpler function than 1/;. For instance, when the kernel is ezt the relation is called the Laplace transform. This can be quickly obtained from the Fourier transform
Integral Representations
§5.3]
above ; by setting J.L = -it, we have if;(z) =
fi'"
JI
585
= z, F( -it) = -iv(t) , and f(JI) = if;(z),
v(t) ezt dt ; v(t) =
- i'"
i. 211"
f'"
if;(z) e- zt dz
(5.3.8)
-oo
And when the kernel is z', the relation is ca lled the Mellin transform. It also can be obtained from the Fourier transform by setting J.L = -it, JI = In z, f(ln z) = if;(z) , and F( - it) = -iv(t) , giving if;(z) =
fi~
v(t)zt dt;
v(t) = 2i 11"
-,oo
Joroo if;(Z)Z-t-l dz
(5.3.9)
Still other transforms can be devised, not all of them as closely related to the Fourier transform as these two . The transform f v(t) (z - t)" dt, for instance, is called the Euler transform (see also Sec. 4.8) . The rest of this section will be devoted to the study of a number of examples of integral representations, both to illustrate the techniques of obtaining such solutions and the methods of using the results and also with an aim to make more familiar certain fun ctions which will be generally useful later in this volume. Two types of kernel will be studied in considera ble detail: (z - t),,; ezt ;
the Euler transform the Laplace transform
Other types of transforms, of less general utility, will be discussed in less detail. The Euler Transform. As mentioned previously, we should expect that the kernel (z - t)" would be a satisfactory representation for solutions of equations having only regular singular points, such as the Papperitz equation (5.2.36) or the corresponding canonical form , the hypergeometric equation (5.2.42). This restriction of the form of equation for which the Euler transform is applicable evidences itself in a somewhat unfamiliar way when we come to apply the differential operator £ to the kernel (z - t)". The form for £ is that given in Eq. (5.3.6), the form in which the hypergeometric equation (5.2.42) is already given. When £ is applied to the kernel (z - t)"H, there results a complicat ed algebraic function of z and t: £z«z - t),,) = J.L(J.L - l)f(z)(z - t),,-2
+ J.Lg(z)(z
- t),,-l
+ h(z)(z -
t)"
This is now to be converted into some differential operator mrt, operating on some power of (z - t). The form of mr could be worked out painfully by dividing out (z - t)r 2, expanding the rest in a bilinear form of z and t and then trying to work out the form for mr which would give the result. A more orderly procedure results when we expand the functions
586
Ordinary Differential Equations
[CR. 5
g, h in a Taylor's series about z = t (since t. g, and h are polynomials in z this ca n always be done) . For instance, f(z) = f(t) + (z - t)f'(t) + ·H z - t)2f"(t) + , and so on. The result is
f,
.e.«z - t),,) =
p.(p. - 1)f(t)(z - t)r 2 + [p.(p. - 1)f'(t) + p.g(t)](z - t),,-l + [ip.(p. - 1)f"(t) p.g'(t) h(t)](z - t)"
+
+
+
where the fourth term in the series involves the third derivative of f, the second derivative of g, the first derivative of h, and so on. In order to have this form represent a second-order operator, operating on (z - t)", we must arrange that the fourth and all higher terms in this series be zero. There are many ways of adjusting the functions f , g, h so that this is so, but the simplest way (and sufficient for our needs here) is to require that all derivatives of f higher than the second, of g higher than the first , and of h higher than the zeroth be zero. In other words, if f(z) is a second-order polynomial in z, g(z) a first-order polynomial, and h(z) is a constant , then the expression above will have only the three terms which are written out; all higher terms will vanish. We see that this automatically restricts us to an equation with three regular singular points ; for f , being a quadratic fun ction, will have two zeros, and when the equation is written in the form of Eq. (5.2.1)
we can soon verify that, in general, there are three regular singular points; two at the roots of f(z) = 0 and one at infinity. The hypergeometric equation is of just this form (as, of course, it must be) . Referring to Eq. (5.2.42) we see thatf = z(z - 1), g = [(a + b + 1)z - c], and h = abo But having gone this far, we can go further toward simplifying the equation, for we have the liberty of choosing the value of u, The coefficient of (z - t)" is now independent of z and t, for f", g', and h are constants. Consequently if we set ~p.(p.
- 1)1"
+ p.g' + h =
0
(5.3.10)
we obtain two solutions for p., either of which can be used in the integral representation. The differential form ~t is then ~t
where
a = f(t);
d2
d
= a(t) dt 2 - f3(t) dt
f3 = (p. - 1)f'(t)
+ g(t);
+ 'Y 'Y
= ip.(p. - 1)1" + p.g' + h (5.3.11)
§5.3]
Integral Representations
587
and the adj oint equation and the bilinear concomitant are d2 mtt(v) = dt 2 [j(t)v]
P(v,K)
=
jv
~ (z
+ ded [,B(t)v] + 'YV
- t)" - (z - t)"
= -p.fv(z - t),,-I - (lJ.f'v
= 0
~ (fv)
- (3v(z - t)"
+ jv' + gv)(z
If jJ. is adjusted so that Eq. (5.3.10) holds, then 'Y for v can quickly be solved : 2y
y = jv;
d d dt 2 = - de
vet) =
or
1
({3) ] y ;
exp {-
dy {3 dt = - ] y;
f[
(IJ. -
1)
7+ 1]
=
- t)"
0 and the equation
In y = -
f
(3
] dt
+ In A
dt} = Aj-"e-f Re b > and as long as z is not a real number larger than unity. For cases where Re b ~ 0, we can interchange a and b in this formula and obtain another, equally valid representation, since F(a,blclz) = F(b,alclz). The integral of Eq. (5.3.14) can then be considered to be the "real" hypergeometric function, from which one obtains series expansions about any desired point. It is interesting to note that this integral representation has a very different appearance from the one given in Eq. (5.3.4), which is equally valid. The difference is only superficial, however, for there is a close connection between gamma functions and the Euler transform, as will be indicated in some of the problems. An interesting extension of this formula is the expression for one of the solutions of the Papperitz equation (5.2.36), corresponding to Eq . (5.2.46) . Setting t = [(u - a)(b - c)/(u - c)(b - a)] (where a, b,
§5.3]
Integral Representations
589
and c are now the positions of the singular points, not the indices as above) we have
(: =~ ~ ~y(: =~: =~y ==
+ JL + v, A + JL + v'l A (-I)I' - A' + 1) r(A + JL + V')r(A + JL' + v) . F (A
A'
+ 11 zz --
a bb c
c)
- a
Hr(A
i
. (z - a)A(z - b)l'(z - c)v(a - cV(c - W"(b - a)"' . C
(u - z)-X-I'-v(u - a)-X-I"-v'(u - b)-X'-I'-v'(u - c)-X'-I"-V du
(5.3.15)
where A + A' + JL + J.L' + v + v' = 1. This has a suggestive symmetry with respect to singular points and to indices. From it, by setting a, b, c t Plane
A
o
Contour Fig. 6.6 Limiting contour for obtaining joining relations for hypergeometric function. Integrand is real at point A when z is real.
equal to 0, 1, 00 in different orders, one can write out a whole series of useful integral representations for different solutions of the hypergeometric equation about each of the singular points. The integral of Eq. (5.3.14) may be modified and manipulated to give a large number of useful and interesting relationships. For instance, Eq. (5.2.49) joining the solution about z = 0 with the solutions about z = 1 can be derived in a much more satisfactory manner than was given on page 546, with different restrictions from these which seemed to limit the validity of the result there. We start with a contour integral :f(z - t)-ata-c(I - t)C-b-l dt = 0 where the contour is the one shown in Fig. 5.6, carefully omitting all the singularities of the integrand. We make the integrand as shown real at the point A, for t on the real axis a little smaller than z but larger
[cH.5
Ordinary Differential Equations
590
than zero (we assume here that z is real, which is not necessary but makes the arrangements somewhat less complicated to describe; all that really is necessary is that z not lie on that portion of the real axis between 1 and +00) . Now if Re b :::; Re c :::; 1 + Re a :::; 2, we can use the methods described on page 410 to show that the equation above is equivalent to
+
sin(7ra)
f
(t - z)-ata-c(l - t)c-b--l dt
It
- sin 7r(a + b - c) - sin 7r(a - c)
00
10-
(t - z)-ata-c(t - 1)c-b--l dt 00
(z - t)-a( _t)a-c(1 - t)C-b-l dt
=0
We now make use of the formula f(z)f(l - z) = 7r/sin(7rz) to convert sin 7r(c - a) . t f(c sin7r(c-a-b)mo f(a b - c)f(c into f(a)f(l _
- a - b)f(a
+
(w
+z -
+
+
b- c 1) d sin (7ra) f(c-a)f(a-c+1) an sin7r(a+b-c) a - b 1) a) . In the first integral we set t =
+
l)/w, and in the third integral we set t
=
1 - U; this eventually
gives us
roo } 1
(t _ z)-ata-c(t _ 1)c-b--l dt
f + f(a
= f(c - a - b)f(a
+ b - c + 1) . + 1)
I'(c - a)f(a - c 00
(u - 1
+ z)-auc-b--l(u
+b -
c)f(c - a - b f(a)f(l - a)
f
00
-
1)a-c dt
+ 1) (1 _
(w - 1
z)c-a-b .
+ z)a-cwb--l(w -
1)-a dw
which can quickly be converted into Eq. (5.2.49) by using Eq. (5.3.14) in reverse. In this case our restrictions are that Re b :::; Re c :::; 1 + Re a :::; 2. Since the result is obtained by rearrangement of contours of integration and substitution of gamma functions and other finite procedures, it has, perhaps, a more dependable "feel" to it than does the juggling with infinite series required in the earlier derivation. Between the two derivations and by use of analytic continuation, we can extend this joining formula to . cover a still wider range of a, b, and c. For instance, our previous derivation is valid for Re b > Re c Re a, whereas the present one is valid for Re b :::; Re c so that, between the two, the relation is valid over the whole range of b for which the fun ctions are analytic. As long as z is not equal to unity, the function F(a,blclz) is an analytic fun ction of a , b, and c, except for a and b = 00 and for c zero or a negative integer. The right-hand side of Eq. (5.2.49) is an analytic function of a, b, and c except for c zero or a negative integer
591
Integral Representations
§5.3]
or for a + b - c a positive or negative integer or zero. For anyone of these parameters we can find a region for which one or the other of the restrictions on the derivation of Eq. (5.2.49) is complied with and which at the same time overlaps the wider range over which F is analytic. Consequently we can extend the equation, by analytic continuation, eventually to cover the whole range of values of a, b, and c for which the left- and right-hand sides are analytic. The other joining formula (5.3.5) may likewise be extended. Between the two , it is possible to express any solution of the hypergeometric fun ction around one of the three singular points in terms of the solutions about any of the other two points. Thus we have completely solved the problem of the joining factors, mentioned on page 577, for the case of equations with three regular singular points. Analytic Continuation of the Hypergeometric Series. Another set of formulas, useful in later calculat ions, may be obtained by further contortions of the integral expression (5.3.14). Let t = l /u and u = 1 - w, F(a blclz) ,
=
= r(c)(1 - z)-a r(b)r(c - b) =
e e Jo
. r(c) ub-l(1 - u)c-b-l(1 - uz)-a du r(b)r(c - b) [« we-b-l(1 _ w)b-l (1 _
~)-a dw
z - 1
(1 - z)-a F (a , c - bl c] z
~
1)
(5.3.16)
Finally, by applying this equation to the right-hand side of Eq. (5.2.50), we can obtain the further specialized relation
I + ~ + 1 11 ;
F (a,b a
z) = z-aF
(~, a ~ 1 Ia + ~ + 1 /1
-
~)
(5.3.17)
The integral representation. for F can also be used to derive various recurrence relations between contiguous functions . For instance, since (z - t)-ata-e(t -
1)c-b-1
= z(z - t)-a-1ta-e(t -
l)e-b-1
- (z - t)-a-1ta-e+l(t -
therefore -F(a,blclz )
Also, since we have
(bz/c)F(a
+ 1, b + 11 c + 11 z)
(d /dz)(t - z)-ata-e(t - 1)c-b-l dd F(a,blclz)
z
= ab F(a c
- F(a
1)c-b-1
+ 1, bl e] z)
= aCt - z)-a-1ta-e(t - 1)c-b-l
+ 1, b + 11 c + 11 z)
both of which, of course, can be easily obtained by manipulation of the hypergeometric series . Before we finish our discussion of the hypergeometric function, it would be well to verify a state~ent made on page 584 that the different independent solutions for a differential equation may be obtained by
592
Ordinary Differential Equations
[CH. 5
changing the allowed limits of integration of the integral representation without cha nging the form of the integrand. In the case of the hypergeometric equation, the second solution about z = 0 is, according to Eq. (5.2.45) , Y2 = zl-c F(b - e 1, a - e 11 2 - e/ z)
+
+
when the first solution is F(a ,blelz). Use of Eq. (5.3.14) to obtain an integral representation for the new F gives us
AY2 =
Ar(2-e) zl-e rea - e + l j I'(I - a)
j'" (u -
Z)c-b-1Ub-l(U - 1)-a du
1
which has a different integrand from the one for the first solution given in Eq. (5.3.14) . However, by letting u = zit and
A = r(e)r(a - e + l)r(1 - a) r(b)r(e - b)r(2 - e) we finally obtain an integral representation for the second solution
r(e) (" (z - t)-ata-e(1 - t)e-b-l dt r(b)r(e - b) )0 _ r(e)r(a - e + l)r(1 - a) I - e _ r(b)r(e _ b)r(2 _ e) z F(b e + 1, a
AY2 =
_
e
+
1-
1 2
c] z) (5.3.18)
which is valid for Re e < Re a + 1 < 2. This formula for the second solution differs from the formula (5.3.14) for the first solution only in the difference in limits of integration, thus verifying our earlier statement, at least as far as the hypergeometric equation goes. Changing 2 - e to e, b - e 1 to a, and a - e 1 to b we find that
+
+
r(e) Zl-e ~z F(a blclz) = (Z - t)e-b- 1tb-l(1 - t)-a dt , r(b)r(e - b) 0
(53 19) . •
thus obtaining still another representation for the first solution, for Re e > Re b > 0 as before. This integral can be changed into a contour integral about 0 and z if we can arrange the contour so as to return the integral to its initial value at the end of the circuit. This requires a double circuit, as shown in Fig. (5.7), going around both points once in each direction. Denoting the integral in Eq. (5.3.19) by J , we have
¢e(t =
Z) c-b- 1tb-l(1 - t)-a dt = eirb[eir(e-2b) - eire
+ e- ir(e-2b)
- e-irejJ irbJ 2 . . . 41r e 4e'rb sin (1rb) sin 1r(e - b)J = r(b)r(1 _ b)r(e _ b)r(1 b _ e)
+
Therefore a contour integral for F is e- irbzl-e F(a,blelz) = 41r 2 r(e)r(1 b - e)r(1 - b) .
+
. ¢e (t - z)c-b-ltb-l(1 - t)-a dt
(5.3.20)
§5.3]
Integral Representations
593
This equation may now be extended by analytic continuation to the whole range for- a, b, and c except that which makes the gamma functions go to infinity. Finally, by making the substitution t = (u - z)/(1 - u) in Eq. (5.3.19) we can obtain r(c)zI-c(1 - z)c-a-b
F(a,blclz) =
r(b)r(c - b)
.
lZ
(z - U)b-1 u c- b- l (1 - u)o-c du
(5.3.21)
which may also be converted into a contour integral similar to Eq. (5.3.20) . We have therefore found integral Contour C representations for both solutions of the hypergeometric equation which are valid over rather wide ranges of values of the parameters a, b, c. We could obtain other representaz tions which are valid over other ranges by juggling the interrela- Fig. 6.7 Contour C for integral repretions between F's. This is usually sentation of hypergeometric function by not necessary, however, for the in- Euler transform. Integrand is real at point A when z is real. tegral representations are chiefly useful in obtaining other formulas, such as recursion relations, series expansions, and the like. Once these formulas are obtained by using the integral representations, they can be extended for other ranges of the parameters beyond the range of validity of the representation, by analytic continuation, if need be and if the formulas themselves allow it . Rather than go into further detail in the properties of the general hypergeometric function, it will be more useful to study some of the special cases of particular interest later-in this book. Legendre Functions. The Gegenbauer functions, defined on page 547, represent a specialization of the hypergeometric function, since there are only two parameters, a and {3, instead of the three, a, b, and c. However, these functions include the great majority of the functions of hypergeometric type actually encountered in mathematical physics, so it is important to discuss their special properties in some detail. The equation is (Z2 - 1)1/;" + 2({3 + 1)z1/;' - a(a + 2{3 + 1)1/; = 0 (5.3.22) having three regular singular points, at -1, + 1, and 00, with indices (0,-{3), (0,+{3), and (-a, a + 2{3 + 1), respectively. It is interesting and useful to note that direct differentiation of the equation yields (Z2 - 1)1/;'" + 2({3 + 2)z1/;" - (a - l)(a + 2{3 + 2)1/;' = 0
594
[CR. 5
Ordinary Differential Equations
which shows that, if T~(z) is a solution of Eq. (5.3.22), T~~Hz) is a solution of the equation directly above. This indicates that solutions for integral values of f3 can be obtained from the functions T~(z) by differentiation. We also can see that, if T~(z) is a solution of Eq. (5.3.22), T~"'_26_1(Z) is likewise a solution of the same equation. The equation for T~(z), (Z2 - 1)1/;"
- a(a
+ I)1/;
(5.3.23)
= 0
If a is a positive integer, it can be obtained
is called Legendre's equation. from the equation (Z2 -
+ 2z1/;'
l)(dV/ dz) - 2azV
= 0
where V = (Z2 - I)"', by differentiation (n + 1) times, the final equation having d"'V / dz'" for 1/;. Therefore one solution of Legendre's equation is proportional to d"'(Z2 - l)"'/dz'" when a is a positive integer. With these simple properties pointed out, let us use the machinery of the present section to determine solutions of Eq. (5.3.22) for all values of a and f3 and expansions for these functions about the three singular points. Reverting to page 586 for the general procedure in setting up integral representations for equations with three regular singular points, we set down, for this equation, J(t) = (t2 - 1), get) = 2(f3 + l)t, h = -a(a + 2f3 + 1). The equation corresponding to (5.3.10), determining p., is p.2 + (2f3 + 1)p. - a(a + 2f3 + 1) = 0 so that p. = a or -a - 2{3 - 1. From Eq. (5.3.13) we then have the two alternative integral expressions for the solution and for the bilinear concomitant:
f f
1/; = A 1/;
=B
(z - t)'" (t 2 _ 1)"'H+1 dt;
(t2 - 1)"'H (z _ t)"'+26+1 dt; P.
=
(z -
0"'-1
P = -a (t 2 _ 1)"'+6;
P = (a + 2{3 -a - 2{3 - 1
(t
2 -
+ 1) (z _
p. = a
1)"'+6+1 0"'+26+ 2
as is usual with such representations, one integral can be changed into the other by suitable transformation of the integration variable and the limits of integration. We might as well consider that the real parts of a and {3 are positive to start with, for we can obtain the negative cases by analytic continuation later. When Re a and Re {3 > 0, proper limits for a line integral for the first alternative are t = z and t = 00, or the integral can be a contour integral about -1, + 1 and z in the proper order so that P returns to its original value after the circuit. Starting with the case {3 = 0, we wish first to find the solution which is proportional to the ath derivative of (Z2 - I)'" when a is an integer.
Integral Representations
§5.3]
595
Referring to Eq. (4.3.1) we see that a contour integral, using the second form, enclosing the point t = z and (for instance) the point t = 1, is the simplest means . We accordingly set T~(z)
1-.
(t2 - 1)"
= Perez) = A 'YD (t _ z)"+l dt =
n
21riA d ---nrdz
n
(Z2 - I)"
when a = n = 0, 1, 2, . . . , where the contour goes counterclockwise about both t = 1 and t = z, as shown in Fig . 5.8. The integrand is supposed to be real at point A when z is on the real axis between + 1 and -l. We note that for z on the real axis to the left of -1 the integral has a t Plane
Contour D -I
z
+1 A
Fig. 6.8 Contour D for integral representation of Legendre function of first kind .
different value (when a is not an integer), depending on whether the contour is passed above or below the point t = -1; therefore we make a cut along the negative real axis from -1 to - 00 to keep the function T2(z) single-valued. For -1 < Re a < 0 we can change the contour integral into a line integral from t = z to t = 1. Then by letting t = 1 - 2u and by use of Eq. (5.3.19), we obtain a series expansion for Per:
T~(z)
1 1
= -2iA
sin(1ra)
21riA2"
(t - z)-a-1(1 - t)"(1
«I- z)/ 2
=
rea + nr( -a) )0
=
21riA2" F[ -a,a
+
(1-2_z -
+ 0" dt
)-,,-1 u
u"(1 - u)" du
1111(1 - z)j2]
which final result may then be extended by analytic continuation to all ranges of a for which the hypergeometric series is analytic. Since it is convenient to haveT~(I) = 1, we set A = Ij2,,+I1ri , so that 1 1-. (t2 - 1)" ( Perez) = T2(z) = 2"+~i 'YD (t _ Z),,+I dt = F -a,a
+ 11 11-1 -2-z) (5.3.24)
which can be taken to be the fundamental definition of the Legendre function P a(z). We note, because of the symmetry of F , that P _a_I(Z) = P a(Z).
596
Ordinary Differential Equations
[CB .
5
This formula also allows us to calculate the behavior of P a(Z) for-very large values of z. Referring to Eq. (5.3.17), P a(z) = F ( - a, a + 1111 1 ; z) = za F ( _
~, 1 ;
alII 1 -
~)
By using Eq. (5.2.49) and the equations r(x)r(l - x) = 7I"/sin(7I"x) and .y:;;: r(2x) = 22x - 1r(x)r(x + ~), we finally arrive at Pa(z)
=
rea + ~)
(2z)a F (_
.y:;;: rea +
1)
+ ;(a +
1)
71"
so that, when
a
rea +
i)
t~nj7l":; Z a
is positive, P a
0::
~, 1 -
2
2
all -2 2al.!.) Z2
F(2 +2 a, 1 +2 a/2a :
31.!.)
(5.3.25)
Z2
za when z is very large.
t Plane
-1
Fig. 5.9 Contour E for integral representation of Legendre function of first kind.
Another useful integral representation can be obtained from Eq . (5.3.24) by setting u = i(t 2 - 1)/(t - z) or t = u + vu 2 - 2uz + 1, where t is real for z and ureal, Izi < 1 and u > 1. The transformed integral is _ 0 _ 1ur du. Pa(z) - Ta(z) - 2 ' 71"1,
¢
---;= /=====.2 E V 1 - 2uz u
+
where the contour E is the one shown in Fig. 5.9 ; the part A, corresponding to the circuit about t = 1 going into a circuit about u = 0, and the part B , which was the circuit about t = z going into a circuit a very large distance about u = 0, outside u = 1 or u = z. The two zeros of VI - 2uz + u 2 are at u = z ± ~ (the points P and P' in Fig . 5.9), so the contour reduces to the one marked E, about these two points. Since P a(Z) = P _a_l(Z), we also have Pa(z)
=
T~(z)
1
= 2-' 71"1,
¢ -_ E V
du 2uz + 1
U-a-l
/-;:=;:=====::==:==7-
u2
-
(5.3.26)
Integral Representations
§5.3]
597
When a is a positive integer, the integral about the large circle B goes to zero and E reduces to the circle A about u = 0, so by Eq. (4.3.1)
[d
n
1 Pn(z) = n! dun vu 2
1 2uz
-
+ 1 ] u=o
and, by use of Taylor's series (4.3.3), we have
L 00
VI -
1 2hz
+ h2 --
hnP
(5.3.27)
Z
n( )
n=O
where Ihl must be smaller than the distance from the origin to the points P or P' of Fig. 5.9 and Izi ::; 1. Also, by use of the Laurent series and the integral with ur, we have
Lh:+ 00
VI -
1 2hz
+h
2
1
Pn(z)
n=O
where Ihl must be larger than either [z ± we obtain the useful general formula
vz
2 -
11 or z
::; 1.
Therefore
00
1 vrf - 2rlr2 cos 0
+ r~
= \' ( rn2 ) P (cos 0) · r2 < rl
L...t
rl
n=O
+!
n
(5.3.28)
,
Legendre Functions of the Second Kind . The second solution of Legendre's equation must be found by using a different contour for the integral expression (5.3.24). We cannot simply use the second solution of the hypergeometric functions given in Eq . (5.3.18), for with c = 1, Y2 = Yl. In the present case, since the bilinear concomitant for (5.3.24) goes to zero at t = ± 1, we can use the integral
to form a second solution.
Consequently we define the function
1 Qa(Z) = 2a+I
fl -1
2
(1 - t )a (z _ t)a+1 dt
(5.3.29)
as the Legendre function of the second kind. For this form Re a > -1 and z cannot be a real number between -1 and + 1. For negative values of «(Re a < -1), we must take a contour integral about + 1 and -1. In order to bring the bilinear concomitant back to its original value, we make it a figure-eight contour, going around t = -1 in a positive direction and around.z -= + 1 in a negative direction.
598
Ordinary Differential Equations
[cH.5
Hence, working out the details
unless a is an integer or zero. The remaining case, a a negative integer, can be taken ca re of by setting Q_n(a) = Qn-I(a), which is allowable according to our discussion on page 595. Both integral formulas make it clear that, in order that Q be single-valued, a cut must be made between Z = 1 and Z = - l. When a is a positive integer or zero, Eq. (5.3.29) may be integrated directly, giving
+
Qo(z)
= tIn [(z + l)/(z - 1») ; Ql(Z) = tz In [(z + l)/(z --:- 1») -1; etc.
Even when a is not an integer, Qa(z) has logarithmic singularities at ± 1. For large values of z, we can develop a series in powers of l /z as follows:
z=
..
_ 1 Qa(Z) - (2z)a+!
fl dt {\' r(a + m + 1) (~) m _ 2a} ~ r(a + l)m! Z (1 t) -I
m=O
The integrals for m odd are zero, so that we ca n set u = t2 and obtain
..
\ ' I'(o + 2n + 1) (1) (I n- t Qa(z) - (2z)a+I ~ r(a + 1)r(2n + 1) z2n Jo u (1 _
1
_
a
•
u) du,
n=O
m
= 2n
Using Eq. 4.5.54 for evaluating the integrals and using the formula .y; r(2x) = 22z-- l r ( x ) r ( x + t) several times, we finally arrive at an expression for Qa which is useful for large values of z.
.y; r(a Qa(z) = (2z)a+I r(a
+ {)1) F (2- 2 + -'a -1 2 + -a la + ~I z21) + 3
(5.3.30)
as long as a is not a negative integer (in which case Q-n = Qn-I, a special case) . We see, therefore, that Qa(Z) ~ 0 whenever Re a > -l. z--> 00
Compare this with the corresponding Eq. (5.3.25) for P a(Z). There are several interesting interrelations between the Legendre functions of the first and second kind. One may be obtained by manipulating the contour for P a ( -z) as shown in Fig. 5.10. We first set t = -u in the contour integral for P a ( -z) and then change the contours as shown:
Integral Representations
§5.3]
599
But the contour C is just that for the function Qa and contour B is that for P a' In addition (z - u) in the second integral must be changed to (u - z) in order to compare with that in Eq. (5.3.24). If Im Z > 0, as shown in the figure, (z - u) = eir(u - z); if Im Z < 0, then (z - u) = e-ir(u - z) . Therefore we obtain (5.3.31) with the negative exponential used when Im Z > 0 and the positive when Im Z < O. This equation shows the nature of the singularity of P a(z). Unless a is an integer or zero, P a(Z) (which is unity t Plane at Z = 1) has a logarithmic singularity at z = -1; if n is an integerv Psf-e-L) = (-I)"P n(l) = (-I)n. This formula also allows us to obtain an expression for Q-a-l in terms of Qa and P a for any values of a . Since for all a we have P a(z) = P_a_l(Z), the above equation can be rearranged +1 to obtain -I Q_a_l(Z) = Qa(Z) - 7f' cot(7f'a) Pa(z)
(5.3.32)
which is valid for all but integral values of a . For a = n = 0, 1, 2, .. . , we simply use the formula Q_n_l(Z) = Qn(z). Since P and Q are independent solutions, their ' Wronskian P aQd - P~Qa should not be zero. Referring to Eq . (5.2.3), since p = 2Z/(Z2 - 1) = din (Z2 - 1)/dz, we obtain l:.(Pa,Qa) = C/(Z2 - 1)
-I
Fig. 6.10 Modification of con t 0 u r to relate P a( -z) with Pa(z) and
where the constant C can be obtained by calculating the value for some specific value of z, We Qa(z). choose the point at infinity, where we can use Eqs. (5.2.25) and (5.2.30). Since, for Z very large,
P a(z) ~ 2ar(a + ~)za / V; rea + 1) Qa(Z) ~ V; rea + l) /2 a+lr(a + i)za+l
and
we calculate tha t [Pa(Z)Q~(z)
- P:(z)Qa(Z)]
~
(-a - 1 - a)/[(2a
+ l)z 2];
Z~
00
so that C = -1 and we have, for all values of z, l:.(Pa,Qa)
= [Pa(Z)Q~(z)
Consequently, from Eq. (5.2.4),
- Qa(z)P:(z)]
1 [Pa~:)j2
= 1/(1
00
Qa(z)
=
Pa(z)
(u 2 - 1)
- Z2)
(5.3 .33)
Ordinary Differential Equations
600
[cH.5
Finally, we can utilize the integral representations (5.3.24) or (5.3.29) for P or Q to obtain recurrence formulas for the Legendre functions. Let p,,(z) be P,,(z) or Q,,(z) or any linear combination of these two functions (when the coefficients of P and Q are independent of a) . Then
where the integral is over any of a number of allowed contours, depending on the linear combination involved. In any case the contour is such that fd[(t2 - l)"+I/(t - Z)"+I] is zero. Therefore
°=
f
K
2,,+1
= K(a
[(t2(t _- z)"H l),,+IJ dt
d
dt
+ 1)
2"H _ K(a
-
+ 1)
2"
f f
l)"H} dt
2 2 {2t(t - 1)" _ (t (t - Z)"+1 (t - z)a+2 (t 2 - 1)" K(a 1) (t - z)" dt z 2"
+
+
f
(t 2 - 1)" (t _ Z)"H dt
K(a -
Differentiating by z and dividing out by (a
+
2"+1
+
1)
f
(t 2 - 1)"+1 (t _ Z)"+2 dt
1), we have
where the prime, as usual, indicates differentiation by z, For another equation, we expand fd[t(t2 - l)"/(t - z)"] = 0, obtaining (a
+ l)P"+I(Z)
- (2a
+ l)zpaCz) + azP"_I(Z) =
°
From these, by combination and differentiation, we obtain zPa(z) P~+I(Z) -
P~_I(Z) p~(z)
= [1/(2a + l)][(a + l)p"+I(z) + ap"_I(z)] = (2a + l)Pa(z) (5.3.34) = [a/(z2 - l)][zPa(~) - P"_I(Z)]
where Pet is P" or Q" or a linear combination of both. Finally, using Eq. (5.3.33), we have another relation between P and Q: a[P,,(z)Q"_I(Z) - P"_I(Z)Q,,(z)] = 1
Gegenbauer Polynomials. It is now fairly easy to extend our calculations to the more general functions defined in Eqs. (5.2.52) et seq. We could use, as primary solution, either the function T~, which is finite when a is an integer, or the function (1 - z2)-lmp~+lh which simplifies when 13 is an integer (the two functions are equal when both a and 13 are integers) . We prefer to make the first choice, since we are more often interested in the case a an integer 13 not an integer than we are in the
.
Integral Representations
§5.3]
inverse case.
T~(z)
=
601
We accordingly define (see Eq. 5.3 .21)
2~~~a . : I~~~ ~ 1) F(a + 2~ + 1, -all + ~I t ei..(aHlr(a + 2~ + 1) ,f, (t2 - l)a+~ +~+
= 2aH+2.II'T(a
1) sin [1J"(a
+ ~)] 'f c
tz)
(t _ z)a+l dt
(5.3 .35)
where the contour C is the one given in Fig. 5.7, going around t = 1 and t = z in both positive and negative directions. When a is an integer, the integral may be changed into the differential form given in Eq. (5.2.53). On the other hand, if we had preferred the other solution, we could define (z - l)-~ (1 - Z2)-t~p~+~(z) = r(1 _ {3) F( -a -~, a + {3 + 111 - ~I t - tz) e-i..(aHlr(a + 2{3 + 1) ,f, (t 2 - l)a+P = 2aH+21J"r(a + {3 + 1) sin [1J"(a + {3)] 'f o (t _ z)a+2Hl dt (5.3 .36) which, when {3 is a positive integer m, is equal to T,::(z) , which, in this cas e, equals the mth derivative of the Legendre function Pm+a(Z). When both a and ~ are integers, dm T';(z) = (1 - z2)-t mP::'+m(z) = dz m T?n+n(z)
(n + 2m)! 1 ~ (Z2 _ l)n+m 2n+mn!(n + m)! (Z2 - l)m dz" 1 d n +2m ~-,--------;------,--. - - (Z2 - I) n+m n+m(n 2 + m)! dz n+2m
(5 .3.37)
as indicated in Eq. (5.2.53). These polynomials are called associated Legendre polynomials. From the symmetry of the hypergeometric functions we have
T~----a-2~-1()Z
--
[sin1J"(a+2~)JT~()' . () a Z, P~a+~ () Z -P~ - -a-~-l () Z sm 1J"a
and for {3 = m, an integer, we use the relation F(a,blclw) = (1 . F(c - b, c - al c] w) to show that
T-;;m(z) = (1 -
= =
(Z2 - l)m 2mm! F(2m - a, 1
=
rearea - 2m + 1)+ 1) (z \!
.
z2)tmp~m(z)
(z - I)m r(I + m) F(m - a, a - m
1._
w)c-a-~
2
+ all + ml t -
tz)
tz)
m r - I)mTa_2m(z)
)m -m() _ r(-y - m P., z - r(-y + m
1
+ 111 + ml t -
+ 1) 1) pm( ) + ., z
(5.3 .38)
602
[CH. 5
Ordinary Differential Equat ions
An analogous set of formulas to Eqs. (5.3.26) and (5.3.27) may also be found, showing that the generating function for the Gegenbauer polynomials is '"
2dr(,8 + i)/.y;;: _ '\' n d • + h2 _ 2hz)d+i - ~ h Tn(z) ,
Ihl < 1
(l
(5.3.39)
n=O
By using Eq. (5.3.5) and the relation
which is closely related to Eq. (5.3.17) , we obtain the expansion for these solutions for large values of z : Td(Z) = a
rea + 2{3 + 1) sin(7ra) z-a- 2d-1 .y;;: 2a+d+1 cos[7r(a + ,8)] rea + ,8 + i)
+ :: + 1, a + :: + 2 I a + ,8 + il ~) + 2a+dr(a + ,8 + i) zaF (-i a i - ial i-a - ,81 2) .y;;: rea + 1 ) ' Z2 rea + 2,8 + 1) sin[7r(a + 2,8)] z-a- 2d-1 2)-idpd () Z a+d Z = . V7r 2 cos[7r(a + (3)] rea + (3 + i) . F (a + :: + 1, a + :: + 2 I a + (3 + il ~) +2 + ,8 + i) zaF (_la lal l _ a .y;;: rea + 1) '2" ,'2" '2" '2" .F ( a
(1 -
(5.3.40)
a+l1+ 1
a+l1r(a
I
_
_
(.II ~) fJ
Z2
which shows that both of these solutions go to infinity at z ---+ 00 unless 0>a>-2{3-1. If we consider the function T~(z) to be the first solution of Eq. (5.2.51), we now have collected an embarrassing richness of second solutions. The function T~a-2d-1(z) is, of course, proportional to T~(z) and thus is not independent, but the function variously labeled
= (2 (1 _ Z2)-idpda+d () Z Z
_
+++
+
1)-d rea 2{3 1) sin[7r(a 2,8)] T-d ( ) rea 1) sin (7ra) -a-I Z
= (1 - z2)-ldPd R (z) = (Z2 _ 1)-8 rea -a-,,-I
+ 2{3 + 1) T-da+2"R(Z) + 1)
rea
is independent of T~(z) (unless ,8 is an integer) as are likewise T~( -z) and (1 - z2)-idP~+d( -z). In addition, there are the functions with the sign of the argument reversed. Equation (5.2.49) can be used to show that sin(7r,8)
T~(z)
= sin[7r(a
+ ,8)1 T~( -z)
- sin(7ra)(l - z2)-id P~+d( -z)
I nlegraL Representations
§5.3]
603
which also shows the special relationships occurring when either {3 or a is an integer. ~
All these functions, however, go to infinity at z (except when °which > > 1). We shall often prefer to have a second solution when a > 0, as do the Q functions defined goes to zero as z a
00
-2{3 -
~
00
in Eq. (5.3.29). Also we should prefer a solution which stays independent of T~(z) even when (3 is an integer. Such a function may be obtained from Eq. (5.3.39). We set
(5.3.41)
The contour for the integral is the figure eight, about + 1 and -1, which was used for Qa(z). The second line shows that V goes to zero when z ~ 00 as long as Re (a + 2{3 + 1) > 0, as was desired. The third line shows the relationship between V and the two independent T solutions. The fourth line, to be compared with Eq. (5.3.31), relates the solutions for +z with those for -z; the upper sign is to be used when Im z > 0, the lower when Irn z < O. The last line shows that this second solution is adjusted so that change of sign of superscript does not produce an independent function, in contrast to T~, where Eq. (5.3.38) holds only for (3 an integer. On the other hand, V~a-2~-1 is not proportional to V~ but V~(z)
=
V~a_2~_1(Z)
+ [ll'ei..~ cos ll'(a + (3) /sin(ll'a)lT~(z)
When (3 = m, an integer, the third relation in Eqs . (5.3.41) goes to a finite limit, but in this case the following, simpler relations hold : m _ V -a-2 m-1 -
rea + 2m + 1) V-m rea + l)(z2 _ l)m -a-1
(5.3.42)
The case of {3 = ±i is of some interest, both because of the resulting polynomials and also because of the special properties of the hypergeometric functions which are displayed :
604
Ordinary Differential Equations
I~ cosh [a cosh- l z] =
T;!(z) =
'\J7r
T!(z) = a
a
+ 1) cosh'\J;~ sinh [(ayz2 _1
V;!(z) = .y2;.
(f) exp[
! VZ2=1 p!
[cH.5
a (1 - z2)i a-i
l
(z)
z] = a (1 - z2)i p-i (z)
VZ2=1
-a cosh-' z]
= -
~ yz2 -
a-i
( ) 5.3.43
1 VLl(Z)
which, for a an integer, are proportional to the Tschebyscheff polynomials. The Confluent Hypergeometric Function. The hypergeometric equation for F(a, b + e] e] z/b) is z (1 -
~) F" + [c - (a + b + c+ 1) ~ ] F' -
a (1
+ ~) F
=0
This equation has regular singular points at 0, b, and ao with indices (0, 1 - c), (0, -a - b), and (a, b + c), respectively. If we let b go to infinity, we shall at the same time merge two singular points (the ones originally at band co ] and make one index about each of the merged points go to infinity (the index -a - b at the point b and the index , b + c at the point eo }, This double-barreled process is called a confluence of singular points ; it has been mentioned before on page 537. The resulting equation zF" + (c - z)F' - aF = (5.3.44)
°
is called the confluent hypergeometric equation [see Eq. (5.2.57)]. The solution analytic at z = is given by the limiting form of F(a, b + e] c] z/b) when b goes to infinity;
°
_ I - 1 F(alcz)
a a(a + 1) a(a + I)(a + 2) + CZ + 2!c(c + 1) z + 3!c(c + I)(c + 2) z + 2
3
(5.3.45) is called the confluent hypergeometric series. To see how the confluence affects the integral representation, we start with a form derivable from Eq. (5.3.14) by changing t into I/t :
F
(a, b + c] c] ~)
=
F (b
+ c, al c] ~)
=
r(c) r(a)r(c - a)
e (1 _~)-1>-C ta-l(I _ t)c-o-l dt b
)0
Letting b go to infinity changes the nature of the kernel [1 - (zt/b)]-1>-c from an algebra ic to an exponential fun ction, for [
1 -
~J-b ~eZ b
b->
00
The singularity changes from a branch point at z = b to an essential singularity at x = ao, and the resulting integral representation for the
Integral Representations
§5.3]
605
confluent hypergeometric function is then
F(alclz) =
e e" ta-1(I _ t)c-a-l dt
r(c) r(a)r(c - a)
)0
(5.3.46)
as long as Re c > Re a > O. For other ranges of a and c we can devise a corresponding contour integral which will be valid except when c is a negative integer or zero, in which case even the series expansion "blows up ." In these cases, which will be discussed later, the solution has a logarithmic branch point at z = O. The second solution of Eq. (5.3.44) may be found by performing the confluence on the second solution of the hypergeometric equation ZI-c F(a - c 1, b + 112 - c] zjb) or may be found by remembering that Zl-c j(z) is a solution and, by inserting this in Eq. (5.3.44), showing that the equation for j is also of the form of (5.3.44) . In either case the second solution is found to be
+
Zl-c F(a - c
+ 112 -
c] z)
which is valid unless c is 2, 3,4, . . . . If c = 1, this series is convergent but it is identical with the first solution F(aIIlz), so that the above expression does not represent a second solution whenever c is a positive integer. The second solution in these special cases will be obtained later. There is no need to investigate the case of c < 1 separately, for if c = 2 c' < 1, we can multiply our solutions by zc-l = Zl-c' and call the original second solution the first solution, and vice versa, and then c', which is the new c, is greater than 1. An interesting interrelation, first pointed out in Eq. (5.2.62), may be derived from the integral representation, for since
101 c,t ta-1(I
- t)c-a-l dt
=
c'
101 C-'" u c-a-1 (1 -
u)a-l du
we must have F(atclz) = c'F(c - alcl-z). Similarly another form of the second solution is zl-cc'F(I - al2 - cl-z) . The Laplace Transform. But before we go further with a discussion of the solution, we should examine more carefully the new form of the integral representation, for we have now changed from an Euler transform to a Laplace transjorm, mentioned on page 584. The new kernel is c't, not (t -z )1'. Referring to page 583, where we discussed integral representations in general, we see that the exponential kernel has certain advantages. For instance, since dc,tjdz = te" and zc,t = dc'tjdt, if our original equation is d1/l \' d n1/l o = £,(1/1) = j d2if; dz2 + g dz + h1/l = ~ A mn zm dzn m,n
j
=
L
A m2zm ; g
m
=
L
Am1zm ; h
m
=
L
Amozm
m.
606
[cH.5
Ordinary Differential Equations
then the corresponding transformed expression will be C'tl'I'
and from thence back to t = 1. Thus again the integral splits into two integrals. For the first let t = -we-io/> Ilzl = wei(r-l/lzl, and for the second let t = 1 - ue-io/>/lzl, where both u and ware real. The integral representation then becomes
610
Ordinary Differential Equations
F(alclz) =
r(c) {elzlei4>-iCc-a)4> roo e-"Uc>-a-l r(a)r(c - a) Izlc-a Jo
eia ("- 4» roo + ~ Jo e-wwa-
1
[CH. 5
(1 _z~)a-l du
( w )C-a-l} 1+ dw
z
(5.3.50)
for 0 < tP < 71". Writing only the dominant terms in the expansions of [1 ± (w or u/z)] (i.e., assuming that these quantities are practically equal to lover the range of values of w or u for which the integrand is not negligible) we have for the asymptotic formula for z = Izlei4>,lzllarge,
0 /lzl = we-i(H")/lzl and t = 1 - ue- i4> /lzl, respectively. Here we have let 1/-z = e-i(H")/lzl in the expression for w, instead of the expression ei("-4» /lzl used before , because we want w to be real and positive when t is along the dashed line and because we have taken the cut for the integrand from t = 0 to t = - 00. With this change we find that
/
F(alclz)
=
r(c) Izla-cei(a-c)4>e1zlei4> r(a)
+
r(c) Izl-ae-ia('-++> I'(c - a)
when z = Izle i4> and 0 > tP > -71". We note that, as tP goes negative from 0 to -71"/2, the second term emerges from its" eclipse" at tP = 0, differing from the corresponding term at tP = +71"/2 by a factor e- 2ria . This sudden change in the second term at tP = 0 does not require a discontinuity in F as tP goes through zero, for it is just at tP = 0 that the
Integral Representations
§5.3]
611
second term is smaller than the inherent error in the first term, so any change in the second term does not count. However, the change is just what is required to make the second term real for both q, = 1T' and q, = -1T'. If the additional factor e- h i4 had not been included in the expression for the range 0 > q, > -1T', the asymptotic formula for q, = -1T' would have been e2ria times the formula for q, = +1T', which would have been wrong because F is analytic about z = O. In order to maintain this analyticity the two terms in the asymptotic expansion play" hide and seek" with each other; as each goes into " eclipse," becoming smaller than the error in the other, it changes phase, emerging from obscurity with enough of an additional phase factor to ensure continuity over the range of q, where it is large . This "hide-and-seek" behavior of the terms in an asymptotic expansion is called the Stokes' phenomenon, after the person who first called attention to it. It must be displayed by the first term at q, = 1T', where the first term is smaller than the error in the second. For instance, in the range 1T' < q, < 21T', in order that F for q, = 21T' equal F for q, = 0, we must have F(alclz) ---+ r(c) Izla-cei(O
(5.3.76)
8= 1
~ [In (.:) + I'J ; 2
n = 0
7r
1 (2)n • z->O - 7r(n - I)!
------?
s + s +1 n J
\ ' [1
n
z'
>
0
°
When n = 0, the finite sum (from to n - 1) is not present. This expansion has the same general form as Eq. (5.3.61) for the confluent hypergeometric fun ction of the second kind; it has a term involving the product of J n with In z, a finite series involving negative powers of z (except for n = 0 when this is absent) , and an additional infinite series which starts with a term in zn+2. Approximate Formulas for Large Order. The asymptotic formulas given in Eqs. (5.3.68), (5.3.69), (5.3.70), and (5.3.75) assume that the argument z is very much larger than the order II of the Bessel functions. Useful formulas for J .(z), when both z and II are large, may be obtained by using the method of steepest descents (see Sec. 4.6). When both z and II are large, the integral J~ ezt (1 + t 2).- i dt , from which we obtained the earlier asymptotic expansions, does not have the largest value of its integrand near t = i, as we assumed heretofore. Consequently the first term in the asymptotic expansion is not a good approximation, and we must look for another formula. It turns out that the integral representation (e- i r i'/7r)fe i (zco, w+ l'W) dw
.
mentioned on page 623 with the contours shown in Fig. 5.13 is the most suitable for our present purpose. For instance, if the contour is A, the integral is 2J .(z); if it is B, the integral is H~1)(z); and if it is a double contour Band C (with B taken in the reverse order, from top to bottom), then it is 2N.(z) . Each of these contours goes from a region where the int egra nd is vanishingly small (the unshaded areas in Fig . 5.13, far from the real axis) across a region where the integrand is large to another region where the integrand is vanishingly small. If we draw a contour of arbitrary shape between the specified limits, the integrand will vary in a complicated manner, reversing sign often and often having large imaginary parts, but these additional complexities all cancel each other out in the integration, and the result is the same no matter how we have drawn the
628
Ordinary Differential Equat ions
{cH.5
intermediate part of the contour (for the integrand only has singularities at w = 00). If we are to calculate an approximate value for the integral, we should first pick out the path along which the integrand has the simplest behavior possible. For example , since the integrand is an exponential function , the real part of the exponent controls the magnitude of the integrand and the imaginary part controls the phase. To eliminate undue fluctuations of the integrand we should arrange to perform the integration along a contour for which this phase is constant. We should next try to find a route for which the integrand is large only over one short range of the route, so that calculation in this restricted range is sufficient to give a good approximate answer. Anyone of the contours shown in Fig. 5.13 is a path over a mountain range. We wish to stay in the plains as long as possible in order to cross via the lowest pass and to regain the plains on the other side as fast as possible. It turns out that the path which does this is also the path for which the phase of the integrand stays constant, as was shown in Sec. 4.6. For our integrand is the exponential of a function of a complex variable w = u + iv; the real part of this function controls the magnitude of the integrand and the imaginary part is the phase F(w) = i(z cos w
+ vw) = f(u,v) + ig(u,v);
f = z sin u sinh v - vv;
= z cos u cosh v + vu The white areas of Fig. 5.13 are where f becomes large negative, and the shaded areas are where f becomes very large positive. We wish to go g
over the lowest pass in the f(u,v) surface. Since f is the real part of a function of a complex variable, it therefore is a solution of Laplace's equation in u and v (see page 353) and it can have no maxima or minima in the finite part of the plane. This means that the top of the lowest pass must be a saddle point, where the surface curves downward in one direction and curves upward in a direction at right angles . In other words the top of the pass is the point where dF/ dw = 0 and near this point W.)2 + . . . y2) cos 2t1 - 2xy sin 2t1 J i g. + ia[(x 2 - y2) sin 2t1 + 2xy cos 2t1 + .. .J where b = ae2i" = (d 2F/ dw 2). and x = u - u., y = v - v.. Along the lines y = -x tan t1 and y = x cot t1 (at angles -t1 and -hr - -t1 to the real axis) the imaginary part of F is constant (g = g.) and the real part has its greatest curvature. Along the line y = -x tan -t1 the real part of F is Re F = f = f. + iax 2 sec - t1 + . . . = f . + iad 2 + . . .
F
= F. + b(w = f. + a[(x 2 -
+
+ . ..
where d = v'x 2 + y2 is the distance along the line y = - x tan tJ. Therefore along the line at an angle - t1 with the real axis, f (and there-
629
Integral Representations
§5.3]
fore the integrand) rises as we go away from the saddle point W = weHence this line is along the axis of the mountain range and is not the direction to take, for it will lead us among the high peaks. However, along the line y = x cot {} the real part of f is
f
=
f . - iax 2
esc- {}
+ ...
=
f. -
taS + . . . 2
where s = V x 2 + y2 is the distance along this line from the saddle point. Therefore along this path we head downward toward the plane in either direction from the top of the pass . If we have no other mountain ranges to cross with our contour, an approximate expression for the integral will then be
f
eF(w) dw
~ el.+ig.
f-.. .
i-i~ ds = ~ el.+ig.-i~H ..i
e-!a.2+!..
If the contour (in the direction of integration) makes an angle ~ - {} with the real axis near W = W" the element dw = e! ..i-s» ds where ds is real. [If the integral is in the opposite direction, a factor (-1) must be added to the final result or the term ~i in the exponent should be changed to -Pi.] This expression will approach the correct value for the integral the more closely the higher and narrower the pass over which we must go (i .e., the larger f. and a are). To apply the technique to the Bessel function integral we set F(w) = iz cos w + ivio. The saddle points, where dF j dw is zero, are the points where sin w. = IIjz. When z is real , there are two cases : one where z > II when w. is along the real axis at the points sirr' (IIjz); the other where z < II when w. is complex, at points (2n + t)1r + i cosh-1(IIjz) . The two cases are shown in Fig. 5.14, the drawing at the right for z = 0.86611 and that at the left for z = 1.54311, both plotted on the w = u + iv plane. The light lines are contours for equal magnitudes of f = Re F ; the heavy lines are the contours which go through one or both of the saddle points S , S' and along which g = Im F is constant. The contour A, coming from ioo -~, going through S (or through S and S' on the right) and going to i 00 + %Jr, gives 2J .(z) . The contour C, going from i 00 - ~ through S' (on the left) or S (on the right) to -ioo + ~ gives H~ll(z), and the contour D gives H~2)(Z) . If both C and D are used, starting at i 00 - ~ and i 00 + j?r and ending both at -ioo +~, we obtain 2N.(z). The contours Band B' are for g constant, but for these cases f increases as we go away from the saddle point. If both II and z are large, the value of f at the saddle point is much larger than its value anywhere else along A, for instance, and we can neglect all the integrand except that quite close to S. Taking first the case of z < II (we set z = II sech a for convenience), we find that dF jdw = 0 at w = w. = ~ ± ia . The upper point, S, is the one crossed by the contour A, suitable for 2J .(11 sech a). The
630
Ordinary Differential Equations
[CR. 5
value of I at w. is I. = v[tanh a - a] , and the value of g there is g. = j-,r". The second derivative, d 2F/ dw 2 , at w = w. is - v tanh a, so near the point S an approximate value for F is F ~ v[tanh a - a] + tiV7l" - iv tanh a(w - W.)2 = F. + iae 2i"(w - W.)2; a = v tanh a; t'J = j-,r By our earlier discussion the contour which comes down off the pass is at an angle j-,r - t'J = 0 to the real axis and the approximate value
v=2
v=-2
I
U=7T
Case for z =1.543v
Case for z =0.866v
Fig. 6.14 Paths of integration for asymptotic forms of Bessel functions for both z and v large.
of the integral is 2J v( v sech a)
~!. -i.-iv 71"
f
e F (w )
A
dw =
I, 2 :\J 7I"V tanh a
e:(taDha-a)
The contours for the Hankel functions go through S', where
I. = v(a - tanh a) ;
g. = j-,rv; a
=
(d2F /dw 2) . = v tanh a; v tanh a
t'J
= 0;
For the first Hankel function the contour goes in the negative direction through S'; for the second function it goes in the positive direction, so H (2)(V sech a) v
~
-H(v sech a) v
=
'-_ _2 __ '\j7l"V tanh a
e v (a - ta Dh a ) + i.-i
and therefore the corresponding expression for the Neumann function is - N v(v sech a)
~ 1
2 '\j7l"V tanh a
ev(a-taDha)
When z > v, we set z = v sec {3, and the saddle points are at w. = The second derivative (d 2F/dw 2) . = ±iv tan {3, so that
ir ± {3.
631
Integral Representations
§5.3]
± (11/4); also f. = 0 and g. = tn-v =+= v(tan (3 - (3). Contour 0, for the first Hankel function, crosses Sew. = j-,r - (3) in the negative direction, so that
a = v tan f3 and v =
H v
The Coulomb Wave Function. The radial part of the Schroedinger equation for an electron of mass m, charge - e, and total energy E in a
632
[cH.5
Ordinary Differential Equations
centrally symmetric potential field V(r) is
~ ~ (r2
r 2dr
+
dR) _ l(l 1) R dr r2
+ 2m2 [E h
- V(r)]R = 0
where the term l(l + 1) is the separation constant coming from the angular factor ; if the potential is a function of r alone, then 1 is an integer. If the potential V is the coulomb field dueto a nucleus of charge Ze (heavy enough so that the nucleus is at the center of gravity), .V is then equal to - (e2Zlr) . Setting E = - (me 4Z 2/2h 2)K2 and x = 2KZ = (2Kme 2Zlh 2)r and R = e-~ xxlF(x), the equation for F is 2F d x dx 2 + [2(l
dF [ x] dx - 1 +
+ 1) -
1-
1] °
-;z F =
+
which is a confluent hypergeometric equation with a = [l 1 - (11K)] and c = 2l + 2. The solution of the above equation, which is finite at z = 0, is therefore
y; (2KZ)Ie-«z ( 1 C1(K,Z) = 2 21+Ir (l + j) F 1 + 1 - -; 12l ~ z ..... 00
+ 212Kz
)
+ 1) (2KZ)-1-O/' )e«Z (11K)] r(l + l)e ir [I+I- 0 / . ) ] r(l
r[l
+
+
1 -
r[l
+ 1 + (11K)]
(2KZ)-1+(1;.)e-"
(5.3.78)
When K is real (i .e. for energy E negative), this solution, though finite at Z = 0, becomes infinite at Z ---+ 00 unless r[l + 1 - (11K)] is infinite. If 11K is a positive integer n larger than l, then r[l + 1 - (11K)] is the gamma function of zero or a negative integer, which is infinite. Only in these cases is there a solution which is finite from r = 0 to r = 00 . Thus the allowed negative values of energy are
En = - (me 4Z 2/ 2h2n 2)
n = 1 + 1, 1 + 2, l
;
+ 3,
...
which are the values first computed by Bohr. When K is imaginary ( K = ik, for positive energy) , C1 is finite for the whole range of z for a ny real value of k (any positive value of energy) . In this case the first (finite) solution takes on the various forms
. .y; (2ikz)1 [ C1(tk,z) = 221+1r(l + j) = (2ikz)/r(l
+
+ 1) [r[l + 11+
rll
r(l
+
~ kzlr[l
+ 11_
1 -
(ilk)] U 1 (l
(i lk)] U 2(l
l)e(irIl2)-(r/2k) . (i lk)rl sin
+ 1+
Z
!
1 - l + 1 + 21(l +
+ 1+
[1 + kz
+
(l l)k 2 2 1)(l j) z -
+
]
. . .
+ 1+ ~ 12l + 212ikZ)
~ 12l + 212ikZ)Je- ik• 1
kIn (2kz) - pl - 4>1(k)
(5.3.79) ]
§5.3]
Integral Representations
where
[r (1 + + i)/r(1 + i)] = e2i~ICk)
and
Ir (1 +
1
+ ~)l2
633
1
1-
=
[1 + bJ [(I - 1)2 + bJ .. . 2
.[1 + ~J
7rlk k 2 sinh(7rlk)
The second solution, which has a singularity at z = 0, can be written
. 0 (2ikz)Ze-ikz ( Dz(tk,z) = 22Z+lr(l + -I) G I ~ z-.oo
r(I +
ikzlr[I
. ) + 1 + ik 12l + 212tkz
l)ei(..z/2)- cr / 2k) [1 1 ] + 1 + (ilk)] I cos kz + kIn (2kz) - -vI - z(k)
The series expansion about z = 0, showing the nature of the singularity there, can be obtained from Eq. (5.3.62). Other solutions, suitable for other equations of interest, will be discussed when the problems are taken up. Thus the wave functions for parabolic and parabolic cylinder coordinates can be expressed in terms of the F's and G's. They will be treated in Chap. 11. Mathieu Functions, We have intimated on page 587 that the integral representation is most useful for the case of equations with three regular singular points or with one regular and one irregular point. We shall not be disappointed, therefore, to find it not quite so useful with more complex equations. To illustrate the difficulties and to show that a measure of utility still survives, let us apply the Laplace transform to the Mathieu equation (in one of its algebraic forms) : ( Z2 -
1)1/;"
+ z1/;' +
(h 2z 2
-
b)1/; = 0
(5.3.80)
We can see by inspection that the application of the Laplace transform will give an equation for v(t) of the same general form : (t2 + h 2)v"
+ 3tv' -
(t 2 + b - l)v = 0
It thus requires as much work to solve the equation for v as it does to solve for 1/; ; v is also proportional to a Mathieu function, and our integral representation reduces to an integral equation. But an integral equation can be of some value, so it is worth while exploring further. It is more satisfactory to change the scale of t so that the equation for v comes as close as possible to that for 1/;. We set 1/;(z) = Jeihtz v(t) dt and obtain for an equation for v and for the bilinear concomitant (t 2 - l)v" + 3tv' + (h2t 2 - b + l)v = 0 P(v,e ihIZ) = -iheihzl {(t 2 - l)v' + [t - ihz(t 2 - l)]v}
634
[CR. 5
Ordinary Differential Equations
If v is set equal to !(t)/V1 - t2, the equation for! turns out to be just the same as Eq. (5.3.80) for 1/;. Consequently we can say that, if
! is a solution of Mathieu's equation (5.3.80), then another solution of the same equation is 1/;(z)
=A
f
eihztj(t)
dt
VI -
(5.3.81)
t2
If the constant A and the limits of integration are chosen properly, then 1/; can equal! and we have an integral equation for 1/;.
But even in its present form we can use the integral to help us in the analytic continuation of the solution, which is why we need an integral representation. For instance, a solution of Eq. (5.3.80) (see page 565) for a particular value of b( = be2m) is .,
L B 2n cos(2nq,);
z = cos q,
n~O
From Eq. (5.3.81), we see that another solution of Eq. (5.3.80) is ., ., 1/;(z) = A fo27r eihzco.'; B 2n cos(2nq,) dq, = 2?rA (-I)nB 2nJ 2n(hz)
L
n=O
L.
n=O
•
where we have used Eq. (5.3.65) to obtain the Bessel functions. This is an extremely interesting and important relation; that a solution of the Mathieu equation can be expressed in terms of a series of Bessel functions is interesting in itself but that the numerical coefficients for the series are equal (with alternating signs) to the coefficients in the corresponding Fourier series solution is one of those satisfactory denouements which renew one's faith in the essential unity and simplicity of rnathemati cs. We could, of course, have given ourselves this pleasant surprise in the last section, when we were talking about the series expansion of Mathieu functions. We could have tried using a series of Bessel functions to solve Eq. (5.3.80) (see page 575), and we should have found that the recursion formula for the coefficients is exactly the same as the formula (5.2.69) for the Fourier series coefficients (with alternating signs for the a's). Then we should have said that the recurrence formulas (5.3.64) for the Bessel fun ctions , which lead to the recursion formula for the coefficients, are closely related to the recurrence formulas for the trigonometric functions, so that for an equation of the particular symmetry of Mathieu's equation the recursion formulas turn out to be equivalent. Since we waited to demonstrate the property until the present (in order to be better acquainted with Bessel fun ctions), we should now be inclined to say that the close integral relationship between trigonometric and
§5.3]
Integral Representations
635
Bessel functions, via the Laplace transform, is the property which is responsible for the coincidence. The two statements are but two aspects of the same general property of course. As yet we have shown only that the Bessel function series is a solution of Eq. (5.3.80); we have not shown how it is related to Sem(h,z), which is also a series (the series which is analytic at z = ± 1). To do this we change from z = cos rp to z = cosh 0 (0 = irp) in S e and then use Eqs. (5.2.54) relating the hyperbolic functions to the Gegenbauer fun ctions
l: ~
Se 2m(h,z) =
l: ~
B 2n cosh(2nO) =
n=O
~
B 2nT;;,i(z)
n -O
By expanding this in powers of z and expanding the Bessel function also in powers of z, we can show that the two series are proportional and are therefore the same solution. Thus we have obtained an analytic continuation of the solution Se, defined for the range -1 < z < 1, into the range 1 < z < co. For the range 1 < z < co we shall prefer to use a function which has simple asymptotic properties. Using the asymptotic behavior of the Bessel functions [Eq. (5.3.68)], we define
l: ~
J e2m(h,z) =
~
(-l)n-mB 2nJ 2n(hz) ~
~ cos [hz -
p(2m
+·m
n=O
(5.3.82) since "};B 2n = 1. The asymptotic expression is valid for the argument of z between -p and +p. Since this solution is proportional to S e2m, we can find the factor of proportionality by comparing values at any satisfactory values of z, The value z = is most satisfactory, for
°
l: ~
J e2m(h,0 ) = (-I)mB o
Therefore
[l:
~~
and
Se2...(h,0) =
(-I)nB 2n
n=O
(-I)nB 2n] J e2m(h,z) = (-l) mBo ~Se2m(h,Z)
(5.3.83)
n
We are also now in a position to obtain a second solution for Eq. (5.3.80), by replacing the Bessel by Neumann functions:
l: 00
N e2m(h,z) =
~~
(-1)n- mB2nN2n(hz)
~ ~sin
[hz - p(2m
+ i)]
n=O
(5.3.84)
636
Ordinary Differential Equations
[cH.5
This solution has singularities at z = ± 1 and is, of course, independent of Ses« and J e2m. The series of Eq. (5.3.84) cannot be used for Izi ::; 1 (as a matter of fact it does not converge well for small values of z just greater than 1), and a different expansion must be used . Since the second solution for the range Izi ::; 1 is rarely needed in physical problems (since they are not periodic in q,), we shall not go into the matter further. The- Laplace Transform and the Separated Wave Equation. Before we continue with our discussion of the Mathieu and spheroidal functions, it will be illuminating to introduce an extremely important technique of solution of the wave equation, which will be used extensively in later chapters. Here we shall use it only to bring out a new point of view with regard to the Laplace transform which will enable us to set up new integral representations practically at will. To do this we return to the discussion of the separation of the wave equation in two dimensions, discussed on pages 498 to 504. If the coordinat es ~I and t2 are separable and represent a conformal transformation from x, y, the Helmholtz equation can be put into the form (5.3.85) Any solution of the Helmholtz equation, whether it separates into factors in ~I, t2 or not, must be a solution of this equation. For instance, ei kx or Jo(kr) , when x or r is expressed in terms of ~I, t 2, satisfies Eq. (5.3.85). If the solution happens to separate in these coordinates, if; = XI(~I)X 2(t2), then X I and X 2 are solutions of the separated equations (d2X I/dW + k 2[gl(h) - a 2jX I = 0 ; (d 2X 2/dW + k2[g2(~2) + a 2]X 2 = 0
where a 2 is the separation constant. The point of reviewing this material here is that we can now show that any solution of the Helmholtz equation is a suitable kernel for the integral representation of one of the separated solutions X I in terms of the other X 2• Suppose that K(z,t) is a solution of the Helmholtz equation V 2K + k 2K = 0, expressed in terms of the coordinates (z = ~I, t = ~2, to obviate subscripts) in which the solutions X I, X 2 separate. For instance, K can be eikx , and the coordinates for separation can be polar coordinates (z = In r, t = q,); then K = eikr co. ', Then K satisfies Eq. (5.3.85)
a2K az 2
+k
2g
l(z)K = -
a2K at2
-
k 2g 2(t)K
If now the separated function XI(z) is to satisfy the equation £.(X I ) = (d2Xtldz 2 )
+- k
2(gl
- ( 2)X I = 0
§5.3]
Integral Representations
then the integral representation X 1 (z) £.(K)
=
637
fK(z,t)v(t) dt can be used.
= mlt(K) = - (o2Kjot 2) - k[g2(t)
For
+ a 2]K
and because of the absence of the first derivative term and the fact that the second derivative has no factor in t, the adjoint operator fir t = mlt • Therefore the equation for v, (d 2vjdt2) + k 2[g2(t) + a 2]v = 0 is just the equation satisfied by the separated factor X 2, which goes along with XI to form the factored solution of the Helmholtz equation. What we have just demonstrated is that, if K(z,t) is any solution of k 2)K, expressed in terms of separable coordinates z and t, and if a (V2 separated solution in the coordinate tis X 2 (t) , a separated solution in the coordinate z is given by the representation
+
(5.3.86) and X I(Z)X 2(t) is a separated solution of the equation (V2 + k 2)X IX 2 = 0 in the coordinates z, t. If X 2 is a simpler function than XI, we have obtained an integral representation of a complicated function in terms of simpler ones (if K is sufficiently simple). As a first example of this we can take the polar coordinates ~1 = In r, and ~2 = ¢, where Eq. (5.3.85) takes the form [see Eq. (5.1.13)]
o~
o~i
+ o~ + k2e2~ .. 0 ~~
t,
."
=0
A solution of (V2 + k 2)K = 0 is K = eik % = exp (ike~1 cos ¢) = ei kr oo•• and a solution of the equation for the ¢ factor is cos(a¢) (where a is the separation constant) . Therefore a solution of the equation for the r factor is and from Eq. (5.3.65), we see that this is, indeed, an integral representation of the Bessel function J a(kr) if the integration is between 0 and 211". A complete separated solution for the separation constant a is Ja(kr) cos(a¢) . We could, of course, have used any of the other solutions for X 2 , such as sin(a¢) or ei a , etc ., to obtain other representations of Bessel or Neumann functions. Or we could have used other wave solutions for the kernel. Or we could have turned the factors around and expressed the trigonometric functions in terms of an integral of Bessel functions of the sort
638
Ordinary Differential Equations
[CR. 5
which turns out, if the integration is from zero to infinity, to be proportional to eia• • To express a simple exponential in terms of a Bessel integral, however, is somewhat of a work of supererogation. Moving on to the Mathieu functions, we can use the coordinates iii = ~l - 13, ep = ~2 given in Eq. (5.1.16) where x = d cos iJ cos ep, Y = id sin iJ sin ep. We again use the kernel ei kz = ei h O O. This solution is of some interest, for the wave equation for parabolic coordinates, which can be written as
F"
+ ~Z F' + ('Y 2 -
has a solution F = Ae-/lz2I 2
Finally, the equation (Z2 -
1),p" +
(z -
2 11
Z2
fJ2Z2)
-
fo" t-Y'!2{Je-
n ,p'
t ' / 4{JJ
F=0
,(tz) dt
+ ( - J.! 2 + ~) ,p
(5.3.99)
= 0
which has three regular singular points, at z = 0 and irregular point at infinity, also has a solution of the form
±], and one
,p(z) = f J ,(zt)v(t) dt In this case the operator m1:(J) is 2J
m1:(J) = t 2 d dt
+ t dJ + (t2 dt
J.!2)J
and the adjoint equation m't(v) = t2
[~:~ + ~ ~~ + (1
_ 1) v] = 0 J.!2 ;;
is related to the Bessel equation again, having a solution (l jt)J,,(t). The bilinear concomitant is
and as long as Re
J.!
+ Re
II
> -1, we can have that
This quantity is known to have a dis continuity at z = 1.
646
Ordinary Differential Equations
[CR. 5
There are a number of other integral representations which are of some value in quite special cases but are of little use on any other equation . Such solutions usually have to be found by trial and error or by "hunch." In any case there is little point in our spending time cataloguing them or in trying to give recipes for when to use which. We have covered the field of the most useful transforms, and the others can be discovered in the literature.
Problems for Chapter 5 6.1 Set up the Helmholtz equation in conical coordinates and separate it . What are the shapes of the coordinate surfaces? For what physical problems would the equation be useful? 6.2 Set up the Laplace equation in bispherical coordinates and separate it . Show that the constant k~ of Eq. (5.1.47) is t and that R = (x2 + y2)i. 6.3 Set up the Schroedinger equation for an electron in a diatomic molecule in the prolate spheroidal coordinates ~
= (rl
+ r2) /a;
71
= (rl - r2)/a;
I{'
= tan-l(y/x)
where rl is the distance from one nucleus and r2 the distance from the other and the nuclei are supposed to lie at the points z = ± ia, x = y = O. Express z, y, z in terms of ~, 71, I{' , obtain the scale factors ; set up the Schroedinger equation and the Stackel determinant. Show that for the potential fun ction - (CI/rl) - (C 2/r2) the Schroedinger equation separates. Obtain the separated equations. 6.4 The exponential coordinates of Prob. 1.9 are ~ = In (x 2 + y2) - z; 71 = i(x 2 + y2) + z; I{' = tan-l(y/x) Sketch the surfaces, find the scale factors, set up the wave equation, and show that the equation does not separate. 6.6 Hyperboloidal coordinates are defined by the equations X4 = Z2(X 2 + y2); J.1.2 = i(Z2 - x 2 _ y2); I{' = tan-l(y/x) Sketch some of the coordinat e surfaces, compute the scale factors, set up the wave equation, and show that it does not separate. 6.6 Rotational coordinates are characterized by having an axis of rotational symmetry (the x axis, for example), the coordinates being X(r,x), J.1.(r,x), and I{' = tan-1(z/y) (r 2 = y2 + Z2). The I{' factor, when I{' is unobstructed, is sin(ml{') or cos(ml{'), and if we set the solution .y,. of the three-dimensional Laplace equation equal to e±im.pl{'(r,x)/r, the equation for I{' is
Problems
CH.5]
iJ2rp iJx2
iJ2rp
+ iJr2 +
(i--r2- m 2) rp
647 = 0
Investigate the separability of this equation, for the coordinates Xand p., as follows: Set z = x + ir and w = X + ip., so that z is a function of w or vice versa. Use the techniques of Eqs. (5.1.6) et seq. to show that the requirement that Iz'12/r2 = -4z'z'/(z - Z)2 be equal to f(X) + g(p.) gives rise to the equation where
F + 2zG + Z2 H = F + 2zG + Z2 H F = (l/z')[z"'z2 + 6(Z')3 - 6z llz'z] G = [3z" - (Z"'Z/Z')]; H = z"'/z'
and z' = dz/dw, etc. Show that the most general solution of this equation is for d 2F/dz2 = c.: d 2G/dz2 = C2; d 2H/dz 2 = C3 which eventually results in the solution (dz/dw)2 = ao
+ alZ + a2z 2 + aaZ 3 + a4Z
4
Solve this equation for the various different sorts of coordinate systems z(w) which allow separation of the X, p. part of the Laplace equation : iJ2rp iJX2
2) + iJp.2 iJ2rp + \dZ\2 (i - m = dw r2 rp
0
Show that the case z = w is the usual cylindrical coordinate system; (z') 2 = Z, Z = iw2 is the parabolic system; z' = z, z = e" the spherical system; (Z')2 = Z2 ± 1 the two spheroidal systems; and z' = 1 ± Z2 the bispherical and the toroidal coordinate systems. Sketch the coordinate systems corresponding to (Z')2 = Z3 and (z') = Z2. 6.7 For the rotational coordinates discussed in Prob. 5.6 discuss the case (Z')2 = a(l - z2)(1 - k 2z2), which results in z = a sn(w,k) [see Eq. (4.5.74)]. Sketch on the z = x + ir plane, for k = 0.6, enough of the coordinate lines X = constant, p. = constant to indicate the nature of the system. Set up the X, p. part of the Laplace equation and separate it. For what physical problem would this coordinate system be useful? 6.8 Analyze the rotational coordinate system (see Prob. 5.6) corresponding to the relation z = a cn(w,k) (z = z + ir ; w = X + ip.). Sketch the shape of the coordinate lines, X, p. = constant on the z plane. Separate the Laplace equation in these coordinates. For what physical situation would this system be useful? [See Eq. (4.5.77) for definition of en]. 6.9 Analyze the rotational coordinate system (see Prob. 5.6) corresponding to the relation z = a dn(w ,k) (z = x + ir; w = X + ip.) [see Eq . (4.5.77)]. Sketch the coordinate system, separate the Laplace
648
Ordinary Differential Equations
[CR. 5
equation, and indicate the physical situation for which this system would be appropriate. 6.10 Set up the Laplace equation in the polar coordinates r, cP, and separate it. Find the basic fundamental set of solutions of these two equations at the points r = a, cP = O. 6.11 The Schroedinger equation for an electron in the one-dimensional potential field V = (h2/ 2M )x 2 is v/' + (k - x 2)1/1 = 0, where k = 2MW/ h2 • One solution of this equation, for k = 1, is 1/11 = exp( -ix 2) . Find the basic fundamental set of solutions at x = O. 6.12 Find the general solution of (d 2if;/dx 2) - (6/x 2)if; = x In x 6.13
A solution of the Legendre equation (1 - x 2)if;" - 2xif;' + 2if; = 0
is if; = x. Find the basic fundamental set at x = O. What is the Wronskian for this set? What solution has the value 2 at x = I? What solutions have the value 2 at x = O? Why is the answer unique in one case and not in the other? 5.14 The Lame equation is
.1." + [ Z2 'Y
J' +
Z + z _ a2 Z2 _ b2 if;
k - m(m + 1)z2 .t• 0 (Z2 -a2)(z2 _ b2) 'Y =
Locate the singular points of this equation, and give the indices of the solutions at each point. What is the basic fundamental set of solutions about z = O? What is the Wronskian for this set? 6.16 Show that the only singular point of if;" - 2azif;'
+ [E + 2bcz -
(a 2 - b2)Z2]if;
=
0
is the irregular point at z ~ 00. Show that, by setting if; = exp (az + (3z2)F(Z) and adjusting the values of a and {3, an equation for F will be obtained which will allow a series expansion of F about z ~ 00 . Write out three. terms of this series. Write down an equation which will have only an irregular singular point of infinity, for which the solution will be if; = exp (az (3z2 'YZ3)G(z) where G(z) is a series of the sort aoz' + alz·+1 + . . ' . Compare this equation with the above and with if;" + k2if; = 0, which also has an irregular singular point at z ~ 00. What can you suggest concerning the classification of irregular singular points? 6.16 In Eq. (5.2.26), with one regular singular point, the exponents of the two solutions about this point add up to -1. For the equation with two regular singular points (5.2.28) the sum of the exponents of the solution about one point plus the sum of those for the solutions about the other (X p. - X - p.) add up to zero. What is the corresponding
+
+
+
649
Problems
CH.5]
statement concerning Eq. (5.2.36) for three regular singular points? What is the corresponding equation for four regular singular points and no irregular points, and what is the corresponding statement? By induction, what is the sum of the exponents about all singular points for an equation with N regular points and no irregular points? 6.17 Show that a solution of
1//'
+ {-He - 1)2 + lab - tea
is
+b-
!e(a e
+ b - e+ 1)]
+ l)(a + b -
C~-: .) C~-: .Y}
e - 1)
if;
=0
if; = [1 - e-z]i(a+b-C+l)e-Hc+l)zF(a,ble!e-z)
For what values of a, b, and e is the hypergeometric series a finite polynomial and is if; finite in the range 0 ~ x ~ oo ? 6.18 Set up the Schroedinger equation in one dimension for a particle of mass M in a potential field - (h 2A212M ) sech 2(xld) . Changing the independent variable into z = i + i tanh(xld), show that the resulting equation in z has three regular singular points. Express the solutions of this equation in terms of hypergeometric functions of z. What solution stays finite when x ~ - 00 ? Find the values of the energy for which this solution is a finite polynomial in z. Is this solution finite at z ~ oo? 6.19 Show that the equation for if;(x) = [z;ce-izlVZ'] F(alelz), where z is a fun ction of x, is if;"
(z") -
(z')
2
+ {-Hz')2 -
i-?
2
e(e - 1) Z
+ (0 -
(Z')2 a) -z-
.
+ i (z~:')}
if; = 0
where z' = dzldx, etc . Show that the equation for if;(x) = zi c(l - z)i;(z)
=
imf(n + m + 1) 1I"f(n + 1)
10t: [z + (Z2 -
By
I)! cos ",,]n cos (m",,) d""
Show that this function is equal to the one defined in Eq. (5.3.36) . Show that these functions are involved in all potential and wave problems in spherical coordinates. 6.33 By use of Eqs. (5.3.33) et seq., show that, for n an integer, Qn(Z) = jPn(z) In [(1
+ z) j(1
- z)] - Wn_1(z)
where Wn-1(z) is a polynomial in z of degree n - 1. when x is a real quantity between -1 and + 1, lim [Qn(X
......0
6.34
+ iE)
- Qn(X - iE)]
Thus show that,
= riPn(X)
Use Cauchy's theorem to show that
where contour C1 is around w = ± 1 but is inside w = z and contour C2 is a circle of radius R» Izl and 1. Show that the integral around C2 is zero for n = 0 or a positive integer. Reduce the contour C1 to a circuit close to the line between ± 1, and by use of the second result of Prob. 5.33, show that
CH.
Problems
5] 6.36
653
Prove that polynomial Wn-1(z) of Prob. 5.33 is
2n - 1 Wn-1(z) = - n - Pn-1(z)
2n - 5
+ 3(n _
1) Pn-a(Z)
2n - 9
+ 5(n _
2) Pn_S(z)
+
6.36 We may define the second solution of the hypergeometric equation about Z = 0 as _ r(c)r(a
o
Y2(a,blclz) -
+b-
r(a)r(b)
c)
F(a, bla
+ b _ c + 111
_
z)
_ r(c)r(c - a - b) (1 _ z)c-a-bF(c - a c - blc - a - b I'{c - a)r(c - b) ,
+ 111 -
z)
Show that this is equal to
s~n 1l"(c [ sm 1l"(c -
a) s~n 1l"(c - b) + s~n 1l"a s~n 1l"bJ F(a blclz) a) sm 1l"(c - b) - sm 1l"a sm 1l"b '
+[
21l"Z l - C sin 1l"C r(c) r(c - 1) J . sin 1l"(c - a) sin 1l"(c - b) - sin 1l"a sin 1l"b F(a - c + 1, b - c + 112 - clz) r(a)r(b)r(c - a)r(c - b)
Show that the limiting form of this function, as c ~ 1, is the series given on page 668. 6.37 Prove that Jm(z) = -1.n ~2r ei o Go. " cos(mu) du 21l"t 0 6.38 Find the asymptotic series for the Whittaker function U 2 (alclz). From it, by the use of Eq. (5.3.3), obtain the relation U 2(alclz) =
eiraz~a 2m
fi'" -i'"
r(t
+ a - c + l)r(t + a) I'( -t) z-' dt I'(e - c
+
l)r(a)
What is the exact path of the contour? Duplicating the procedure used to obtain Eq. (5.3.5), obtain Eq. (5.3.58) (discuss the limits of convergence at each step). 6.39 Show that the radial factor for solutions of the Helmholtz equation in polar, spherical, and conical coordinates satisfies the Bessel equation
~ ~ (z ~:) + (1 - ~:) J
=
0
Locate and describe the singular points; work out the first three terms in the series expansion for solution J >., regular at z = 0, about each. Use the Laplace transform to derive the integral representation of Eq. (5.3.53). Show that a second solution of this equation is N>.(z) = cot(1l"X) J>.(z) - csc(1l"X) J_>.(z)
654
[CR. 5
Ordinary Differential Equations
Compute the first three terms in the series expansion of N x (for X not an integer) about z = and the first three terms in ita asymptotic expansion. Show that, for X = 0,
°
[3:...
No(z) = .!.1im Jx(z) - (-l)x3:...J_x(z)] 7r x-o dX dX m
~
=
~ [In (tz) + ~]Jo(z) - ~
L \~~j; m=l
(tz)2m
[L 0)] 8=1
6.40 Show that a solution of the Schroedinger equation in parabolic coordinates x = ~ cos /p, y = ~ sin /P, z = t(X - J.l), r = tex + J.l) for a particle of mass M in a potential V = T/2/r is if;
= N eim'l'(XJ.l) !me-!ik('A+I') .
+
. F(tm + t - O'im llikX)F(tm + t - rim + llikJ.l) where 0' + r = -i(MT/ 2/h 2k) and where k 2 = (2ME/h 2) = (Mv /h)2. Show that, for m = 0, a = -t, and N = r[l - i(T/2/hv)]e ("~'/2hv), the
solution has an asymptotic form if; --+ exp [ikz - i(T/2/hv) In k(r - z)]
+
2 {T/ exp
[i(T/2/;~2~~ ~ ~ z/r)
- 2iO]} exp [ikr
+ (~:2) In (kr)]
where I'[I - i(T/2/hv)] = Irlei 6• Discuss the physical significance of this result and obtain the Rutherford scattering law. 6.41 In a manner analogous to that indicated in the text for Je2m(h, cosh J.l), prove the series expansions for the "radial" Mathieu function J e2m+l :
L ~
J e2m+l(h, cosh J.l) =
~
(1)n-mB2n+1J2n+1(h cosh J.l)
n-O
v;: L ~
=
(1)n-mB 2n+1[Jn(the-l')Jn+l(thel') - I n+1(t he- I')J n(t hel')]
n=O
where B 2n+l are the coefficients of the Fourier series for the "angle" function Se2m+l(h, cos 11) as defined on page 565. 6.42 By the use of the Laplace transform, show that, if u(s) = fa ~ e-B1K(t) dt and v(s) = fa ~ e-81/p(t) dt, then f(x) = JX K(x - t)/p(t) dt where u(s)v(s) =
fa ~ e-B1f(t) dt. n
~x J m(X _
From this, prove that
t) J net) (~t) = J m+n(X)
CR.
Separable Coordinates
5]
655
Table of Separable Coordinates in Three Dimensions The coordinate system is defined by the interrelations between .t he rectangular coordinates x, y, z and the curvilinear coordinates ~1, ~2, ~3 or by the scale factors h n = V(OX/O~n)2 + (OY/O~n)2 + (OZ/0~n)2, etc ., having the property (see page 24)
d8 2 = dx 2 + d y2 + dz 2 =
Lh~(d~n)2 n
The expressions for the Laplacian, gradient, curl , etc., in terms of the h's are given in the table on page 115. The standard partial differential equation V2if; + kiif; = 0 becomes
~
_1_
3 oif; ] + k2if; = 0 .s. [h 1h2h h;" es; }
~ h 1h2h3 O~m m
where k~ = 0 for the Laplace equation, k~ = constant for the wave equation, and k~ = E} - V(~) for the Schroedinger equation for a single particle in a potential field V. For separation the quantity h}h2h3/h~ must factor as follows: h}h2h3/h~
g}a2,~3)h(h);
=
etc .
The Stackel determinant is 12(h) 22(b)
13(h)
S = 21(~2)
ll(h) 31(~3)
32(~ 3)
33(W
23(~2)
The first minor of S for the element m}(~m) is related to the scale factors by the equation M m = OS/Om1 = S/h;" where M 1(~2, ~3) = 2233 - 2332;
Therefore
M 2(~} ,~3.) = 1332 - 1233 ; M 3(h,~2) = 1223 - 2213
Lmhi~m) ~ L =
m
and also
mnMm = On}
m
gl(~2,~3) = h}h2~3 = hh}
(M1) (Sld?!3) S h
= M 1(~2,~3)/2(~2)f3(b); etc. The standard partial differential equation then becomes
L~m[L ~m (im :tJJ + 0
'"
kiif;
=
0
656
Ordinary Differential Equations
and the three separated equations, for the wave equation are , '" = XI(~I)X2(~2)X3(~3),
[cH.5 (k~
= constant)
fm(~m) d~m [fm(~m) ~~: ] + ~ 'Pmn(~m)k~Xm = 0 n
where k~ and k~ are the separation constants. The partial differential equation is obtained by multiplying the m = 1 equation by (M t/8)X 2X 3 , and so on, for m = 2 and 3, and then summing over m. For the Schroedinger equation to separate the potential V must have the form V =
~ V~fm)
=
m
where case,
Vm
is a function of
~m
~ (~m) Vm(~m) m
alone . The separated equations are, in this
Ld~m [fm ~~:] + [~'PmnEn -
vm]
x; = 0
n
where E2 and E3 are the separation constants. The following table lists the scale factors h-«, the related functions I-, and the Staekel determinant for the 11 different separable three-dimensional coordinates for the wave equation. The singular points of the three separated equations in their canonical forms are also given . In the cases where alternate scales for the coordinates are sometimes used, these alternate expressions are also given. The general forms for the potential function V for separation of the Schroedinger equation are also given. I Rectangular Coordinates x = h; Y = b ; z = ~3 ; hI = h 2 = h 3 = 1; it = f2 = h = 1 1 -1 -1 8=0 1 0=1 o 0 1 Irregular singular point at infinity all three equations. General form for V = u(x) + v(Y) + w(z) II Circular Cylinder Coordinates (rotational) (Fig . 5.15) x = h~2 ; Y = ~l ~~; Z = ~3; hI = h 3 = 1; h 2 = ~l/~~ it = h; h =~; 13 = 1; ~l = r; b = cos ep; ~3 = z 1 -(11m -1 1 8 = 0 1/(1 - W 0 - 1 _ ~~ 001 Equation for ~l , regular singular point at 0, irregular singular point at 00 . Equation for ~2, regular singular points at -1, + 1, 00 . Equation for h irregular singular point at 00 . General form for V = u(r) + (1/r 2)v(ep) + w(z)
ClJ·
5J
657
Separable Coordinates
z
r:.5,z:I.O
~+--;>(--------
Fig. 5.15.
III
Elliptic Cylinder Coordinates Z
y Fig . 5.16.
x
=
hi =
~1~2;
Y
vim -
h = vI~~
d 2)(1 z = ~3 ; d2m/a~ - d 2); h 2 = ~;:-~---d=2~=i'---)/=(1 -
- d2; 12
h = d cosh
Jl
= j(rl
=~ ; fa = 1 + T2) ; ~3 = z; h = cos
page 556) 1 -l /(~i - d 2) S = d2 1/(1 -
o
m;
= vlW -
0
m
-1 -d 2 1
~~ (~~ -
- d2~i d 2)(1 -
'P
m;
h3 = 1
= (1/2d)(rl - r2) (see
m
Equation for b has regular singular points at -d, +d, irregular singular point at 00 . Equation for ~2 has regular singular points at -1 , + 1, irregular singular point at 00 . Equation for ~ 3 has irregular singular point at 00. General form for V = I [u(rl + r2) + v(rl - r2)l!rlr2} + w(z) .
658 IV
Ordinary Differential Equations
[cH.5
Parabolic Cylinder Coordinates Z p.=+2.0 z=o
p.=O,z=O >.=+J.O,z=O
p.=+2.0 z=O
x
p.=O,z=O
x
>'=+ 1.0, z=O
Fig. 6.17.
y
y
x=iW-W ; Y=~1~2; z=h; hi=h2=V~~+~~; ha=l;
/l = f2 = f3 =
o
~~
1 -1
~i 1 = ~r + ~~ 1 -1 0 Equations for h , b, ~3 have irregular singular points at 00. General form for V = {[u(h) + V(~2)l/vx2 + y2} + w(z) V Spherical Coordinates (rotational) Z Z
S = 0
f-----1~~++-J---i--X
JL--H-HH-H----,/L-X 8=90!-O"'\"""~~~~
1:2
I /~i
o 1/m -
= r 1
1)
=
'>
+ z and ~i
m = r - z.
+ '>
1:2 2
1 - ~i
Equations for h , ~2 have regul ar singular points at 0, irregular singular points at co . Equation for ~ 3 has regular singular points at - 1I + 1I co General form for V
=
[u(h)
+ v(b) ] +
V x 2 + y 2 + Z2
w(b ) . x 2 + y2
Separable Coordinates
CH.5]
VIII
661
Prolate Spheroidal Coordinates (rotational)
z
z
1'=1.0
= O'
1'=1.0 8=90'
Fig. 6.21.
x = ~3
_
Vm I~~
-
'\j ~~ _
d 2)(1 - m;
d2~L d2 '
_ h2
hl
-
1
=
~l
= d cosh 1J = i(rl
~~
-
d 2; f2
1 ~~ - d 2 1 d2 1
s=
~~
0
- 1
0
=
-
I~~
y = V(~~
-
d2~L
'\j I _ ~~ '
1 - ~~;
+ r2);
fa ~2
=
-
d 2)(1 - m(1 -
_ h3
VI -
-
1m
'\j
W;
- d 2)(1 (1 - W
z = ~l~2
m
~~;
= cos tJ = (l j2d) (rl
-
r2);
~3
= cos
d2
m- d 2)2 1 m-d2W 2 m-1)2 m-d )(1 - W(l - m 1 m-1)
Equation for h has regular singular points at -d, +d, irregular singular point at 00. Equation for ~2 has regular singular points at -1 and + 1, irregular singular point at 00 . Equation for ~3 has regular singular points at -1, +1, 00.
662
[CR. 5
Ordinary Differential Equations
IX Oblate Spheroidal Coordinates (rotational) Z Z
fL=1.0 4>=90·
Fig. 6.22.
~l
~2
= d sinh J.l ; 1
s=
-d 2
0
= cos t?-;
1 ~~
+d
2
1 ~~ -
0
1
~3
-d 2 W d 2 )2 1
= cos '"
+
m-
1)2
(~~
+ d2~~)
- W + d 2)(1
-
W(l -
m
1 (~i
-
1)
Equation for ~l has regular singular points at -id, +id, irregular singular point at 00 . Equation for ~2 has regular singular points at -1, + 1, irregular singular point at 00. Equation for ~3 has regular singular points at -1, +1 , 00 .
CH .
Separable Coordinates
5]
663
X Ellipsoidal Coordinates
z
z ~~~-
x,=2.5 / xa=I.O x,=2.5 x2=I.O
-f--t+thf'-ttt+--;//---X
X, =2.5 x2=1.5
X,=2.5
x2=1.5
X,=2.5 X2'2.0
Fig. 6.23
m-
~n
=
bx n , a
= 2b.
b2) (a2 - b2) 1
m- b2)( a2 -
b2)
1
Equations for h,
~2, ~3
= Genera1 form for V dn(}.,k)
h = a cn(}.,k);
t: 0
..
r(c) \ ' r(a + n)r(b + n)zn r(a)r(b) '-' r(c + n)n !
o = F(a blclz) =
Yl
,
n=O
(hypergeometric series) F(a,blclz) = F(b ,alclz) = (1 - z)c-a-bF(c - a, c - blclz)
=
(1 - z)-aF (a, c =
bl c] z
~ 1)
(1 - z) -bF (c - a, bl c] z ~ 1) (Re z < i)
F(a,blcll) = [r(c)r(c - a - b)JI[r(c - a)r(c - b)] F(a,bI2blz) = (1 - iz)-aF[ia, ia -lIb -llz2j(2 - Z) 2] F(2a,213Ia 13 ilz) = F(a,l3la 13 il4z - 4z 2) zF(a,blclz) = (c - l)j(a - b)[F(a - 1, bl C - 11 z) - F(a , b - 11 C - 11 z)] F(a,blclz) = 1/(a - b)[aF(a + 1, bl e] z) - bF(a, b + 11 c] z)] (d/dz)F(a,blclz) = (ab/c)F(a 1, b 11 c 11 z)
+ + + +
+ +
+
+
+
+
+
1, b - c 112 - c] z), A second solution about z = 0 is zl-cF(a - c but this is not independent of y~ when c = 1, 2, 3, .. . . An independent solution for all values of Re c ~ 1 is o = G(a blclz)
Y2
,
= (s~n 1r(c - a) s~n 1r(c - b)
+(
s~n 1ra s~n 1rb) F(a blclz) Slll1r(C - a) Slll1r(C - b) - SIll1ra SIll1rb '
+
21rz 1- C sin 1rC r(c)r(c - 1) ) . sin 1r(c - a) sin 1r(c - b) - sin 1ra sin 1rb F(a - c 1, b - c 112 - cl z) r(a)r(b)r(c - a)r(c - b)
+
+
The series can be constructed from the definition of F except when c = 1,2,3, .. . . When c is an integer, by a limiting process we can show that G(a,blllz) = 2 sin~1ra) tn~1r~~ {[In z + 2-y -1r SIll1r a + ~ cot (1ra)
+ if;(a) + if;(b) + ~ cot (1rb)]F (a,bll lz)
n-l
CIO
+
+
\ ' r(a n)r(b n) [ \ ' ( 1 r(a)r(b)[n!]2 '-' a r
+ n=O '-'
r=O
1
2
)J }
+ +b+r - r+1
zn
CR.
5]
Second-order Differential Equations
669
For m = 2, 3, 4, G(a,blmlz)
=
2
sin~1ra) tn:~~ a
{lIn z + v + y;(a) +. y;(b) -
-1rSlll1r
y;(m)
+ -b- cot(1ra) + -b- cot(1rb)]F(a,blmlz) + + n)f(b + n)f(m) . Lt f(a)f(b)f(n + m)n! n=O '" ~ f(a
n-l
.[l: (a ~ r=O
r
+ b~r
-
r
~ 1 - r ~ m)]zn
m-l
_
~ f(a - n)f(b - n)f(n)f(m) (-z)-n}
Lt
f(a)f(b)f(m - n)
n=1
Integral representations for the first solution about z = 0 :
°
_
f(c) Yl(a,blclz) - f(a)f(b)
fi'"
f(a
+ t)f(b + t) + t) f(-t)(-z)
2rif(c
-i'"
I
dt
where the contour goes to the left of the points t = 0, 1, 2, . . . and to the right of the points -a, -a - 1, -a - 2, .. . ,and -b, -b - 1, -b - 2, .. . (a, b cannot be negative integers) . yO(a blclz) =
1,
yO(a blclz)
or
1,
f(c) f(b)f(c - b)
= f(c)(1 -
Jlt : (t -
z)-at«-c(t -
l)c-b-l dt
z)c-a-b ('" (t _ z)a- ct-a(t -
f(c - b)f(b)
Jl
l)b-l dt
when Re c > Re b > 0 and (for the second representation) when b + 1 is not a real number larger than unity. The letters a and b may be interchanged in these integrals to give representations of Yl when Re c > Re a> O. yO(a blclz) = 1
,
=
f(C)Zl-c ~. (z - t)c-b-ltb-l(l - t)-a dt f(b)f(c - b) 0
e:'..b 411"2 f(c)f(1 - b)f(1
+b-
C)ZI-C
Se2m(h, cos 4»
..
+
l
D 2n sin(2n4» } ;
n=l 'Y;m = [ 1
+ l2nD
..
2n
Tl
n
l D 2n+1sin[(2n + 1)4>]} ; n=O 'Y;m+l = [ + l + 2n+lTl .. 'Y2m {4>S0 2m(h, cos 4» + l D 2n cos(2n4»} ; n=O
Fe2m+l(h, cos 4» = 'Y;"'+1 {4>Se2m+1(h, cos 4»
+
1
(2n
1)D
n
F02m(h, cos 4» =
..
F02m+l(h, cos 4» = 'Y2m+l {4>S02m+l(h, cos 4»
+
l D 2n+ cos[(2n + 1)4>]} ; n=O = [l D 2n r 1
'Y2m+l
+1
n
For further details of comput at ion of the coefficients Band D, see pages 565 et seq. For behavior of solutions for complex values of 4>, see pages \ 635 et seq., and also the tables at the end of Chap. 11.
Bibliography Pertinent articles and books related to the problem of the separation of variables: Bocher, M.: " Uber die Reihenentwickelungen der Potentialtheorie," Leipzig , 1894 (dissertation). "I Eisenhart, L. P.: Separable Systems of Staeckel , Ann . Math., 35, 284 (1934).
CH.5]
Bibliography
675
Eis~hart,
L. P.: Separable Systems in Euclidean 3-space, Phys . Rev., 46, 427 (1934). Eisenhart, L. P.: Potentials for Which Schroedinger Equations Are Separable, 'Phys. Rev., 74, 87 (1948). Michel: Exhaustion of Neumann's Mode of Solution for the Motion of Solids of Revolution, etc., Messenger of Mathematics, 19,83 (1890). Redheffer, R. M.: " Separation of Laplace's Equation," Massachusetts Institute of Technology, Cambridge, 1948 (dissertation). Robertson, H. P. : Bemerkung uber separierbare Systeme in der Wellenmechanik, Math. Ann., 98, 749 (1927). Additional material on the solution of ordinary differential equations : Bateman, H .: "Partial Differential Equations of Mathematical Physics," Cambridge, New York , 1932. Forsyth, A. R. : " Theory of Differential Equations," Vol. 4, Cambridge, New York ,1890. Ince, E. L.: " Ordinary Differential Equations," Longmans, New York , 1927, reprint Dover, New York , 1945. Riemann-Weber, "Differential- und Integralgleichungen der Mechanik und Physik," Vieweg, Brunswick, 1935. Schlesinger, L. : "Theorie der Differentialgleichungen," Goschen, Leipzig, 1922. Whittaker, E. T ., and G. N. Watson : "Modern Analysis," Cambridge, New York, 1927. Works containing further details about the special functions discussed in Secs. 5.2 and 5.3: Gray, A., G. B. Mathews, and T . M. Ma cRobert: "Treatise on Bessel Functions," Macmillan, London , 1922. Hobson, E. W. : " T heory of Spherical and Ellipsoidal Harmonics," Cambridge, New York, 1931. Klein, F.: "Vorlesungen uber die Hypergeometrische Funktion," Springer, Berlin , 1933. MacRobert, T . M. : "Spherical Harmonics," Methuen, London, 1927, reprint, Dover, New York, 1948. McLachlan, N. W. : " Bessel Functions for Engineers," Oxford, New York, 1934. McLachlan, N. W. : "Theory and Application of Mathieu Functions," Oxford, New York, 1947. Stratton, J. A., P. M. Morse, L. J. Chu, and R. A. Hutner: " Elliptic Cylinder and Spheroidal Wave Functions," Wiley, New York, 1941. Strutt, M. J . 0 .: "Lamesche, Mathieusche und verwandte Funktionen," Springer, Berlin, 1932, reprint Edwards Bros., Inc ., Ann Arbor , Mich. Watson, G. N .: "Treatise on the Theory of Bessel Functions," Cambridge, New York, 1944. Books containing tabulations of formulas relating the functions of interest, supplementing the tables at the ends of the chapters in th e present work : Jahnke, E., and F. Emde: "Tables of Functions," B. G. Teubner, Leipzig, 1933, reprint Dover, New York , 1945. Madelung, E. : "Mafhematlschen Hilfsmittel des Physikers, " Springer, Berlin , 1936, reprint Dover, New York , 1943. Magnus, W., and F. Oberhettinger : "Special Functions of Mathematical Physics," Springer, Berlin, 1943, reprint, Chelsea, New York , 1949.
CHAPTER
6
Boundary Conditions and Eigenfunctions
We have now discussed the methods of solving the ordinary differential equations which will come up in the solution of many problems in the behavior of continuous media. As we have seen, a problem is not uniquely specified if we simply give the differential equation which the solution must satisfy, for there are an infinite number of solutions of every equation of the type we have studied. In order to make the problem a definite one, with a unique answer, we must pick, out of the mass of possible solutions, the one which has certain definite properties along definite boundary surfaces. Any physical problem must state not only the differential equation which is to be solved but also the boundary conditions which the solution must satisfy. The satisfying of the boundary conditions is often as difficult a task as the solving of the differential equation. The first fact which we must notice is that we cannot try to make the solutions of a given equation satisfy any sort of boundary conditions ; we should not try to "squeeze a right-hand foot into a left-hand shoe," so to speak. For each type of equation which we have discussed in Chap. 2, there is a definite set of boundary conditions which will give unique answers, and any other sort of conditions will give nonunique or impossible answers. Now, of course, an actual physical problem will always have the right sort of boundary conditions to give it a unique answer (or, at least, so we all hope!) , and if we make our statement of the problem correspond to the actualities, we shall always have the right boundary conditions for the equations. But it is not always easy to tell just what boundary conditions correspond to " act uality," and it is well for us to know what conditions are suitable for what equations so we can be guided in making our mathematical problems fit the physical problems as closely as possible.
6.1 Types of Equations and of Boundary Conditions Let us first discuss a two-dimensional example in order to bring out the concepts without confusing by complexity. All the two-dimensional 676
§6.1]
677
Types of Equations and of Boundary Conditions
partial differential equations for scalar fields, which we discussed in Chaps. 2 and 3, and many of the equations for components of vector fields have the general form a;p A (x ,y) ax 2
2if;
+ 2B(x,y) axa;pay + C(x,y) aa y
2
= F
(
aif; aif;) x,y,if;, ax' ay
(6.1.1)
where, if the equation is linear in if;, F has the form D(x,y) ::
+ E(x,y)
:t +
G(x,y)if;
+ H(x,y)
This is, of course, the most general linear partial differential equation in the two variables, x and y. These two coordinates may be either two space coordinates or one space coordinat e plus time.
Boundary Boundary Fig. 6.1 Boundary conditions in two dimensions. Surface z = 4(x,y), boundary curve z = ~(s), y = ,,(s), unit vectors a, and an in x, y plane ; vector a tangent to surface at boundary.
There is a nondenumerable infinity of solutions of this equation; the additional conditions imposed by the problem, which serve to fix on one particular solution as being appropriate, are called boundary conditions. Usually they take the form of the specification of the behavior of the solution on or near some boundary line (or surface, in three dimensions). (From this point of view, initial conditions are just boundary conditions in time.) It naturally is of interest to see what sort of curves these boundary curves may be and what sort of specification of the field along the line there must be in order that a unique answer result. For a two-dimensional problem the solution if;(x,y) may be represented by the surface z = if;(x,y) . The boundary is a specified curve on the (x,y) plane, not the edge of the surface z = if;(x,y) which is above the boundary curve. The boundary conditions are then represented by the height of the if; surface above the boundary curve and/or the slope of the if; surface normal to the boundary curve (see Fig . 6.1). The edge of the if; surface just above the boundary curve (which is in general not a plane curve) is sometimes called the supporting curve for the boundary. If the distance along the boundary from some origin is s and the parametric equations for the boundary curve are x = Hs) , y = 7/(s),
678
Boundary Conditions and Eigenfunctions
[CH. 6
then the equation for the supporting curve is z = if;(~"T/) = if;(s). The unit vector tangent to the boundary at the point s on the boundary is at = i(d~/ds) + j(d"1/ds) , and the unit vector normal to the curve is an = at X k = [-j(d~/ds) + i(d"1/ds)] . The expressions for these vectors are particularly simple because we have said that s is the distance along the boundary curve, so that V(dUds)2 (d"1/ds)2 = 1 (why?) . Since an is an axial vector, having a choice of directions, we choose axes and directions (for this section, not for Chap. 7) so that an points inwa rd (on the side of the boundary which contains the solution). In terms of these vectors and derivatives the gradient of if; normal to the boundary at s is an. grad if; = aif; d"1 _ aif; a~ = N(s) ax ds ay ds
+
where aif;/ay and aif;/ax are taken at the points x = Hs), y = 1](8). In terms of these definitions we can now visualize the various types of boundary conditions. Types of Boundary Conditions. In every case, of course, we must specify the shape of the boundary. It may be a closed curve for the Laplace equation in two space dimensions, or it may be an open, If-shaped boundary consisting of a line parallel to the space axis and two lines parallel to the time axis for a string (wave equation in time and one space dimension) fixed at the ends and given a specified start at a given time, and so on. As mentioned on page 690, the boundary is closed if it completely surrounds the solution (even if part of the boundary is at infinity); it is open if the boundary goes to infinity and no boundary conditions are imposed along the part at infinity. In a one-dimensional case, the solution of a second-order equation is uniquely spec ified if we specify its initial value and slope. By analogy we might expect that, if the boundary were parallel to one of the axes, specification of the value of if; along the boundary [i.e., specifying if;(8)] and of the gradient of if; normal to the boundary [i.e., specifying N(s), in this case aif;/ay] will uniquely fix the solution. This is correct, as will be shown later, but it is too spe cial a case to satisfy us. We should sometimes like to have a boundary which is not contiguous with a coordinate line but is free to take any, reasonable shape. It is not quite so obvious that spe cification of value and normal gradient on a boundary of any shape will give a unique result (nor is it true!), and we shall have to determine the sort of boundaries which are satisfactory. The type of boundary condition mentioned in the last paragraph, the specifying of value and normal gradient, is called the Cauchy boundary condition, and the problem of determining the shape of boundary and type of equation which yields unique and reasonable solutions for Cauchy conditions is called the Cauchy problem, after the investigator who first studied it in detail. Specifying the initial shape and velo city of an
§6.1]
679
Types of Equations and of Boundary Conditions
infinite flexible string corresponds to Cauchy conditions along the line t = constant. As we know, this uniquely specifies the solution. On the other hand if the solution is to be set up inside a closed boundary, it might be expected that Cauchy conditions are too much requirement and might rule out all solutions. Perhaps one only needs to specify the value if;(s) alone or the normal gradient N(s) alone along the boundary in order to obtain a unique answer. The specifying only of values along the boundary is called Dirichlet conditions, and the specifying only of slopes is called Neumann conditions . A potential problem, such as the determination of electric potential inside a sequence of conductors at specified potentials, corresponds to Dirichlet conditions. On the other hand the determination of velocity potential around solid bodies, where the fluid must flow tangential to the surface of the solids and the normal potential gradient at the surface is zero, corresponds to Neumann conditions. Alternatively we may, at times, need to specify the value of some linear .combination of value and slope, a single boundary condition which is intermediate between Dirichlet and Neumann conditions. In terms of our supporting line in Fig . 6.1, Cau chy conditions correspond to our specifying not only the line if;(s) = z but also the normal slope at the edge of the surface if;(x,Y) = z. It is as though, instead of a line, we had a thin ribbon as a support to the if; surface, a twisted ribbon which specified slope perpendicular to its axis as well as height above the z axis (but not higher derivatives). For Diri chlet conditions the supporting line is really a line, not a ribbon. For Neumann conditions the ribbon is free to move up and down, only the "slant" of the ribbon is fixed. Sometimes these two conditions are homogeneous, when exif;(s) + (3N(s) = 0, for ex, (3 specified but independent of s, and sometimes the conditions are inhomogeneous , when exif;(s) + (3N(s) = F(s) . The distinction will be of interest in Sec. 6.3. But we now must go back to our general equation (6.1.1) and see under what circumstances Cauchy conditions along the curve x = Hs), Y = 1](s) will result in a unique solution. Cauchy's Problem and Characteristic Curves. In order to compute if; at some distance away from the boundary we can have recourse to some two-dimensional power series, analogous to Taylor's series : if;(x,y)
=
if;(~,1]) + [ (z
-
~) ~~ + (y -
+ i [(X - ~)2 a'lif; + 2(x ax 2
_
~)(y
1])
~t]
_ 1]) a'lif; ax ay
+ (y
2if;]
_ 1])2 a ay 2
+ (6.1.2)
where if; and all its derivatives on the right-hand side of the equation are evaluated at the boundary point (~,1]). Once these partial derivatives
680
Boundary Conditions and Eigenfunctions
[CH.
6
of 1/; are all evaluated at the boundary, then 1/; is uniquely specified within the radius of convergence of the series, i.e., over all of a strip contiguous to the boundary line, which may be infinite in width depending on the nature of the equation. If we can work out a recipe for computing the partial derivatives, we shall have the Cauchy problem well along toward solution. This is not so straightforward as it may at first seem, for we are given only the equation for 1/;, the parametric equations for the boundary, and the values of 1/;(s) and N(s) on the boundary and from these data are to compute all the double infinity of values of the partial derivatives for each point (~,"1) on the boundary. It is not too difficult to express the first derivatives in terms of known quantities. There are two of them, and there are two equations, one giving the specified normal gradient N(s) and the other the rate of change of the known value 1/;(s) along the boundary:
(~~) (:~) - (~;) (:t) = an· grad 1/; ; = (~;) (~~) + (~~) (:t) = at · grad 1/;;
N(s) = %s 1/;(s)
Since the determinant of the coefficients always a solution for these equations:
(d~/ds)2
+
at x =
~,
y = "1
at x =
~,
y = "1
(d"1/ds)2 = 1, there is
(:~)E.n = N(s) (~~) + (~;) (~:) = pes); (*)E,n = (~~) (~:) - (~;) N(s)
=
q(s)
(6.1.3)
But the next step, to obtain the second derivatives, is not so simple. It is also the crucial step, for if we can find the three second partials, we shall find that solving for the higher derivatives is simply" more of the same ." Now that we have solved for the first derivatives, we know p and q, given in Eq . (6.1.3) as functions of the parameter s. Two of the needed three equations for the second derivatives are obtained by writing down the expression for the known rate of change of p and q with s in terms of these second derivatives; the third equation is the differential equation 1/; must satisfy, Eq. (6.1.1) itself:
(~;) (:~) + (~~) (a:2~y) = ~~ (~;) (a:2~y) + (~:) (:~) = ~; A(s)
(:~) + 2B(s) (a:2~y) + C(s) (:~~) = F(s)
§6.1]
Types of Equations and of Boundary Conditions
681
where A(s), etc ., are the known values of the coefficients at the point ~(s), 7/(s) on the boundary. These three equations can be solved, to find the three partials, unless the determinant of the coefficients d~
ds A
= 0
d7/ ds
o
d~
~~
ds 2B
A
= C
(~;Y - 2B (~;) (~~) + A (~~y
(6.1.4)
C
is zero. If the determinant A is not zero, all the higher partials can be solved for by successive differentiations of known quantities with respect to s, the distance along the boundary, and the resulting Taylor's series will uniquely specify the resulting solution, within some finite area of convergence. Thus we have shown that Cauchy conditions on a boundary do choose a particular solution unless the boundary is such that the determinant A is zero along i t. The equation A = 0 is the equation of a curve C(X,y)(dX)2 - 2B(x,y) dx dy
+ A(x,y)(dy)2
=0
(6.1.5)
(where we have changed the differentials d~, d7/ into the more familiar dx, dy) or, rather, of two families of curves, for this equation may be factored, giving A dy
= (B
+
yB2 - AC) dx;
A dy
= (B - yB2 - AC) dx
(6.1.6)
These curves are chara cteristic of the partial differential equation (6.1.1) and are called the characteristics of the equation. If the boundary line happens to coincide with one of them, then specifying Cauchy conditions along it will not uniquely specify the solution; if the boundary cuts each curve of each family once, then Cauchy conditions along it will uniquely specify a solution. Hyperbolic Equations. In order to have this statement mean anything physically, the two families of characteristics must be real curves . This means that our statement (as it stands) applies only to those partial differential equations for which B2(X,y) > A(x,y)C(x,y) everywhere . Such equations are called hyperbolic equations. The wave equation
is a hyperbolic equation if t is considered as being the second coordinate y . Equation (2.3.29), for supersonic flow, is also a hyperbolic equation. For hyperbolic equations the natural coordinate system is formed from the two families of characteristics, which are real. Integration of the
682
Boundary Conditions and Eigenfunctions
[CH. 6
first of Eqs. (6.1.6) gives a solution 'A(x,Y) = constant ; integration of the second gives J.L(x,y) = constant; and 'A and J.L are the natural coordinates. Since motion along one of the characteristics 'A = constant corresponds to (iJAjiJx) dx + (iJAjiJy) dy = 0 (the gradient of 'A is perpendicular to the vector i dx + j dy for motion along the characteristic), substituting from this equation back into Eq. (6.1.5) shows that A
(:~Y + 2B (:~) (:~) + c (:~Y = 0
(6.1.7)
with a similar equation for the partials of J.L , for the other family. We now go back to the original equation (6.1.1) and express it in the new coordinates. For instance,
(:~) = (:0) (:~Y + 2 (iJ~2~J.L) (:~) (:~) + (::~) plus terms in iJt/t/iJx and iJt/t/iJy.
(::Y
We finally obtain
:~~ [A (:~Y + 2B (:~) (:i) + c (:~YJ
+2~ [A (iJ'A) (iJJ.L) + B (iJ'A iJJ.L + iJ'A iJJ.L) + C (iJ'A) (iJJ.L)] iJ'A iJJ.L iJx iJx iJx iJy iJy iJx iJy iJy + ::~ [ A
(:~Y + 2B (:~) (:;) + c
(::YJ
=G
(:~, :~, tIt, x, y)
But the first and third expressions in brackets are zero because 'A and J.L are characteristic functions of the equation. If the equation is homogeneous, G can be put into the form a(iJt/t/iJ'A) + b(iJt/t/iJJ.L) + ct/t, and the second bracket expression (which is not zero) and a, b, and c can be made functions of 'A and J.L . We thus arrive at the normal form of the hyp erbolic equation iJ2ift iJ'A iJJ.L
=
P (iJt/t) iJ'A
+ Q (iJt/t) + Rt/t iJJ.L
(6.1.8)
where P, Q, and R are functions of 'A and J.L . If these quantities (P,Q,R) are zero, as they often are (for the wave equation in one space dimension and the equation for supersonic flow, for instance), the solution of Eq. (6.1.8) is (6.1.9)
where f can be any function of 'A and g any function of J.L. For the wave equation, for example, 'A = x - ct and J.L = X + ct, so that t/t = f(x - ct) + g(x + ct), corresponding to waves of arbitrary shape traveling to the right and to the left with velocity c and -c. We shall discuss a case where P and Q are not zero on page 687.
§6.1]
Types of Equations and of Boundary Conditions -,
683
We have thus shown that solutions of at least some hyperbolic equations are similar to traveling waves and that the families of characteristics correspond to the wave fronts. When the normal form of the equation has the particularly simple form (6.1.10)
waves of any shape may be present with fronts along Xand along u, When the boundary crosses both families of characteristics (as in the first of Fig. 6.2), then the Cauchy conditions will uniquely determine both j'(X) and g(p.) . Each point on the boundary, labeled by the distance s .from an origin, corresponds to a given value of X and u, Specifying if;(s) and N(s) at this point gives two simultaneous equations which serve
Boundary
Boundary
Fig. 6.2 Intersection of families of charact erist ics, for a hyperbolic equation, with boundary line .
to determine bothf and g for this pair of values of Xand u , If the boundary crosses all of both families of characteristics, then f and g will be specified for all values of X and p. and the field will be uniquely determined everywhere. When the characteristics are everywhere real, if if;(s) and N(s) are bounded and continuous, then f and g will also be bounded and continuous and so will if;(x,y) . Cauchy Conditions and Hyperbolic Equations. We now can see why Cauchy conditions do not specify the solution when the boundary coincides with a characteristic. If the boundary is along p. = P.o, say, then Cauchy conditions give data concerning only g(p.) and the slope of g(p.) at p. = P.o and nothing at all about the behavior of g for any other u, In this case f(X) is determined, because the line p. = P.o cuts all the family of X characteristics, so the value of if;(s) [which in this case can be written y;(X)] is f(X) . The normal slope N(s) determines dg/dp. at p. = P.o but no higher derivatives can be determined, and so g(p.) , for any other p., is wholly undetermined. In general the values of f and g are determined only for those values of X and p. which are crossed by the boundary line . To put it still another way, a boundary which coincides with a characteristic is a boundary which travels along with a wave front . Since it
684
Boundary Conditions and Eigenfunctions
[CR. 6
never comes in contact with any of the rest of the wave traveling its direction, it can affect only the wave traveling in the opposite direction (i.e., it can determine only f, not g). The close relation between these statements and the discussion on page 168, of shock waves which appear when a fluid flows past a boundary at a speed greater than the speed of sound, should be apparent by now. We also can see now what happens when the boundary curves around so that it crosses a family of characteristics twice, as in the second part of Fig. 6.2. At the point P(At,J.Lt) the boundary is tangent to the chara cteristic J.L = J.Lt ; for all values of J.L > J.Lt the boundary cuts the J.L chara cteristics twice and the characteristics for J.L < J.Lt are not cut at all. Suppose that Cauchy conditions are given on the PA part of the boundary. This determines f(A) for A < At and g(J.L) for J.L > J.Lt. It does not matter that g(J.L) is undetermined for J.L < J.Lt, for these values of J.L are outside the boundary, but we do need to know values of f(A) for A > At. These must be determined by boundary conditions along the PB part of the boundary. If Cauchy conditions [both if;(s) and N(s)] are given along the portion P B, the solution will be "overdetermined," for along this portion g(J.L) is already fixed, being determined by the Cauchy conditions on PA, and only f(A) for A > At needs to be determined by the conditions on PB. This may be done by specifying either if;(s) or N(s) along PB (or a linear combination of if; and N) but not both. Consequently Dirichlet or Neumann conditions (or the intermediate combina tion) are sufficient for PB. Of course, we could also obtain a unique answer by setting Cauchy conditions on PB and Dirichlet or Neumann conditions on PA . In general, we can say that, if the boundary is curved so that it cuts a family of characteristics twice, then Cauchy conditions are needed on the part of the boundary on one side of the point which is tangent to a characteristic and Dirichlet or Neumann conditions on the other side are sufficient. It is not difficult to carry this reasoning on to the case where there is more than one point of tangency. For instance, for aU-shaped boundary, Cauchy conditions are needed along the base of the U and Dirichlet or Neumann conditions are enough along the sides, and for a Z-shaped boundary, Cauchy conditions along the top and bottom of the Z, with Dirichlet or Neumann conditions along the diagonal part, will be proper. It is also not difficult to see that, when the boundary is closed, so that every interior characteristic crosses the boundary twice, Cauchy conditions on any finite part of the boundary overdetermine the solution. It is not too easy to see, however, whether Dirichlet conditions (or Neumann) all around the boundary are sufficient, and we shall have to postpone our discussion of this until later in the chapter. It might be well to discuss a simple case of the above problem in
Types of Equations and of Boundary Conditions
§6.1]
685
order to see how it works out in practice. The case of the flexible string is the simplest.hyperbolic equation, where the displacement I/; of the string depends on x and t according to the equation (see page 124)
The characteristic functions are X = x - ct and J-L = x tion having its normal form a21/;jax aJ-L
+ et, the
equa-
°
=
with solution I/; = I(X) + g(J-L). Specifying initial value and slope of an infinite string corresponds to Cauchy conditions on a simple, open boundary cutting the characteristics but once. If the initial shape, at t = 0, is I/;o(x) [= I/;(B)] and the g ~~
f(x-ct)
/.::::::.;:-- -'q
__ _ _~fltt
'
'"
~' ..........
\,\. ........
-
,,","
"
~,
g[itctl
~f---
Fig. 6.3 Initial shape (solid line) and initial velocity (arrows) of string. Subsequent motion is given by sum of dotted lines f and g, moving in opposite directions.
initial velocity is Vo(x) [= N(B)], we must arrange the functions I and g so that I(x)
+ g(x)
= Vto(x)
and
-f'(x)
+ g'(x)
=
(l je)Vo(x)
where the prime indicates differentiation with respect to the argument. It is not difficult to see that f(X)
= il/;o(X) - 2-1 ~h Vo(w) dw; g(J-L) = #o(J-L) co·
+ -21c 11'0 Vo(w) dw (6.1.11)
The solution is then I/;(x,t) = I(x - ct) + g(x + et), consisting of the sum of two waves traveling in opposite directions with velocity c and -c. This is shown in the first part of Fig. 6.3. If now the string is clamped at x = 0, the boundary is L-shaped and cuts the J-L characteristics at two points. The values of I(X) and g(J-L) for X and J-L larger than zero are fixed by the Cauchy conditions of initial value and velocity along the t = 0, x > part of the boundary, and the value of f(X) for X < is determined by the Dirichlet condition that I/; = for the x = 0, t > part of the boundary. Values of g for J-L < are not fixed by the boundary conditions, nor are they needed .
°
°°
°
°
686
[CR. 6
Boundary Conditions and Eigenfunctions
The procedure for fitting these boundary conditions is to choose a value for f( -X) which is an "image" of g(p.), so that for any value of t the two will just cancel at x = O. For initial shape and velocity Yto(x) and Vo(x) we can see that the proper solution is Yt = f(x - ct) + g(x + ct), where g(p.)
= iYto(p.)
l
1 r" Vo(w) dw; + 2c)0
i YtO(X) -
f(X) =
;c LA 1
Vo(w) dw;
t:
-#o( -X) - 2c )0
p.
>0 X> 0
Vo(w) dw;
(6.1.12) X
a;'An+l or a~ < [O(f) /An+l] But O(f) is a positive quantity, independent of n, and we have proved that An+l approaches infinity as n approaches infinity. Therefore an approaches zero as n approaches infinity, and we have, by this procedure, proved that the series (6.3 .21) is a least-squares fit for the function f in the range a < z < b. Therefore if if;n is a sequence of eigenfunctions, solutions of Eq. (6.3.12) (which can be adjusted so that AO = 0), and satisfying boundary conditions such that [pif;(dif;/dz)]~ vanishes, then the set if;n is a complete set. This proof completes our study of the fundamental characteristics of eigenfunctions. We can now turn to special cases and to the discussion of general techniques which will be useful later. Asymptotic Formulas. It is sometimes useful to obtain approximate expressions for the eigenfunctions and eigenvalues of high order (n large) . We return to the Liouville equation :z (p
~~) + (Ar + q)if; =
0
and make the substitutions y
== Cpr)lif;;
~
= J-11'~r -p de; a
J
= 7r-11b~r -p dz a
to obtain the transformed equation (d2y/d~2)
where
k 2 = J2 A and
+ [k2 -
wW]y = 0
(6.3.22)
(9.)]
w = [_1_.!!!.... (pr)i - J2 (pr)i d~2 r
[cH.6
Boundary Conditions and Eigenfunctions
740
when expressed in terms of the new variable j . The range of the new independent variable is 0 ::; ~ ::; 7r. When A is large, k2 is large compared with w, and we would expect y to approximate a sinusoidal dependence on~. To make this specific, we can rearrange the equation as though it were an inhomogeneous equation : (d 2ylde) + k 2y = wy and solve it as though wy were the inhomogeneous part. Eq. (5.2.19) we see that an expression for y is
y(~)
=
A
sin(k~) + B cos(k~) + ~ ~~ sin[k(~ -
Referring to
t)]w(t)y(t) dt
(6.3.23)
where ~o is some suitable lower limit for integration. This is, of course, an integral equation for y and not a solution. In Chap. 8 we shall study its solution in greater detail. Here, however, we are concerned only with the solutions when A is very large, large enough so that we can neglect the integral compared with the first two terms (if A is large , k is large , and the integrand, times 11k, vanishes) . The values of A and B and of k must be adjusted so that the boundary conditions are satisfied. If neither a nor b is a singular point, the problem of fitting our asymptotic form to the boundary conditions is fairly simple . If the requirements are that Yt be zero at a and b, the first approximation, obtained by neglecting the integral in Eq. (6.3.23), is simply k = n;
Ytn ~ [1/(rp)l]
sin(n~);
An
= (nIJ)2
(6.3.24)
which is valid as long as n 2 is larger than w everywhere in the range o < ~ < n: A better approximation can then be obtained by setting this back into the integral in Eq. (6.3.23) and computing a correction term, but we shall avoid such details for a moment. If the boundary conditions are that 00/; + f3(dYtldz) = 0 at z = a and a similar relation, with a and f3 possibly different constants, at z = b, the procedure is somewhat more complicated. We set, for our first approximation,
Yt ~ [l/(pr)l]
cos(k~
+ 0)
where ~ is a phase angle to be determined from the boundary conditions. Since k2 is to be larger than wW everywhere in the range (0 < ~ < 7r) , we have dYt 1 d~ d dz '" (pr)l dz d~ [cos(k~
+ 0)] =
1
-
(pr)l
k
rr sm(k~ . + 0)
J '\jp
since the derivative of the (pr)-l factor is small compared with the derivative of the cosine factor for large k , The solution for 0 is obtained
741
Eigenfunctions and Their Uses
§6.3] from the equation
cot 8 ~ ~ J
[~a '\jI"£J p
z =-a
If k is large enough, this reduces to 8(k)
~ ~ [~.Jl
evaluated at z = a
Similar fitting at z = b determines the allowed values of k and therefore of A = (kfJ)2: i; ~ n
J [a fPJba; + rn i3 V;:
1
1/;n = (pr)t cos [kn~
+ 8(n)]
(6.3 .25)
where n is an integer. Thus when a and b are ordinary points, the higher eigenvalues of the Liouville equation are approximately equal to the square of 7r times a large integer n divided by the square of the integral of vr/p between a and b.
The corresponding eigenfunction is approximately equal to the cosine of (rn) times the ratio between the integral of VT7P from a to z and the integral of VT7P from a to b [with an additional "amplitude factor" (pr)-t modifying the amplitude]. When a or b, or both, are singular points, where p goes to zero, the integrals defining J and s [given just above Eq . (6.3.22)] may diverge, and we may have to modify our definitions somewhat. In addition the function q may go to infinity at such points ; in any case the function w will become infinite there. We can, of course, solve the Liouville equation near the singular point in terms of a power series and find the nature of the singularities of the two solutions there. If our equation is sensible physically, one of the solutions will be finite there. To see the way things go, let us take a specific example : that of the Bessel equation, resulting from the separation of the wave equation in polar coordinates (see pages 550 and 619),
-d (d1/;) zdz dz
2 )
m + ( AZ-z
1/;=0
(6.3.26)
where z = r = h, where m is an integer (determined by periodic boundary conditions on the ~2 factor) and where A is the separation constant. In this case we have p = z, r = z, q = - (m 2 /z) . Suppose we take the singular pointz = 0 as the limit a. Making the substitution for Eq. (6.3.22) we have
1/;=~Y ; J=~lbdZ=~ ; s=~z;
k 2 = (b2 /7r 2)A
742
Boundary Conditions and Eigenfunctions
(d2y /ds 2)
+ (k2
y(s)
[CB. 6
- (m 2 - t)!S2]y "'" 0 and the integral equation for y is
= cos(ks + 0)
-
k1 }roo f sin[k(s -
t)] y(t)
[2m t-: 1] dt T
where the limits of the integral were chosen to ensure convergence. For large values of k and z, the first term is sufficiently accurate and we need not worry about the integral except when s is near zero. As long as we can find the value of the phase angle 0 which corresponds to the requirement that if; is finite when z = 0, we need never use the asymptotic form for S small (where it is inaccurate) . We find the value of 0 by means of the Stokes' phenomenon (see page 609), by requiring that the change in the asymptotic formula for if;, as S is taken in a circle about s = 0, correspond to the change in the series solution about S = O. Examination of the indicial equation (see page 532) for Eq. (6.3.26) shows that the finite solution at z = 0 has the form zm times a Taylor's series in z. Consequently, if if; is real for z large along the positive real axis (phase angle q, = 0), it is eim'll" times a real function when q, = 11" (z negative and large) , and its leading term should be eim4> times a real function when q, is not an integral multiple of 11" (z complex and large) . Setting if; ~ (1/2 Vz)[eikf+i6 + e-ikf-i6], which is real for q, = 0, we first examine the behavior for q, = ~ (s = ilsl). Here the first term in the brackets is negligible for S large, and the asymptotic formula gives
= eim'll"/2 . real function
if;
,....., (1/2
$\
ViZi) eklfl-i6-lir;
z
=
Izleii'll"
so that 0 = -~(m + -j). To check this we also try q, = (s = -iISl)· Here the second term is negligible and if;
-~
= e- imr / 2 • real function ~
(1/2
ViZi) eklfl+i6+1i'll";
z
=
Izle-F'II"
so again 0 = -~(m + -j), which checks. Therefore the asymptotic form for the eigenfunctions in this case is if;
~ ~z cos [1I"~Z
-
~(m + -j)
l
-~ < tP < ~
which is the asymptotic behavior given in Eq. (5.3.68). If the boundary condition at z = b is that if; = 0, the asymptotic expression for the eigenvalues is k« ~ n
+ -j(m + -j) ;
An = (1I"k n / b) 2; n large
This same technique can be carried through for other cases where one or both limits are singular points. A more complete discussion of
Eigenfunctions and Their Uses
§6.3]
743
asymptotic formulas will be given in Chap. 9. From our discussion here we can say that any of the usual sorts of boundary conditions (whether at singular or ordinary points) can be fitted to the order of approximation suitable for asymptotic formulas by adjusting the phase angle 8 in the expression if;
~ (p~)i cos [ 0 faz ~ dz + 8]
(6.3.27)
Likewise the asymptotic formula for the higher eigenvalues will turn out to be
where the value of ex depends on the specific kind of boundary conditions at both a and b. Comparison with Fourier Series. We see from this discussion that every eigenfunction series behaves like a Fourier series for the higher terms in the series. In fact we can prove that the difference between the first n terms of an eigenfunction series and the first n terms in a Fourier series, covering the same range and for the same function, is uniformly convergent as n is increased indefinitely. To show this we transfer to the new independent variable t and new dependent variable y as defined in Eq. (6.3.22) . The function we wish to represent is F(z) , which is multiplied by (pr)i , and this function, expressed in terms of r, will be called !(t) . No new singularities or discontinuities are introduced in ! by this procedure. The expansion of ! in terms of t is given by the formula [see Eqs. (6.3.17) and (6.3.18)] ., !(t)
=
[p(z)r(z)]t
l
b
F(t)
L~;,.
if;m(z)if;m(t)r(t) dt
m~O
(6.3.28)
where
if;m(z) .= Ym(t) /(pr)i;
t E;"
F(t)
= !(r) /(pr)i
it
j faz ~~~;) dz ; r = j ~~ dz ; J = J"r = fab if;;"r dz ; N;" = y;,,(t) dt = 7l'~~
=
l"
IE. dz ~p .
744
Boundary Conditions and Eigenfunctions
[CB. ,6
But the function f can be expressed in terms of a Fourier series in the range 0 < t < 7r. For instance, we can use the cosine series fm
=
~... f(r)
.
l: ~
cos(mt) cos(mr) dr
(6.3.29)
m=O
where Eo = 1, En = 2 (n > 0) is called the Neumann factor . If the boundary conditions on !f are such that the asymptotic form for !fn is that given by Eq. (6.3.25), then the form of the y terms in the series will approach the form of the cosine terms as n increases. The two series will approach each other, term by term, and eventually the individual terms will be alike, a much more detailed correspondence than one might expect. Many series of eigenfunctions are only conditionally convergent (i .e., converge only because the terms alternate in sign and partially cancel out) . Many of the difficulties attending discussions of the representation of peculiar functions by means of eigenfunctions come about due to this weaker convergence. These difficult cases have been worked out in some detail for Fourier series, so if we can show that what holds for Fourier series also holds for other eigenfunction series, we shall have saved ourselves a great deal of work. If it turns out that the difference between the first n terms of series (6.3.28) and the first n terms of series (6.3.29) is absolutely convergent, as n goes to infinity (i .e., the magnitude of the difference converges), then we shall have established the detailed correspondence we need. For example, according to Eq. (6.3.18) the series
somehow represents the delta function 5(t - r). Such a series is far from being absolutely convergent, but we can show that, if the boundary conditions are those resulting in Eq. (6.3.25), then the function n
(fjJ)
obtaining (rJ2M/dJ.l2)
+ [h2 cosh- J.l
-
b]M = 0; . (d 2cI> /dfjJ2)
+ [b -
h2 cos" fjJ]cI> = 0
758
Boundary Conditions and Eigenfunctions
[CH. 6
where h = kd and k is related to the vibration frequency v = w/27r (for use of the Helmholtz equation corresponds to an assumption that the motion is simple harmonic, with time factor e-i"'l) by the relation k = w/ c = 27rv/c. The constant b is the separation constant, but of course, h = 27rvd/c is also a separation constant, so both equations contain both constants. Referring to Sec. 5.2, we see that the second of these equations is the same as Eq. (5.2.67). The boundary condition for cI> is that it be periodic with .p, and the boundary condition for M is that it be finite and continuous at J.I. = and that it be zero at J.I. = J.l.o, the ellipse corresponding to the boundary. We must simultaneously adjust band h so that both sets of conditions are fulfilled. To do this we solve the equation for .p, assuming that h is known. As shown on page 562, the periodic solutions of Mathieu's equation are expressible as Fourier series. There are two sorts : one set, Sem(h, cos .p) , even in .p, and the other set, SOm(h, cos .p) , odd in .pi corresponding to the first set of eigenfunctions are the eigenvalues bem(h) and to the second set are bom(h). These eigenvalues are "interleaved," so that, for a given value of h,
°
be;
<
bo,
< bel < . . .
bo.; < be;
<
bO m+l
<
Therefore, if h were fixed arbitrarily, the sequence Se, So , forms a complete, orthogonal set of eigenfunctions. But h is not fixed, so all we can say is that we have obtained a relationship between h and a sequence of values of b.. which will allow the boundary conditions on cI> to be satisfied. The solutions for M are those given in Eq. (5.3.90) or (5.3.91), and we must first decide whether J e or N e fits the requirements of continuity at J.I. = 0. Examination of the coordinates themselves shows that (J.I. = 0, .p = a) and (J.I. = 0, .p = -a) are really the same point (x = d cos a, y = 0), and a few minutes cogitation will show that, in order that Vt be continuous in value and slope across the line J.I. = 0, when factor cI> is even in .p, then factor M must have zero slope at J.I. = 0; when factor cI> is odd in .p, then M must go to zero at J.I. = 0. Reference to page 635 shows that the proper solution to go with Sem(h, cos.p) is Jem(h, cos .p) for the same value of h and of m (i.e ., the same value of b) and the proper one to go with SOm is J Om with corresponding hand m. The functions N e, No are therefore not used in this problem. We finally are supposed to set M = at J.I. = J.l.o. For any given value of b, it is possible to adjust h so that this is obtained, but of course, b also depends on h, so hand b must be determined simultaneously. In practice, we would compute a series of curves giving the sequence of values of J.I. for which J em(h, cosh J.I.), for instance, went to zero for each different value of h. These values could be called J.I.':nn(h), the subscript m
°
§6.3]
Eigenfunctions and Their Uses
759
corresponding to that of J em and the subscript n labeling which in the sequence of zero points we mean (J-Lml can be the smallest value, J-Lm2 next, and so on). . We thus have a twofold array of values of J-L, onefold for different values of m and the other for different values of n. Each of these values is a function of h. We now invert the problem by solving for the value of h to make each one of the roots J-Lmn(h) equal to J-Lo, the coordinate of the boundary. The root of the equation J-L:'n(h) = J-Lo will be called hemn, and that of the equation J-L?.n(h) = J-Lo will be called homn. From these values we can then compute the allowed values of frequency p of free vibration of the elliptic membrane. Corresponding to the particular frequency w~n/271" = (c/271"d)he mn we shall have the two-dimensional eigenfunction Sem(hemn, cos fjJ)Jem(he mn, cos hJ-L), and corresponding to the eigenvalue w?"n/271" = (c/271"d)ho mn we have the other eigenfunction SOm(ho mn, cos fjJ)Jom(ho mn, cosh J-L). The complete solution will thus be
L [AmnSemJem cos(w~nt + amn) + BmnSomJom cos(w?.nt + (jmn)]
m.n
with the A's, B's, a's, and {j's determined by the initial displacement and velocity of the membrane. It is apparent, of course, that the factors Se, So are mutually orthogonal if they all correspond to the same value of h. But the terms in the above series are each for a different value of h, so the factors Se; are not all mutually orthogonal, nor are the factors SOm (though all the Se's are still orthogonal to all the So's) . However, the general arguments which were outlined on page 727 show us that our present two-dimensional eigenfunctions are mutually orthogonal, so it must be that the functions Jem(hemn, cosh J-L) must be mutually orthogonal over the range o < J-L < J-Lo and likewise the functions Jom(homn, cosh J-L), for different values of n, whereas the functions Sem(hemn, cos fjJ) must be mutually orthogonal for different values of m. Therefore all terms in the series are orthogonal to all others, after integrating over the area of the membrane, and the coefficients A , B and phase angles a, {j can be determined in the usual manner. By this example we have shown that the eigenfunction technique of fitting boundary conditions can always be used when the equation separates in the coordinat es suitable for the boundary, even though the separation constants may not be separated in the resulting ordinary equations. Such cases require a great deal more computation to obtain a solution than do the cases where one of the separated equations contains only one separation constant, but the calculations are straightforward, and no new principles need to be invoked. Density of Eigenvalues. The rather haphazard distribution of eigenvalues exhibited by even the simple two-dimensional cases brings up a
760
Boundary Conditions and Eigenfunctions
[cH.6
question we have hitherto neglected: Is there anything we can say about the number of eigenvalues between A and A + E? This number is a discontinuous function of A and E (since the eigenvalues are a discrete set), but we might try to compute the average number of eigenvalues between Aand A + E. Presumably this quantity would be a "smoothedout" approximation to the actual number and would be a continuous function of A and E. Such an approximate density function for eigenvalues can be obtained for the one-dimensional case by using the asymptotic formula following Eq. (6.3.27). We shall actually calculate the density of kn's , where k; = An. This is generally more useful, for if the equation is the wave equation, k is proportional to the allowed frequency of free vibration of the system. The asymptotic formula for the eigenvalues of k is
This formula means that the higher values of k are distributed uniformly along the real k axis. The asymptotic spacing between successive values is 7r j (f vrlP dz) , the number of eigenvalues less than k is n(k)
k ~-
7r
lb~r-dz a
p
and the average number of eigenvalues between k and k
+ dk is therefore (6.3.52)
where the quantity in brackets can be called the average density of eigenvalues of k, for large values of k. We notice that this density is proportional to an integral of a function vrlP, integrated over the interval a, b. If rand p have the same dimensions (in which case Ijk has dimensions of length), then the integral of vrjp has the dimensions of a length. For the wave equation in one dimension, rand p are unity and the average density of eigenvalues is just Ij7r times the distance from a to b. The integrand vrjp is related to the scale factors of the curvilinear coordinates, so that a deviation of vrjp from unity is somehow related to the curvature of the coordinate under consideration. The integral I vrlP dz can be called the effective interval length for the coordinate and the boundaries in"question. Turning now to the two-dimensional case we find here that the density of eigenvalues of k is not approximately independent of k. For example, plotting the values given in Eq. (6.3.48), for a rectangular membrane, we see that the density seems to increase with increasing k .
§6.3]
Eigenfunctions and Their Uses
761
That this is correct can be shown by utilizing the particularly simple relationship between k and the integers m and n . Equation (6.3.48) is analogous to the equation relating the distance from the origin to a point specified by the cartesian coordinates 7rm/a and 7rn/b. These points, for integral values of m and n, are plotted in Fig . 6.11 as the intersections of a rectangular grid . An allowed val~e of k corresponds to the distance of anyone of these grid points to the origin. We can then speak of "the density of allowed points in k space." Since the spacing of the grid in the two directions n is 7r/a and 7r/b, the average density of / points is ab/7r 2 , where ab = A, the area /'+./ enclosed in the boundaries. Consequently for this simple case the average number of l / -f~ eigenvalues of k less than a value k is m-
t
If
t 1--t---+---:>lL-~
n
where the quantity in parenthesis is the area between the lines r = k, y = 0, and x = O. The differential of this
dn
~
[Ak/27r] dk
I-f....j
(6.3.53) Fig.
6.11
m-
Distribution
of
gives the density of eigenvalues of k for eigenvalues for a rectangular and this simple case (the quantity in brackets). triangular membrane. Length We see that it is proportional to k (which of vector k is value 0 . is what we wanted to prove) and also is proportional to the area A inside the rectangular boundary. We can go further than this, however, for we note that the area covered by the dots, in Fig . 6.11, is not quite all of the first quadrant. Since the points for m = 0 or for n = 0 are missing (for the boundary condition ,p = 0), we should remove half a grid strip adjacent to the two axes, and a more accurate formula would be
n(k) or
~
(ab/7r 2)[pk 2 dn
(k7r/2a) - (k7r/2b)] = (A /47r)k2 ~ [(Ak /27r) - (L/47r)] dk
-
(L/47r)k (6.3.54)
where L = 2a + 2b is the perimeter of the boundary. This second term in the expression for the density of eigenvalues depends on the boundary conditions. For instance, if the boundary conditions at the inner surface of the rectangle were that the normal gradient of ,p be zero instead of ,p, then the expression for the eigenvalues of k would be the same but the points for m = 0 and n = 0 would now be allowed. Consequently the density function would be [(Ak /27r) + (L/47r)] .
762
Boundary Conditions and Eigenfunctions
[cH.6
It appears that the first term in the average density is independent of the precise kind of boundary conditions applied as long as some sort of condition is applied; it depends only on the area inside the boundary. The second term depends on the particular boundary condition imposed. Other cases where exact solutions of the Helmholtz equation in two dimensions are known (for instance, the case of a circular membrane) can be worked out. It turns out that, if the shape of the boundary is changed but the area A is not changed, the points representing allowed values of k are moved around on the "k plane " but the average density of points is not changed. In addition one can show that, if the boundary conditions are not changed as the boundary shape is varied, the corre ction term (in terms of L, the perimeter of the boundary) is not changed in form . One can verify both of these statements by taking the case of the triangular membrane. Here the reduction to half the area seems to eliminate half of the points in the first quadrant. Detailed counting shows that, for the boundary condition Vt = 0, Eq. (6.3.54) holds, with A = ja 2 and L = 2a + .y2(i2. Equation (6.3.54) presumably holds for boundaries of any shape, for conditions if; = 0 at the boundaries. It has been proved that the first term (which is the preponderating term for large k) is valid in general. The proof has not been extended to the second term, though there have not been found any contradictory cases among those which can be worked out in detail. In any case, if we are dealing with large values of k, we can neglect the second term in the density, using just the first term, which is known to be valid for all boundaries and all reasonable boundary conditions. Thus we have shown that for the higher eigenfunctions the density of eigenvalues is independent of k for one-dimensional cases, is proportional to k for two-dimensional cases, and, by extension, is proportional to k 2 for three-dimensional cases. In each case the density is also proportional to the" size " of the space within the boundary; length for one dimension, area for two , and so on. If the equation is the Helmholtz equation, and if the interval is not curved, these " sizes " are the actual lengths, areas, etc., but if the coordinates are curved and the line or surface bounded by the boundary is curved, the " sizes" are effective lengths, areas, etc., being the integral of certain combinations of scale factors, similar to the integral f Vr/p dz for the one-dimensional case. Continuous Distribution of Eigenvalues. We see, from the discussion above, that the spacing between eigenvalues diminishes as the size of the boundaries increases. For instance, for the one-dimensional case, the average difference between successive eigenvalues, according to Eq. (6.3.52), is 1r divided by the effective interval length f vr;p dz. Ai3 this length goes to infinity, the separation between eigenvalues goes to zero until all values of A (or k) larger than the lowest value are eigenvalues.
§6.3]
Eigenfunctions and Their Uses
763
In this limiting case we have a continuous distribution of eigenvalues, and our series representations turn into integral representations. Our first example, on page 709, can be extended to show the transition. Suppose that the value of a, the distance between the two ends of the range of x, is increased indefinitely. The eigenfunction sin (lrnx /a) for a given n will have a longer and longer wavelength, and its normalization constant En = va72 will become larger and larger. The Fourier series
will conserve its general form, but each individual term will reduce in size and (for a given finite value of x) will change less and less rapidly from n to successive n until, near the limit, each term is vanishingly small and the rate of change with n of the coefficients in the brackets is vanishingly slow. In fact, in the limit, it is not advisable to use n as the variable of summation, but k = n1r/a, the square root of the separation constant X. The spacing between allowed values of this variable becomes smaller and smaller, until eventually k is a continuous variable and the summation over n becomes an integral over k from zero to infinity. As a is made ever larger, the average number of eigenvalues of k between k and k + dk, (a/7r) dk , given by Eq. (6.3.52), becomes more and more nearly equal to the actual number, as more and more allowed values of k are found in any finite segment dk. At the same time the successive terms in the sum, for the eigenfunctions for those eigenvalues within dk, differ less and less from each other (at least for finite values of x), so that we can finally represent the sum of all the terms between n = akj1r and n + (a dk/1r) by the expression
(a :k) sin(kx) [~la f(r) sin(kr) dr ] and when a goes to the limit, the sum finally becomes the integral
21'"
f(x) = 1r
sin(kx)
0
1'"
f(r) sin(kr) dr dk
(6.3.55)
0
which can be used to express any piecewise continuous function in the range 0 < x < 00 which goes to zero at the ends of the range. This is one form of the Fourier integral (see page 454) . The more general form f(x) = -1
f'"
f'"
(6.3.56) eikz dk f(r)e- i k i dr 27r can be obtained by using the more general boundary conditions that the functions be periodic in x with period a. -00
-00
764
Boundary Conditions and Eigenfunctions
[cH.6
But after all, it should not be necessary to obtain the properties of eigenfunctions for continuous eigenvalues by a clumsy limiting process . From the point of view of function space, we can express a vector F representing any function F in terms of its components F(x) along the unit vectors e(x) (corresponding to each value of x in the allowed range) or in terms of its componentsf(k) along the alternative set of unit vectors e(k) (corresponding to each allowed value of k) . Instead of having one continuous set and one discrete set, we now have two continuous sets, a more symmetric arrangement. The eigenfunctions are still the projections of the vectors e(k) on the vectors e(x) , but since both sets of vectors are preferably unit vectors, these eigenfunctions are now direction cosines and the whole formalism gains considerably in symmetry. The eigenfunctions if!(k,x) are at the same time components of the vectors e(k) on the vectors e(x) but also components of the vector e(x) on the vectors e(k) . Extending Eq. (6.3.17), the component F(x) of an arbitrary vector F in the direction given by e(x) is related to the componentf(k) of F in the direction specified by e(k) by the equations F(z) = ff(k)if!(k,z)r(k) dk; f(k) = fF(z)Vt(k,z)r(z) dz
(6.3.57)
where the function r(k) is related to the density of eigenvalues of k along k, just as r(x) is related to the density of eigenvalues along x. The range of integration for these integrals is over the allowed ranges of k and x , sometimes from 0 to 00, but more often from - 00 to 00 . Letting k go to - 00 does not mean that there is no lower bound to the eigenvalues of A, for A = k 2 and, as long as k is real , A > O. Furthermore when we include negative values of k, we can arrange to have if! a complex quantity (as eik z in the example) if we use the complex conjugate Vt in the second integral. The normalization and orthogonality properties of these eigenfunctions are obtained by extending Eqs. (6.3.18): r(k)fif!(k,z)Vt(K,z)r(z) dz = ~(k - K) r(z)fif!(k,z)Vt(k,t)r(k) dk = ~(z - t)
(6.3.58)
both of the integrals corresponding to delta functions . The functions if! are direction cosines, since both e(x) and e(k) are unit vectors, so the factors Em are unity and do not explicitly appear in the formulas. As always with expressions corresponding to delta functions, they are to be used in integrals, not in differentials or by themselves. The most useful form of the normalization integral corresponds to the usual definition of the delta function
765
Eigenfunctions and Their Uses
§6.3]
In other words , in the limit Ll~O (k.+A lk.-A r(K) d«
f if;(k,z)?t(K,z)r(z) dz = { 0;i,:
Ik - kol > Ll Ik - kol < Ll
(6.3.59)
with a similar equation, reversing z and k, corresponding to Il(z - r). Usually the limits of the second integral are first set finite but large (0 to R, or - R to R, R large) and then later extended to infinity, for ease in calculation. As an example of this we consider the Bessel equation resulting from the separation of the Helmholtz equation in polar coordinates
d(dR) + (k dr dr
-
r-
2
r - -m
r
2 )
R = 0
(6.3.60)
where m is the separation constant coming from the q, factor and is an integer if q, can go from 0 to 21r (in other words , if ~ is periodic in q,). If the boundary conditions for R are that it be finite at the singular point r = 0 and that R = 0 at r = a, the eigenfunctions are the Bessel functions [see Eq. (5.3.63)] if;n(r) = J m(anr/a); k n = (an/a);
J mean) = 0 n = 0,1,2,
(6.:t.61)
These functions are orthogonal (with density function r) l~n
l
= n
so that we have a complete set of eigenfunctions (n = 0, 1, 2, .. .) for each value of m. If a goes to infinity, the eigenvalues of k are a continuous sequence from k = 0 to infinity. Therefore the function Jm(kr) is proportional to an eigenfunction for a continuous k, To normalize, we set if;(k,z) = AJm(kz) and determine A by using Eq. (6.3.59). As mentioned previously, to simplify the calculations we make the upper limit of the second integral R and let R go to 00 later. Using the asymptotic expression for the J's and doing the calculations for m > 0, we have
766
Boundary Conditions and Eigenfunctions
[CH. 6
This final result also holds for m = 0, though some of the intermediate steps are different in form . Therefore the normalizing constant A is equal to unity, the normalized eigenfunction is just J m(kz), and the expression corresponding to the Fourier integral formula (6.3.55) is just (6.3.62) which is called the Fourier-Bessel integral. Eigenvalues for the Schroedinger Equation. In a number of solutions of the Schroedinger equation, cases are encountered where the eigenvalues are discrete for a certain range of values and are continuous for the rest of the range. A consideration of the relationship between the Schroedinger equation [see Eq. (2.6.28)] and the Sturm-Liouville problem will indicate how these come about and will perhaps shed further light on the Sturm-Liouville results [see also the discussion of Eqs. (12.3.25) and (12.3.27)]. The Schroedinger equation in one dimension is
d?,J;
dx 2
2m + 7i2 [E
- V(x)]f = 0
(6:3.63)
where m is the mass of the particle under consideration, E is its total energy in the state considered, V is the potential energy, and h = h/21r is the modified Planck's constant. The probability that the particle is between x and x + dx is proportional to Ifj2dx, and the average" current density" of the particle is proportional to the imaginary part of y;(dljtjdx) (that is, if f is a real function for x real, then the net current is zero). According to classical mechanics the particle would be only where the particle energy E is larger than the potential energy V; in such regions the greater the value of E - V (= kinetic energy), the greater the current density and the smaller the particle density. In fact the probable density of the particle would be proportional to (l /velocity) ex: 1/ y E - V, and the probable current density would be proportional to yE - V. No energy E would be possible which would be less than V everywhere; all energies E would be possible for E larger than V somewhere. If a particle is in one minimum of potential separated from another minimum by a peak higher than E, then the particle cannot go from one minimum to the other. In contrast, the Schroedinger equation is more restrictive in regard to allowed values of energy but less restrictive on location of the particle, as the derivation of the equation (pages 243 and 314) has suggested. Let us take a case where Vex) has a minimum (we can place the minimum at x = 0 for convenience) and where the asymptotic value of V is larger
§6.3]
Eigenfunctions and Their Uses
767
than this minimum and analyze the equation from the point of view of the Sturm-Liouville problem (see page 722). When E is less than V everywhere, the solution of Eq. (6.3.63) is not oscillatory ; it behaves more like the real exponentials e'" and e:», and no combination of the two independent solutions will produce a solution which is finite both at - co and at + ca , As indicated in Fig. 6.8, if the solution goes to zero at - V, it curves toward the axis, behaving like a trigonometric function. Starting 1/1 out at zero at - "-I)] is not necessarily equal to [1/(
772
Boundary Conditions and Eigenfunctions
[cH.6
tion it is a combination of the operators CP just defined, operating on the "vector" corresponding to the function t/;(x) [which can be considered as the component of the vector along the direction given by the unit vector e(x)] ; in the case of the Dirac equation the operator is a combination of operators which interchange the four components t/;l(X) .. • t/;4(X) and of differential operators which act on the x dependence of the four t/;'s. In any of these cases we can talk about a vector F and an operator ~ which, in general, changes F into another vector E, as was discussed in Sec. 1.6. The vector F can be described in terms of its components along suitable coordinates [x, Y, z components for strain displacement, F(x) for each x for differential equations, different spin states for the Dirac equation, etc .]. The operator must correspondingly be given by a matrix of components
(see Eq. 1.6.33)
~. F =
I [In AmnFnJ
em
=E
m
the matrix (A mn) representing ~. When the axes of reference are rotated, the components of F and of ~ are modified according to the usual rules of transformation given in Chap. 1. The fiction of a discrete set of axes, represented by the unit vectors en, can be used even when the "subscripts" are continuously variable [as with e(x)] and the scalar product has to be represented as an integral over x, instead of a sum over the subscript n . In the following it should not be difficult to make the necessary change from sum to integral when this is necessary. We should recollect the discussion following Eq. (6.3.7) in this connection. For instance, corresponding to the operator equation ~ . F = E is the differential equation (1,,,,F(x) = E(x), where (1,,,, is an ordinary differential operator of the sort d2 d (1,,,,
= f(x) dx 2
+ g(x) dx + h(x)
which we have been discussing in this chapter. scalar product (G* . ~ . F) we have the integral
Corresponding to the
lab G(x) (1,,,,F(x) dx (G is the complex conjugate of G) and so on. In nearly all cases of interest the operator ~ is Herm itian, which means that its Hermitian adjoint, ~*, obtained by interchanging rows and columns in the matrix and then taking the complex conjugate, is equal to
§6.3]
Eigenfunctions and Their Uses
·773
~ itself. We worked out some of the consequences of this on pages 83 to 87. Let us see what it means for a differential operator having components corresponding to the continuum of values of x. The Hermitian adjoint of an operator ~ is the operator ~*, such that if ~
•F = E
then
E* = F* .
~*
in other words G* .(~ .F) = (~*·G)*·F
(6.3.71)
When the operator is real, then the concept of Hermitian adjointness corresponds to the concept of adjointness for differential operators given on page 527. For instance, in integral form, Eq. (6.3.71) becomes
lab G(x)o."F(x) dx = lab G(x) [I ~;; + g ~~ + hFJ dx But according to Eq. (5.2.10) this last integral is equal to
Jab [&,.G] F dx + [P(G,F)]~ where &" is the adjoint differential operator given by d2
d
dx 2 (fG) - dx (gG)
+ hG =
_ a"G
and discussed on page 584. If both F(x) and G(x) satisfy suitable boundary conditions at a and b, the bilinear concomitant P(G,F) is zero at a and b and we have
f
Go."F dx
=
f
[&"G]F dx
which corresponds to Eq. (6.3.71) , defining the adjoint of an operator a". This means that our use of the word adjoint in relation to a generalized operator ~ corresponds to our use of adjoint in relation to an ordinary differential operator a" if a" is real. If a" is not real, then its Hermitian adjoint is the complex conjugate of its adjoint, 1* = &. Consequently an ordinary differential operator which is self-adjoint, as defined on page 720, corresponds to a real Hermitian operator, which is correspondingly self-adjoint in the operator sense . (We have already seen that the Liouville equation is self-adjoint.) This will be discussed more fully in Sec. 7.5. Whether it be a differential operator or a more general kind of operator, if ~ is Hermitian, then (G* .
~
. F) = (G* .
~*
. F)
(6.3.72)
This means that the quantity (F* . ~ . F) is a real quantity, no matter what the vector F, if ~ is Hermitian. In quantum mechanics this quantity would be called the average value of ~ in the state characterized
774
Boundary Conditions and Eigenfunctions
[cH.6
by the vector F. If 2( corresponds to a physical quantity (position, momentum, etc .), its average value should be real, of course. In a great number of cases th is average value (F* . 2( • F) is alwa ys positive (as well as being real) for all vectors F. In such cases the operator is said to be positive definite as well as Hermitian. Corresponding to each operator 2( is a sequence of eigenvectors En, such that 2( . En = anEn where an is the eigenvalue of 2( corresponding to En. From what we have said above, it is easy to prove that, if 2( is Hermitian and positive definite, then all its eigenvalues will be real and all of them will be greater than zero. As we have seen earlier, the eigenvalues may be a series of discrete values or a continuous range or a combination of both. We have discussed eigenvectors several times already ; our purpose here is to link up our earlier findings with the findings of the present section on eigenfunctions. For instance, in connection with Eq. (1.6.9), we showed that the eigenvectors for a specific operator were mutually orthogonal, which corresponds to the finding that eigenfunctions, for a given differential equation and boundary conditions, are mutually orthogonal. We can, of course, normalize our/eigenvectors to obtain the mutually orthogonal set of unit vectors en. There are as many of them as there are II dimensions" in the abstract vector space suitable for the operator 2(, Consequently any vector in the same abstract space can be expressed in terms of its components along the principal axes for 2(: F = lFne n;
Fn = (F*· en)
n
This seemingly obvious statement corresponds to the fundamental expansion theorem for eigenfunctions. To show the generality of this theorem, proved earlier for differential operators, we shall here sketch the argument for a general operator 2(. In order that the eigenvectors for a given operator 2( form a complete set, it is sufficient that : 1. 2( is self-adjoint (or Hermitian); i.e ., for any vector F (F* . 2( • F) is real. 2. 2( is positive definite; i.e., for any vector F (F* . 2( • F) is greater than zero. 3. That the equation for the eigenvectors (6.3.73)
corresponds to some variational principle. Such a variational principle can be quite general. For instance, we can compute the scalar (real and positive) quantity D(F) = (F*· 2( . F)/(F* . F) (6.3.74)
Eigenfunctions and Their Uses
§6.3]
775
for an arbitrary vector F. The variational requirement that F be the vector for which D is a minimum gives the eigenvector Eo for Eq. (6.3 .74), and the value of D for F = Eo is just the eigenvalue ao. To prove this we consider the variation of D, as F and F* are varied by arbitrary, small amounts of and of*, and require that oD = 0. Multiplying across by (F* . F), we require that D[(oF* • F)
or
+ (F* . of)]
[oF* • (~ . F - DF)]
=
[(oF* • ~ . F)
+ [(F* . ~ -
+ (F* . ~ . of)]
DF*) . of]
=
°
Since of* and of are arbitrary and independent variations, the variational requirement is thus equivalent to Eq. (6.3.73) and its conjugate, with D equal to a. As we mentioned above, the vector which gives the minimum value of D is Eo and the corresponding value of D is ao, the lowest eigenvalue (which is greater than zero, since ~ is positive definite). The vector which gives a minimum value of D , subject to the additional requirement that it be orthogonal to Eo, is E I , and the corresponding value of D is aI, the next eigenvalue. To prove this statement we have to consider that the operator ~ can be dealt with formally as if it were an analytic function, i.e., that the operators ~-1, v11, etc., have the same meaning as do their algebraic counterparts. We have indicated that this is possible for differential and integral operators. If this is so, we can show that the eigenvectors of any function ~(~) which can be expressed as a series of powers of ~ are the corresponding eigenvectors of ~, and the eigenvalues are the equivalent functions H(a n ) of the corresponding eigenvalue an. From this it follows that the solutions of the variational problem oB = 0, where
B = [F* .
~(~)
. Fl/[F* . F]
are just the eigenvectors En, which are solutions of Eq. (6.3.73) if the related algebraic function H(a) is always real and positive when a is real and positive. The resulting stationary values of B are the values H(a n ) . (The proof of this statement can be left to the reader; it is, of course , true only if ~ is positive definite.) The product K = [F* . (~ - ao) • F], where ao is the lowest eigenfunction of ~, is never negative, no matter what vector F . we choose, as a consideration of the variational equation will show. Nor is the product J = [F* . (~ - ao)(~ - al)F], where al is the next eigenvalue, for the minimum: values of
+
J _ F*[~ - i(ao al))2F _ l( _ )2 F* • F [F* . F] 4" ao al
are for F equal to either Eo or E I • For these two vectors J is zero, whereas K is only zero for F = Eo. Consequently the quantity [J / K ] is
·776
Boundary Conditions and Eigenfunctions
[CH.
6
never negative, and it is zero only when F = E I . A rewording of this last sentence will show us that we have proved the statement, made above, which we wished to prove. Suppose we set up a vector G = ~ . F which is automatically orthogonal to the lowest eigenvector Eo (G* . Eo = 0) but which is otherwise quite arbitrary. We then set up the variational ratio (G*. ~ . G) J D(G) = (G* . G) = K + al The sentence in italics shows that the minimum value of D is for G = val - ao E I , and its value for that G is aI, which is what we set out to prove. Since we have shown that the minimum value of D for a vector orthogonal to Eo is al and the corresponding vector is E I , we can extend the argument and arrive at the statement . D(G)
2::
a.+l
if
(G*. Em) = 0;
for m
= 0, 1, . ..
,-
,8
(6.3.75)
which is equivalent to the statement preceding Eq. (6.3.21). For the general case of an arbitrary operator, there may not be an infinite sequence of eigenvalues. If the vector space has only n dimensions (i.e., if all vectors in the space can be expressed as a linear combination of only n mutually orthogonal vectors), then there are just n eigenvectors and n eigenvalues (if there are degenerate states, some of the eigenvalues will be equal but the n eigenvectors will still be mutually orthogonal), for as long as we can make up a vector orthogonal to the first 8 eigenvectors, we can find still another eigenvector and eigenvalue. Only when s = n will it be impossible to find another orthogonal vector, and at that point the sequence will stop. Therefore for vector spaces with a finite number of dimensions, the number of mutually orthogonal eigenvectors will equal the number of dimensions and this set of eigenvectors will be complete, for by definition any vector in this space can be expressed as a linear sum of this number of orthogonal vectors. For vector spaces with an infinite number of dimensions the proof of completeness is not so simple [see comments following Eq. (6.3.7)]. To make it watertight we should prove that the sequence of eigenvalues tends to infinity as n goes to infinity. But a proof of this theorem for the most general sort of positive-definite, Hermitian operator would carry us too far afield into the intricacies of modern algebra. We indicated that it is true for differential operators of the Liouville type on page 724. It is also true for quantum mechanical operators having an infinite number of possible states. If we assume that it is true in general, then our proof of completeness will follow the same lines as the proof given on pages 736 to 739. Because the arguments using abstract operators are "cleaner" than those for differential equations, we shall run through the proof again.
Eiqenfunctions and Their Uses
§6.3]
777
We wish to express an arbitrary vector G in terms of the unit eigenvectors em, so we set up the finite sum n
S; =
1: Cmem;
Cm = (e: . G)
(6.3.76)
m=O
~
where
• em = ame m and
(e:'.. e.) =
~m.
The vector In = G - S, is orthogonal to the first n eigenvectors of ~, since (F:. J n) = 0 for m ~ n. Therefore by Eq. (6.3.75) we have that
D(Jn~ = (J: ~ In) U:' ~. In] ~ an+l
(J~'Jn) ~ a~+J(G* -
or
t Cme:'.)~(G t -
m=O
Cmem) ]
m=O
But the series which is the second term in the brackets is positive, being the sum of products of squares of quantities times eigenvalues (which are all positive). Therefore
(J:'.. In)
_1_ (G* .
an+l
~ . G)
Since neither of the quantities in parentheses can be negative (nor can an+l), since (G*· ~ . G) is independent of n , and since an+l ---t OC) as n ---t therefore J n) ---t 0 as n ---t OC) and therefore J n, which is the difference between the arbitrary vector G and the first n terms of its series expansion in terms of the eigenvectors em, goes to zero as n goes to infinity. Therefore the complete series (n ---t OC) equals the vector G, and we have proved again that the en'S are a complete set of eigenvectors. Similarly any operator (in the same abstract space) can be expressed in terms of its components along the principal axes for~. In particular, the operator ~ itself has the particularly simple form oc) ,
0: .
~ =
1: enane~;
that is, A mn = an~mn
n
where an is the' eigenvalue of ~ corresponding to the eigenvector en. In other words , when expressed in terms of its own principal axes, the matrix for an operator is a diagonal matrix. Other general properties, applicable alike to abstract vector operators and to ordinary differential operators, have been already discussed in
778
Boundary Conditions and Eigenfunctions
[CH. 6
Sees. 1.6 and 2.6, and still others will be derived later. It should be apparent by now that the abstract vector representation has many advantages of simplicity, because of our familiarity with the simple geometrical analogue, which makes it invaluable as an alternative point of view for nearly all our problems.
Problems for Chapter 6 6.1 A net-point potential (/I(m,n) satisfies the difference equation (6.2.6) and is to satisfy boundary conditions on the boundary lines n = 0, n = 5, m = 0, m = 5. Show that the solution satisfying the requirement that (/I have the value (/Iv at the vth boundary point is given by (/I(m,n) =
LG(m,nlv)(/Iv v
where G(m,nlv) is the solution of Eq. (6.2.6) which is zero at all boundary points except the vth and has unit value at the vth boundary point. Show that all the G's can be obtained from the ones connected with the boundary points (0,1) and (0,2). Compute these two G's, correct to three places of decimals , for each interior net point. 6.2 Show that a solution of the Poisson difference equation [1f(m
+ 1, n) + 1f(m -
+ 1f(m, n + 1) + 1f(m, n
1, n)
- 1) - 41f(m,n)] = F(m,n)
with F(m,n) specified, and with 1f zero at all boundary points, is 1f(m,n)
=
LG(m,nlJ.l,v)F(J.l,v) /lV
where G(m,nlJ.l,v) is a solution of the Poisson difference equation for F(J.l,v) = 1, all other F's = 0, and for G zero at all boundary points. What are the values of G for the 4 X 4 net of Prob. 6.1? How can these results be combined with those of Prob. 6.1 to obtain a general solution of the Poisson difference equation satisfying general boundary conditions? 6.3 The differential equation is the simple parabolic one : a21f /ax 2 = a1f/ at
and the boundaries are x = 0, x = 7r, and t = 0. For the boundary condition 1f = at x = and x = 7r show that the solution for t 2': is
°
°
°
...
1f(x,t) =
L A v sin(vx) exp [-v v=l
2t]
Problems
CH.6)
779
where the A's are chosen to fit the initial value of if; at t = 0. Now consider the net-point approximation to this equation, obtained by dividing the range ~ x ~ 71' into N equal parts with spacing h = 71'/N and dividing t into intervals of length k = 71'/M. Show that the corresponding solution of the equation "
°
(l /h 2)[if;(m
+ 1, n) + if;(m -
1, n) - 2if;(m,n)] = (l /k)[if;(m , n if;(O,n) = if;(N,n) = 0; n ~
°
IS
2:
N-I
if;(m,n) =
B. sin(vmh) exp {n In
[1 - ~~
+ 1) -
Sin2 (t h
if;(m,n)]
l')]}
.=1
What happens to this solution if k has been chosen larger than h2 ? What limitation must be imposed on the size of k in order that the solution be stable? Suppose the initial conditions are such that the coefficients A. in the exact solution are negligible for I' I'max. What can you say concerning the choice of hand k which will result in a net-point solution which is reasonably accurate (to 1 per cent, say) in the range ~ t ~ 71' and which is not, on the other hand, so "fine-grained" (N and MJoo large) as to make the numerical computations too laborious? 6.4 The initial values of if;(x,t) of Prob. 6.3 is
°
if;(x,O)
=
x; tn-(71' {-v
00
x);
(0 ~ x ~ ~)} \' 1 . (~ ~ x ~ 71') = '-' (20" + 1)2 sm [(20"
+ l)x]
,,=0
Compute values of if;(x,t) for x = -br,~ ; t = tn-, ~ for the exact solution. Then compute values of if;, by use of the difference equation if;(m, n +1) = [1 - (2k/h 2)]if;(m,n) + (k/h 2)[if;(m + 1, n) + if;(m - 1, n)]
°
starting at n = and working upward (for increasing n) for the same initial conditions. Take h = -V (N = 4) and make the computations for two choices of k:k = 71'/4 and k = 71'/16. Compare with the four exact values already computed. 6.5 Is the partial differential equation (o2if;/ox 2) - y 2(02if; /oy 2) =
°
elliptic or hyperbolic? What are the equations for the characteristics? Sketch a few of them. Show that, if Cauchy conditions are applied at the boundary y = Yo 0, the solution for y Yo is if;
=
#0 (x + In :0) + #0 (x - In : ) + tyo~o (x + In :0) -
tyo~o (x - In:o)
780
Boundary Conditions and Eigenfunctions
[cB.6
where the value of if; at Y = Yo is if;o(x) and where 'Po(z) = fvo(z) dz, vo(x) being the initial value of aif;/ay at Y = Yo. Why does this solution fail at Yo = O? 6.6 Build up a sequence of mutually orthogonal polynomials in x for the range -1 ~ x ~ 1. Start with Yo = 1, Yl = z, .. . with Yn a polynomial of degree n such that
f
l -1
Yn(x)Ym(X) dx = 0;
m = 0, 1, .. . ,n - 1
Obtain the first four such polynomials. Show that, in general, those polynomials for even n 's have no odd powers of x, those for odd n have no even powers. Show that the polynomials obtained are proportional to the Legendre polynomials P n(X). Is this process of building up a set of orthogonal polynomials a unique one? If not, what restrictions must be added to make the process unique? Is the resulting set of functions a complete set? How can you be sure? 6.7 Repeat the process of Prob. 6.6 for the range 0 ~ x ~ 1 and the orthogonality requirement
10
1
= 0 ; m = 0, 1, .. . ,n - 1
Yn(X)Ym(X)X dx
Start again with yo = 1 and obtain the first four polynomials. Compare this with the set 'Pn(X) = J o(1l'a"nX) , where aO n is the nth root of the equation [dJ o(1l'a )/ da] = O. Show that this is also a mutually orthogonal set of functions for the same range of x and for the same density function x. For what problems would each be useful? 6.8 The Tschebyscheff polynomials Tn(x) are defined by the generating function 1 1 - 2tx
.
t2
+ t2 =
\'
~ Tn(x)t
n
n=O
Obtain the first four polynomials and, by manipulation of the generating function, show that
+
Tl(x) - 2xTo(x) = 0; 2T o(x) - 2xT l(x) T 2(x) = 0 Tn+l(x) - 2xT n(x) Tn_l(x) = 0; n 1
+
and consequently that Tn(x) =
f
l
Tm(x) Tn(x)
-1 .
6.9
En COS
[n cos- 1 X].
dx
V1 -
X2
=
. n1l' Omn
E
The Jacobi polynomials are defined as J n(a,clx)
= F(a
Show that
+ n, -n\ c] z)
CH.
6]
Eigenfunctions and Their Properties
781
Write out the first four polynomials and show that the set is complete (for what range?). By use of the contour integral obtained from Eq. (5.3.21) and subsequent use of Eq. (4.3.1) show that J n(a clx) = ,
x l-c( 1 - x)c-ar(c) d n c+n- I [x (1 - x)a+n-c] I'(c + n) dx»
Show that dd In(a,clx) = - n(n + a) In_l(a + 2, c + 11 x) x c c- 1 xJn(a,clx) = 2n + a [In(a - 1, c - 11 x) - In+l(a - 1, c - 11 x)] and that
Jot
x
c-I
(1
_
x)
a -c _ n![r(c)]2r(n In(a,c!x)Jm(a,clx) dx - (a + 2n)r(a
++an)r(c - c + 1) + n) omn
Express P n(X) and Te(x) in terms of the J's. 6.10 The radial function for the Helmholtz equation in spherical coordinates is jn(kr) = V7r/2kr J n+i(kr). Show that the eigenfunctions for a standing acoustical wave inside a rigid spherical shell are jn(7r{3nmr/a) , where (3nm is the mth root of the equation [djn(7r{3) /d{3] = O. Show that these form a complete, orthogonal set for the range 0 :::; r :::; a. By letting a ~ 00, show that jn(ZU)U 2 du 2~"" 0
f(z) = 7r Show that
6.11
t
0
j(V)jn(UV)V 2 dv
dt = [r(a + n + 1))2 (z - l)n Jo( '" e-zttaLa(t) n n!za+n+1
Prove that
6.12 ""
H ( )H ( ) .E: = 1 n X n Y 2n' _/
L
n.
2 2
2
y exp [2X t - 1 t_(Xt 2 + y ) ] v I - t2
n=O
Table of Useful Eigenfunctions and Their Properties We choose a range of the variable z and a density function r(z) such that r(z), times any positive power of z, integrated over the chosen range of z, is finite. We then choose the lowest eigenfunction to be 1/Io(z) = 1. The next eigenfunction, 1/Il(Z), is chosen to be that combination of 1 and z which is orthogonal to 1/10 in the chosen range and for the chosen density r , Then V;2(Z) is that combination of Z2, z, and 1 which is orthogonal to 1/10
,
782
[cH.6
Boundary Conditions and Eigenfunctions
and 1/;1, and so on. We can thus set up a set of eigenfunctions by a purely mechanical procedure, called the Schmidt method, which will serve as a basis for expanding any piecewise continuous function of z in the chosen range. It is usually found that the eigenfunctions thus obtained are ones which also arise from the solution of some Liouville equation, with boundary conditions, or which are obtained from some generating function. Three useful cases will be summarized here , for three ranges of z and for different density functions r(z) . See also the table of Jacobi polynomials at the end of Chap. 12. I Range -1 :::; z :::; + 1; Density Function (1 - Z2)13 : Gegenbauer polynomials: T~(z) (see page 748). Generating function: 00
213
+ t2 -
(1
'
2tz)l3+!
V; \' = r(13 + -!) L.t tnT~(z) n=O
Special cases:
T~(z)
=
Pn( z), Legendre polynomials
(see Eq. 5.3.24)
(1 - z2)m/2T:;'_m(z) = P,::(z) , associated Legendre functions [see Eq. (5.3.38)]
=
nT;!(z)
~ cosh [n cosh:" z], Tschebyscheff polynomials [see Eq. (5.3.43)]
~ TL1(Z)
=
~~ sinh [n cosh"
z], Ts chebyscheff polynomials
[see Eq. (5.3.43)]
213
V; r(13 + -!) = 1 . 1 ·3 ·5 . . .
Tg(z) = T~(z)
213+ 1
= V; r(13
+ j)z =
(213 - 1); when 13 = 0, 1, 2,
1 · 3 · 5 . . . (213
+ l)z ; when 13 = 0, 1, 2,
Recurrence formulas, relating these polynomials, obtained from the generating function d - TI3(z) dz n
=
TI3+1(Z)' n-l'
d
dz [(Z2 - l)I3T~(z)] = (n
+
l)(n
+ 213)(z2
- l)I3-IT~+Hz)
+ 2n + l)zT~(z) = (n + l)T~+I(Z) + (213 + n)T~_I(Z) d (213 + 2n + l)T~(z) = dz [T~+I(Z) - T~_I(Z)] = T~+l(z) - T~~~(z) (213
+ l)T~(z) + 2z :z T~(z) = T~+I(Z) + T~~~(z) (n + 213 + l)T~(z) = T~+I(Z) - zT~:tHz) (213
nT~(z)
=
zT~~Hz) -
T~~Hz)
6)
CR .
783
Eigenfunctions and Their Properties
(2{3 + 2n + 1)(z2 -
I)T~~:l(z)
=
I)T~-I-l(z)
n(n +
- (n + 2{3)(n + 2{3 +
1)2 '~_1(Z)
T8(z) = 2/lr({3 + i) . " n! .y;;: (2{3 + 1)(2{3 + 3) (2(3 + 2n - 3)(2{3 + 2n - 1) (2(3 + n + 1)(2{3 + n + 2) (2{3 + 2n - 1)(2{3 + 2n) 1 d" . (Z2 _ 1)/l dz" (Z2 - 1),,+/l d2 d (Z2 - 1) dz 2 T~(z) + 2({3 + l)z dz
T~(z)
- n(n + 2{3 +
I)T~(z)
= 0
which is related to the hypergeometric equation, having three regular singularities. {3 = m , an integer, results in associated Legendre polynomials, m 'I'm = P'::+m _ d (P ) n (1 _ z2)im - dzm ,,+m Tg = 1; Tt = 1; '1'5 = 3; T~ = 15; T¥ = z; Tt = 3z; T~ = 15z; T~ = I05z i Tg = i(3z 2 - 1); T~ = j(5z 2 - 1); T~ = ¥(7z 2 - 1) ; Tg = j(5z 3 - 3z); '1'1 = j(7z 3 - 3z) ; '1': = .!.p(3z 3 - z) ; . . . T~ = !(35z 4 - 30z 2 + 3); T~ = !j(21z 4 - 14z 2 + 1); 0
0
0
0
'I'm = (2n + 2m)! [z" _ n(n - 1) Z,,-2 " 2,,+mn!(n + m)! 2(2n + 2m - 1)
+
n(n - 1)(n -2)(n - 3) Z,,-4 .. 2.4(2n + 2m - 1)(2n + 2m - 3)
see also Eqs. (6.3.37) and (6.3.40). Normalization integral: 1 /
(
-1
1 - z
2)/lT/l( )T8() d _ • 2r(n + 2{3 + 1) m Z "z z - Um" (2n + 2{3 + l)r(n + 1)
Special values: 'l'~(z)
=
(-I)"T~(-z)
= r(n + 2{3 + 1). '1'8(1) " 2/ln!r({3 + 1) '
'1'8(0) 0 " = i
.y;;: (ini n+ )j). !'
T/l(O) = (-I)t" 2/lr({3 +
"
n =
when
1 3 5
n=
,
,
, . . .
0 2 4 . , , ,
Relation unth:hyyergeometric junction:
/l _ r(n + 2{3 + 1) F (_ I ,I T,,(z) - 2/ln!r({3 + 1) n , n + 2{3 + 1 1 + {3 2 = (_I)"r(n + 2{3 + 1)
2/ln!r({3 + 1)
z)
F(- n, n + 2{3 + III +{31 1 +z) 2
oJ
784
Boundary Conditions and Eigenfunctions
[cH.6
Addition formula : T~(cos
t'J cos t'J o
+ sin t'J sin t'J o cos cp) n
_ J7C2 ~ ({3 + m)(n - m)! [si . ] = V ~7r ~ r(2{3 + n + m + 1) sm t'J sm t'J o m
•
m=O
. II Range 0
~
z
T~~';.(cos
T~:::(cos t'Jo)T~-l(cos
t'J)
cp)
00; Density Function zOe- Laguerre polynomials: Z
:
L~(z)
Generating function: 00
e-zl/(l-l ) (1 - t)o+1
~
~ i'(n
n=O
tn
+a+
1)
L~(z)
dn
Special cases: LO(z) = e' dz - n (zne-z) n Lg(z) = r(a
+ 1) ;
Li(z) = I'(c
+ 2)[(a + 1) -
z]
Recurrence formulas: d = _La+1l(z)' except for n = 0 n n' dz 'La(z) d
dz
[zoe-zL~(z)]
= (n
L~(z) = ~ [ L~(z) zL~(z) =
(a
- n
l)za-le-zL~+Hz)
+ ~ + 1 L~+l(Z) ]
+ 2n + I)L~(z)
d z dz L:(z) = (z -
a)L~(z)
- a
- (n
~ ~ ~ 1 L~+l(z)
- (a
+ n) 2L~_1(Z)
+ I)L::+t(z)
+ n + 1) ~ d n [zo+ne-z] + 1) za dz + (a + 1 - Z) :z L~(z) + nL:(z) n
La(z) = r(a
I'(n
n
z
+
::2L:(z)
= 0
which is the confluent hypergeometric equation, with regular singular point at z = 0 and an irregular point at 00 . a
=
m, an integer, results in associated Laguerre polynomials .
L:;'(z)
=
dm
(-I)m dzm [L~+m(z)]
Lg = 1; LJ = 1; L5 = 2; L~ = 6; L&, = m! ; = 1 - z ; Lt = 4 - 2z; Li = 18 - 6z; Lf = 96 - 24z; Lg = 2 - 4z + Z2; L~ = 18 - 18z + 3z 2; Li = 144 - 96z + 12z2; L~
6]
CH.
Eigenfunctions and Their Properties
785
+ 9z 2 - Z3; L1 = 96 - 144z + 48z 2 - 4z 3 ; . . . . . . . . .. . . . . . . .. . . . . . .. . . .. . . . . . .. . . . . .
Lg = 6 - 18z
Lm = [em + n)!J2 F( -nlm n n!m!
+ liz)
where F is a confluent hypergeometric series with n
+ 1 terms.
Normalization integrals:
r'" zae-'Lam(z)La(z) dz = 0 [rea + n + 1»)3 n mn I'(n + 1)
Jo
1zPe-ZLf:.-"(z)L~-V(z)
dz = (-I)m+nr(p
+m -
+ l)r(p + n rep + cr + 1) p.
, , \' . u iv: cr!(m - cr)!(n - cr)!(cr
Lt
+ p. -
m)!(cr
- v
+
+ 1) .
v - n) !
" where m, n, J1., v are integers or zero and a takes on all integral values larger than either m - J1. or n - v and smaller than either m or n (if these requirements cannot be met the integral is zero). Relation with confluent hypergeometric series for general values of a: a
+
[rea + n 1»)2 _ n!r(a + 1) F( nla
_
Ln(z) -
=
[rea + n n!r(a
+
liz)
++1)1»)2 e'F(a + n + lla + 11
- z)
~ rea z-
+ ~ + 1)
(-z)n
n.
00
Addition formula and other equations: '"
a
Ln(x
+ y)
_
(-I)m r(n + a + 1) m a+m Lt ----rn! rem + n + a + 1) Y L n (x)
II \ '
- e
m=O
'"
xm
_
\'
- Lt (m n -O
+
mICa m + 1) La (x) - n) ![r(a + n - 1»)2 n
L '"
l(xt)-jaJ
e
l'"
a(2 VXt) =
[rea
+ t: +
1»)2
L~(x)
n=O
e- j uu v+1J
v(iitz)L~v(u) = 8(n
du
+ v + i)r(n + 2v + 1) (Z2 (2z) + l)vH v
1)
(Z2 T~ Z2 + 1
where m and n are integers but a and v need not be integers.
786
Boundary Conditions and Eigenfunctions
m Range nomials: H,,(z)
z<
00
00;
[cH.6
Density Function e- z' : Hermite poly00
Generating function: e- t'+2tz = \' t~ H ,,(z) '-' n . 10=0
Recurrence formulas:
d dz H ,,(z) = 2nHn-l(Z) zHn(z) = nHn_1(z) + iH,,+l(Z) d dz [e-z'Hn(z)] = -2e- z'H n+1(z) n
H n(Z)
d2
=
d e- z' (-I)ne z' dz n
d
dz 2 Hn(z) - 2z dz H n(z)
10
=
_2~ V1r
f
(z
00
+ it)ne-t' dt
-GO
+ 2nHn(z)
=0
which is the equation for the confluent hypergeometric fun ctions F( -inlilz 2) and zF( -in + tlilz 2) H 0 = 1 ; H 1 = 2z ; H 2 = 4z 2 - 2 H 3 = 8z 3 - 12z; H 4 = 16z4 - 48z 2 + 12 H;
n' = (-I)t n (in')!F(-inlil z2); n = 0, 2, 4, = 2( _1)t(n-l ) (in n' ~ i)! zF( -in
+ ililz 2);
n = 1, 3, 5,
Normalization integral and other formulas:
t:
Hm(z)Hn(z)e-"dz = Omn 2nn! Y; (-4 n)n! • H 2n(z) = r(n + i) L n- t(Z2); H 2n+1(Z)
=
2( -4)nn! r(n + i) zL"t(Z2)
2: (2=:1) 00
[e-(Z'+Y'+2ZlIz )/(1+z' )l/Vf+Z2 = e- Z' - lI '
m=O
exp {-i[(u 2 + v2) cos cp
Hm(x)Hm(y) .
+ 2uv sin cpll 00
= sec(icp)
2:
mDO
im tanmCcp) ,"2" eHV"-U'lHm(u)Hm(iv)
m.
CH.6]
Eigenfunctions and Their Properties
787
A variation in these polynomials, similar to the superscripts (3 and a in cases I and II, may be introduced by choosing the density function to be e- z'+2az instead of «r. All this does, however, is to shift the center of the polynomials from Z = 0 to z = a. The new generating function is .,
2: ~~
e- t'+2t(z- a) =
H:(z) = e- 2ta
n=O
2: {;;!
Hm(z)
m=O
where n
H~(z) =
2:
Hn(z - a)
(-2a)n- mn! m.I( n _ m )'. Hm(z)
m=O n
= Z-i n \ ' I( n~ ) I Hm(z V2)Hn-m(-a V2) Ltm.n m . m=O
or n
=
Hm(z) n!(-a)n-m F(n m!(n-m)!
eia%--i z' { \ '
Lt
m=O
+
+ 11 n -
m
+ 11 -ia 2)
.,
2:
Hm(z)
(:!~m~)n! F(m + Jim -
n
+ 11
-ia2)}
m=n+l
The recurrence formulas are all the same, about the new origin. The normalization integral is
With these three sets of eigenfunctions we have covered the various possibilities of singularities of the density function at the end points : The Gegenbauer polynomials correspond to a density having a branch point at both ends of the interval, the Laguerre polynomials to a density function with a branch point at one end and an essential singularity at the other, and the Hermite polynomials to a density function with essential singularities at both ends . The values of the independent variable at the two end points may be changed from the standard values given above by obvious transformations. For instance, for a range from z = -a to z = 00 with density function having branch points at both ends, we use the set of eigenfunctions T~[zj(z + 2a)], with density function r = 22P+1aP+l(z + a)Pj(z + 2a) 2P+2 , and so on.
Boundary Conditions and Eigenfunctions
788
ICH.6
Eigenfunctions by the Factorization Method The fundamental equation is the Schroedinger type (d 21jdx 2 )
+ IA -
V m(X»)1
=0
where A is the eigenvalue and the corresponding eigenfunction 1 is required to be quadratically integrable over the range a ~ x ~ b, where a and b are contiguous singular points of the equation. Parameter m may be continuously variable or may only have discrete values (in which case the scale is adjusted so that these values are integral values) . This equation is sometimes equivalent to the following operator equations: S~+lS;;;+l1n(mlx) = (An - am+l)1n(mlx); S;;;S~1n(mlx) = (An - am)1n(mlx)
where
~
=
[um(x)
+ (djdx)];
S;;; = [um(x) - (djdx)]
are mutually adjoint operators. When V m(X) is such that the factorization can be made, with am a function of m but not of x ; then the eigenvalues are independent of m and when
when
am+!
am;
1n(nlx)
=
1n(mlx)
=
l1m+l
+ 1, m + 2, lab [exp (2JUn+l dx) dx]-ie f un+ (x)dx An
= an+l; n = m, m
. . .
1
1
[Um+l(X)
'V"an+l - am+l lab 1>n(mlx)1>n' (mix)
< am;
1>n(nlx) =
An
= an;
n
+ : ] 1>n(m + llx) X
dx = onn'
= m, m
[lab exp (-2 JUn dX)
.1 . 1>n(mlx) = van _ am [um(x) -
- 1, m - 2, .. dx T1e-fun!X)
dx
(djdx)]1>n(m - llx)
where 1>n is again an orthonormal set. The various sorts of functions V which will allow factorization can be obtained by determining those functions U m which satisfy
for which am is independent of z. Then the corresponding function V for the original equation is
Eigenfunctions by the Factorization Method
CH.6]
789
The trivially simple case is for Umto be independent of x i then am = -u;, and V m = 0, the eigenfunctions being the trigonometric functions. Other possibilities are Um = vex)
+ mw(x)
where , in order that am be independent of x we must have that w2 constant ; v' + vw = constant.
+ w' =
+ mw(x)
Um = (l /m)y(x)
where we must have y = constant and w 2 + w' = constant. Any other choice of dependence on m and x does not allow am to be independent of x. Solving these equations for v, w, and y, for various values of the constants (including zero values), we obtain the following specific forms for Um(x) , am, and V m(X) , which include all the possibilities for the outlined method of factorization :
+ c)b cot [b(x + p)] + d esc [b(x + p)] ; am = b2(m + C) 2 {b (m + c)(m + c + 1) + d + 2bd(m + c + j-) cos [b(x + p)Jl esc" [b(x + p)]
(A) Um = (m
Vm =
2
2
from which, by transformation of variables and by choice of values of the constants b, c, d, and p, one obtains the spherical harmonic functions and other eigenfunctions related to the hypergeometric function.
+
(B) Um = deb'" - m - c; am = b2(m C)2 V m = _d 2e2b'" 2bd(m c i)e bx
+
+ +
from which, by transformation, one obtains the Laguerre functions and 'ot her eigenfunctions related to the confluent hypergeometric function. (C) Um = (m
Vm
= -
+ c)(l /x) + ibx; am = -2bm + ~b + c)(m + c + 1)(1 /x)2 - {-b 2x2 + b(m -
(m
c)
also giving confluent hypergeometric functions. (D)
Um =
bx
+ d i am = - 2bm + d) 2 + b(2m + 1)
V m = -(bx
giving a generalization of the Hermite polynomials.
+
+
(E) Um = ma cot [b(x p)] (q/m); am = b2m2 - (q2 jm 2) 2 V m = -m(m l)b esc? [b(x p)] - 2bq cot [b(x p)]
+
+
+
related to the hypergeometric function [see Eq. (12.3.22)]. Vm
= -
+ (qjm) i
am = - (qjm) 2 (2q/x) - m(m 1)jx 2
(F) Um = (m /x)
+
resulting in Laguerre polynomials [see &[. (12.3.38)].
790
Boundary Conditions and Eigenfunctions
[cH.6
Bibliography References on types of partial differential equations, on types of boundary conditions and on difference equations : Bateman, H.: "Partial Differential Equations of Mathematical Physics," Cambridge, New York, 1932. Courant, R , and D. Hilbert : "Methoden del' Mathematischen Physik," Vol. 2, Springer, Berlin, 1937, reprint Interscience, 1943. Courant, R, K. Friedrichs, and H . Lewy : "nber die partiellen Differenzengleichungen del' mathematischen Physik," Math. Ann., 100, 32 (1928). Hadamard, J. S.: "Lectures on Cauchy's Problem in Linear Partial Differential Equations," Yale University Press, New Haven, 1923. Phillips, H. B., and N. Wiener : Nets and the Dirichlet Problem, J . Math. Phys., 2, 105 (March, 1923). Poeckels, F . : "Uber die Partielledifferentialgleichung V2u k2u = 0," B. G. Teubner, Leipzig, 1891. Sommerfeld, A. : "Partial Differential Equations in Physics ," Academic Press , New York, 1949. Webster, A. G. : "Partial Differential Equations of Mathematical Physics," Stechert, New York, 1933.
+
Books giving fairly complete discussions of eigenfunctions, their properties and uses: Bateman, H . : "Partial Differenti al Equations of Mathematical Physics," Cambridge, New York, 1932. Bibliography of Orthogonal Polynomials, National Research Council, Washington, 1940. Courant, R , and D. Hilbert: "Methoden der Mathematischen. Physik," Vol. 1, Springer, Berlin , 1937. Ince, E. L. : "Ordinary Differential Equations," Chaps . 10 and 11, Longmans, New York, 1927, reprint Dover, New York, 1945. Infeld , L., and T . E. Hull: "Factorization Method," Rev. Modern Phys. , 23, 21 (1951). Kemble, E. C. : "Fundamental Principles of Quantum Mechanics," Chaps . 3 and 4, McGraw-Hill, New York, 1937. Magnus, W., and F . Oberhettinger: "Special Functions of Mathematical Physics," Springer, Berlin, 1943, reprint Chelsea, New York, 1949. Riemann-Weber: "Differential- und Integralgleichungen der Mechanik und Physik," ed. by P. Frank and R von Mises, Vol. 1, Chaps. 7 and 8, Vieweg, Brunswick, 1935, reprint Rosenberg, New York, 1943. Sommerfeld, A.: "Wellenmechanik," Vieweg, Brunswick, 1937, reprint Ungar, New York , 1948. Szego, G. : "Orthogonal Polynomials," American Mathematical Society , New York, 1939.
CHAPTER
7
Green's Functions
In the last chapter webegan the study of the central problem of field theory, that of fitting the solution of a given partial differential equation to suitable boundary conditions. There we explored the technique of expansion in eigenfunctions, a method which can be used in a perfectly straightforward way whenever we can find a coordinate system, suitable to the boundaries, in which the given partial differential equation will separate. But the result usually comes out in terms of an infinite series, which often converges rather slowly, thus making it difficult to obtain a general insight into the over-all behavior of the solution, its peculiarities near edges, etc . For some aspects of problems it would be more desirable to have a closed function represent the solution, even if it were an integral representation involving closed functions. The Green 's function technique is just such an approach. The method is obvious enough physically. To obtain the field caused by a distributed source (or charge or heat generator or whatever it is that causes the field) we calculate the effects of each elementary portion of source and add them all. If G(rlro) is the field at the observer's point r caused by a unit point source at the source point ro, then the field at r caused by a source distribution p(ro) is the integral of Gp over the whole range of ro occupied by the source. The function G is called the Green's function. Boundary conditions can also be satisfied in the same way . We compute the field at r for the boundary value (or normal gradient, depending on whether Dirichlet or Neumann conditions are pertinent) zero at every point on the surface except for r~ (which is on the surface). At r othe boundary value has a delta function behavior, so that its integral over a small surface area near r~ is unity. This field at r (not on the boundary) we can call G(rlro); then the general solution, for an arbitrary choice of boundary values ¥to(r o) (or else gradients No) is equal to the integral of G¥to (or else GN 0) over the boundary surface. These functions G are also called Green's functions. It is not particularly surprising that one can solve the inhomogeneous 791
792
Green's Functions
[cB.7
equation for a field caused by a source distribution, by means of a product of the source density with a Green's function integrated over space, and that the solution of the homogeneous equation having specified values on a surface can be obtained in terms of a product of these values with another Green's function, integrated over the boundary surface. What is so useful and (perhaps) surprising is that these two functions are not different ; they are essentially the same function. For each of the linear , partial differential equations of Chaps. 1 to 3, one can obtain a function which, when integrated over volume, represents a distributed source. When it (or its gradient) is integrated over a surface, it represents the field caused by specified boundary conditions on the surface. Physically this means that boundary condit ions on a surface can be thought of as being equivalent to source distributions on this surface. For the electrostatic case this is, perhaps, not a new concept . The boundary condition on a grounded conductor is that the potential be zero at its surface. Placing a surface dipole distribution just outside the conductor (a surface charge +0- just outside the conductor and another surface charge -0- just outside that) will result in values of the potential just outside the dipole layer which differ from zero by an amount proportional to the dipole density (to 0- times the spacing between +0- and -0-) . Nor is it so new in the case of the flow of an incompressible fluid. The boundary condition at a rigid surface is that the normal gradient of the velocity potential be zero at the surface. Placing a single layer of sources next to this rigid surface will result in values of normal gradient of the velocity potential which are proportional to the surface density of the source layer. As we shall see, the fact that boundary conditions can be satisfied by surface integrals of source functions makes the use of source (Green's) functions particularly useful. It is desirable that we underscore this dualism between sources and boundary conditions by our choice of vocabulary. The equation for a field in the presence of sources is an inhomogeneous partial differential equation (e.g., the Poisson equation V 2!{t = -471"p) , The inhomogeneous term, which does not contain !{t, contains the source density p. Conversely the equation for a field with no sources present is a homogeneous equation (e.g., the Laplace equation V 2!{t = 0). Analogously we can say (in fact we have said) that boundary conditions requiring that the field be zero at the surface are homogeneous boundary conditions (zero values are homogeneous Dirichlet condition ; zero normal gradients are homogeneous Neumann conditions ; the requirements that a!{t + b(dif;/ an) be zero at the surface are homogeneous mixed conditions) . Conversely the requirements that !{t have specified values !{to (not everywhere zero) at the surface are called inhomogeneous Dirichlet conditions; in this case the boundary values may be said to be "caused"
§7.1]
Source Points and Boundary Points
793
by a surface dipole layer of sources, corresponding to an inhomogeneous equation. Likewise the requirements that i:Jif;ji:Jn = No (not everywhere zero) on the surface are called inhomogeneous Neumann conditions, and the requirements that aif; + b(i:Jif;ji:Jn) = F o on the surface can be called inhomogeneous mixed conditions. When either equation or boundary conditions are inhomogeneous, sources may be said to be present; when both are homogeneous, no sources are present. Of course there is another, more obvious , reason why both are given the same descriptive adjective. Solutions of homogeneous equations or for homogeneous boundary condit ions may be multiplied by an arbitrary constant factor and still be solutions; solutions of inhomogeneous equations or for inhomogeneous boundary conditions may not be so modified . The Green's fun ction is therefore a solution for a case which is homogeneous everywhere except at one point. When the point is on the boundary, the Green's function may be used to satisfy inhomogeneous boundary conditions; when it is out in spa ce, it may be used to satisfy the inhomogeneous equation. Thus, by our choice of vocabulary, we are able to make statements which hold both for boundary conditions and source distributions; we have made our words conform to our equations.
v-7.1 Source Points and Boundary Points In the previous chapter we used the concepts of abstract vector space to "geometrize" our ideas of functions. A function F(x,y,z) was considered as being a handy notation for writing down the components of the vector F along all the nondenumerably infinite directions, corresponding to all the points (x,y,z) inside the boundaries. The delta function oCr - ro) represented a unit vector e(ro) in the direction corresponding to the point (xo,Yo,zo) (where r = xi + yj + zk ; we should note that r is a vector in three space whereas e and F are vectors in abstract vector space) . . Formulation in Abstract Vector Space. In Chap. 6 and Sec. 1.6 we discussed the transformation of coordinates from directions given by the unit vectors e(r) to those for unit vectors en, which latter correspond to eigenfunction solutions if;n of certain differential equations £(if;n)
= Anif;n
The corresp.onding vectors en' are eigenvectors for the abstract vector operator 2, corresponding to the differential operator £, such that 2(e n )
=
Ane n
(7.1.1)
We showed that the new unit vectors en were mutually orthogonal and that the vector corresponding to the required solution, fitting the speci-
794
Green's Functions
[cH.7
fied boundary conditions, could be built up in a unique manner by summing the individual eigenvectors
Since the differential operators £ and the corresponding vector operators are linear, solutions may be added and series expansions are possible. A straightforward method of calculating the components An was developed, which allowed our whole abstract picture to become a powerful, practical technique for solving boundary-value problems. But it should be obvious, by now, that other useful resolutions for F are possible . Another possibility is demonstrated by a study of the inhomogeneous equation ~
£(F) = -471"p(x,y,z)
(7.1.2)
To solve this equation by means of eigenfunctions we expand both p and F in eigenfunctions. If the vector corresponding to p is P = 1;Bne n and if we assume that F = 1;Ane n, then the unknown coefficients An may be determined by insertion in the equation
However, we could have resolved the inhomogeneous vector P terms of the unit vectors e(xo,yo,zo) instead of the en'S, P
=
III
L p(xo,Yo,zo)e(xo,yo,zo) xoyozo
corresponding to the equation (which is one definition of the delta function) p(x,y,z) = fJ f p(ro) o(r - ro) dxo dyo dz o We then solve the simpler inhomogeneous equation ~(G) =
-471"e(xo,Yo,zo)
(7.1.3)
(if we can) . The components of the solution G in the (x,y,z) system are solutions of the simpler inhomogeneous differential equation (7.1.4) The components G, obtained from the solution of (7.1.4), are functions, both of the coordinates x, y, z (the independent variables of the differential operator £) and also of Xo, Yo , Zo (the position of the delta function "source") corresponding to the unit vector e(xo,Yo,zo) chosen for Eq. (7.1.3). We shall show, at the end of Sec. 7.5, that the functions G(x,y,zlxo,yo,zo) = G(rlro), for different values of Xo, Yo, Zo, are components, along the directions e(r), of an operator, rather than a vector.
§7.1]
Source Points and Boundary Points
795
This operator changes the vector P, for the inhomogeneous part, into vector F, the solution. Because of linearity we expect that a solution of 2(F) = -471" ;:
L p(ro)e(ro)
is F =
xoyozo
L
p(ro)G(rlro)
(7.1.5)
Xoyozo
a sum of all the individual solutions for unit vectors on the right-hand side, multiplied each by the appropriate amplitude p(ro). Consequently, we should expect that a solution of the inhomogeneous differential equation (7.1.2) would be F(x,y,z) =
f JfG(x,y,zlxo,yo,zo)p(xo,yo,zo)
dxo dyo dz«
(7.1.6)
where G is the solution of Eq. (7.1.4) and is called a Green's function. It is thus apparent that, in terms of abstract vectors, a solution by Green's functions is a representation in terms of the unit vectors e(x,y,z) whereas a solution by eigenfunctions is a representation in terms of the unit vectors en' A much more complete discussion of this representation will be given at the end of this chapter. Of course, it will take the rest of this chapter to outline how the unit solutions G are determined, when the representation converges , and all the other precautionary details analogous to the ones we had to learn, in the last chapter, before we could use the eigenfunction technique with confidence. Boundary Conditions and Surface Charges. We have not yet shown how the solution of an inhomogeneous equation (with homogeneous boundary conditions) will help us to solve a homogeneous equation with inhomogeneous boundary conditions. Perhaps a simple example will clarify the principle before we immerse ourselves in details. It will be shown later that a solution of the Poisson equation V2(]
= -471"o(r - rs)
with homogeneous Dirichlet conditions (G = 0 on the surface S in Fig. 7.1) is a function which goes to infinity as 1/lr - rol for r near the source point roo What we wish to point out here is that this same solution G can be used to build up a solution for an arbitrary charge distribution inside the surface S and also for an arbitrary set of Dirichlet boundary conditions on S (that is, for I/; = 1/;. on the surface). What is done is to replace the inhomogeneous boundary conditions on S by homogeneous conditions, together with a surface distribution of charge just inside the surface. Let us magnify up the situation near the surface, as shown in Fig. 7.2. The charge distribution to replace the inhomogeneous boundary conditions is a surface layer of density rT/E spaced a very small distance E out from the surface S . We make E much smaller than the radius of curvature of the surface and also smaller than the distances over which a varies appreciably. We have replaced the
796
Green's Functions
[CR. 7
inhomogeneous conditions by homogeneous conditions plus this charge layer, so we now require that the potential be zero at the surface. For distances of the order of E, the surface is plane (and may be taken as the y, z plane) and the charge density 0' may be considered to be uniform.
Observation Point
Source Point
Origin
Fig. 7.1 Source point, observation point, and boundary surface for Green's function.
We thus have the situation of a uniform charge sheet ~urface density 0'/ E a distance E in front of a grounded plane conductor" at x = O. From elementary electrostatics we remember that-the normal .gradient of the potential changes by an amount 471"0'/ E when going through such a sheet of charge. Because E is so small, the gradient between charge and boundt-- E ----i
------,rl
I 1
I
S
I
I I I
I I I
x-
I
Charge l.,ayer o:
Boundary Surface S
Fig. 7.2 Potential of a source layer a a small distance outside a grounded surface S .
e
ary must be very much larger than the gradient outside the sheet; in fact we can neglect the latter with respect to the former. Therefore, the gradient between x = - E and x = 0 must equal -(47rO'/E), and the potential in this region must be 1/;
= - (471"0'/E)X; -E <
X
Y gm(yI7J) = m sinh(1I"mbja) sinh(1I"m7J ja) sinh[(1I"m ja)(b - y)]; 71 < Y
The function gm(yI7J) goes to zero for either y or 71 either zero or b, and it has a discontinuity in slope (see page 123) of an amount ( -411") at y = 71 . Finally, inserting all these solutions back in Eq. (7.1.11), we have the simple form for the solution (7.1.12)
where
l:
8s . m (1I"mx) (1I"m~) - s. m -
oo
a
m=O
a
m sinh(1I"mbja) . . { sinh(1I"myja) sinh[(1I"mja)(b - 71)] ; 71 > Y sinh(n·m7Jja) sinh[(1I"mja)(b - y)]; 71 < Y
Since this integral for if; has the same form as the integral in Eq. (7.1.10), the G given here must equal the G given there. A simple (but tedious) process of expansion of gm into a Fourier series in y will show that the two G's are indeed identical. This latest expression for G, 'however, is best arranged to display the relation between the solution of an inhomogeneous equation with homogeneous boundary conditions and the solution, given in (7.1.8), of the homogeneous equation with inhomogeneous boundary conditions ; for if the only charge inside the boundaries is a sheet of charge, of surface density (l j411"E)if;bW a vanishingly small distance E away from the surface y = b, then at 71 = b - E, which is the only place where P differs from zero, G(x , yl~, b - E)
~
\ ' 811"E sin (1I"mx) .
r->O ~ a
a
m
. (1I"m~) sinh(1I"myja) . ·sm a sinh(1I"mbja) '
y ec 2ikh
-r»
(7.2.27)
The integral may also be evaluated exactly in terms of the Fresnel integrals. If C(u)
then
IN°O
=
lU
cos
(~ t
2 )
dt ;
S(u)
=
lU
sin
(~ t
2 )
dt
(7.2.28)
e; dv = ~:n [i - C(~4k:N) + ii - is (~4k:N)]
Employing the simpler expression (7.2.27),
~
becomes (7.2.29)
We thus see that, when kh is much larger than unity, the whole of series (7.2.24) can be expressed in terms of the simple expression (7.2.29) with N = 1. For the wave equation, k = 211"/X where X is the wavelength, so that the simple expression may be used for the whole sum when X is much smaller than h, the spacing between the plates. The only term not included in ~ is then the one for n = 0, the direct effect of the source on the observer. To repeat, when h» X and when Ix - xol «.); then the value of if; at the observation point (x,y) equals the direct term 1I"iHo(klr - rol) plus a small correction proportional to ~. Other Expansions. In the event that either or both of these conditions are not satisfied, more drastic treatment must be applied. If r is at some distance from the source and kh is neither large nor small, the image expansion may be rearranged to obtain a more convergent
§7.2]
Green's Functions for Steady Waves
817
series. This may be performed with the aid of the Poisson sum formula [Eq. (4.8.28)];
. 2: j (27rn ) = - . F(v) =
where
1 27r
.
2:
p=-
f-.. .
F(v) 110
(7.2.30)
j(r)e-il'T dr
To apply the Poisson sum formula to this problem, the Fourier transform of Ho(klr - roD is required. We shall show later on in this chapter (see page 823) that
roD
Ho(klr,-
=
if" dK", f_. .. dK [k2 _ K2
eiK·(r-ro) ]
7r 2
y
_ ..
(7.2.31)
The integral is not completely defined without specifying the manner in which the pole at K = k is to be circumvented. This will be done in the course of the calculation. We must now evaluate
1=
f-.. .
e- i PT {Ho [k
~(x -
xo -
;hY +
(y - YO) 2]
+ H 0[ k ~ (x + xo -
;
h) 2+ (y -
Yo) 2]}
dr
Introducing (7.2.31) into the integrand, I becomes
The r integral may be immediately performed in terms of the delta function (actually we are just using the Fourier integral theorem) :
The K", integral is easily evaluated by employing the l> function property
!-.. .
l>(z)j(z - a) dz
=
j(a).
Then
818
Green's Functions
[cH.7
The final integral can be performed only after the path of integration on the K lI plane is specified. The particular path C, illustrated in Fig. 7.6, is chosen so that I satisfies the boundary condition that the point x = Xo, Y = Yo be a source only , rather than a sink or both source and sink. Ky Plane
c
Fig . 7.6
Contour C for integration of Eq. (7.2.31) .
The evaluation of this integral by means of Cauchy's integral formula [Eq. (4.2.8)] is discussed in Chap. 4, page 415. We find that '
I = ( 411") h
e- ,.,r . x/ h
eiIY-lIoh!k' ("./h)· cos (1I"Vxo) h y'k 2 - (1I"v/h)2
The final expansion for the Green's function becomes
L .,
= ( 211"i) h
E.
.=0
vxo) eilll-yoh!k'-(orv/h)' cos (1I"VX) cos (1I"h h y'k 2 - (1I"V/h)2
(7.2.32)
This result is particularly useful whenever Iy - yol » 1, for as soon as (1I"v/h) > k, terms in the series decrease exponentially. The number of terms required to obtain an accurate value for the series is thus of the order of hk/1I" = 2h/'A. [Note that, when II becomes large , the corresponding terms in (7.2.32) become independent of k.] We see that the expansion given above complements that for 2:. The expansion in images (7.2.24) is feasible if hk» 1 i expansion (7.2.32) applies when hk «1. The expansion in images is appropriatefor short wavelengths and close to the source , for then the effects of the boundary are less important; expansion (7.2.32) is appropriate for long wavelengths and considerable distance from the source. Expansion (7.2.32) is a Fourier expansion quite similar to Eq . (7.1.12) and may be obtained more directly than by the roundabout manner via images and Poisson's sum
Green's Functions for Steady Waves
§7.2]
819
rule, which we have employed here . The importance of the derivation we have given lies in its exhibition of the connection between the two types of expansions. Equation (7.2.32), unlike Eq. (7.2.29), is exactly equal to series (7.2.24), and it always converges . Unless hk /7r = 2h/A is of the order of unity, the series does not converge very rapidly and we must look around for means of improving its convergence. To this end we return to the parenthetic remark of the last paragraph, that the terms in the expansion (7.3.32) for p large are independent of k. This suggests examination of Go(rlro), the Green's function for the Laplace equation, which may often be exhibited in closed form . If we write Gk = Go
+
(Gk
-
Go)
then the expansion for (Gk - Go) will converge more rapidly than that for Gk alone . If we had chosen Dirichlet conditions at x = 0 and x = h, the corresponding static Go would come out in closed form ; Go(rlro) would then be the static potential for a unit charge at Xo, Yo, between two grounded plates. To bring out more of the difficulties, we have picked Neumann conditions, where Go corresponds to the steady flow of a fluid out of a unit source at Xo, Yo. But steady flow requires a sink (in this case at infinity) as well as a source; this has not been included. A slight modification is thus required to take the sink into account, as we shall proceed to show. We start with the series '"
fo(rlro)
~ = 4 Lt
(1);- cos
(7rpx) Ii: cos (7rpx h o) r(n lh)llI-lI.1
(7.2.33)
.=1
which is what Eq. (7.2.32) reduces to when k = 0 (except for the omission of the P = 0 term). By repeated use of the relation
we obtain where
I'e = R(x + xoly - Yo) + R(x - xoly - Yo) R(alb) = - In [1 - 2e- ..Ibil h cos (7ra /h) e-2rlbl/hj
+
(7.2.34)
It is not difficult to show (by computing V 2 f o if necessary) that f o is a solution of the Poisson equation V 2f o
= -47r[o(r - ro) - (l /h)o(y - Yo)]
(7.2.35)
which corresponds to a unit positive charge at (xo,Yo) and a unit negative charge spread uniformly along the line y = Yo, perpendicular to the two
820
[CR. 7
Green's Functions
boundary lines x = 0 and x = h. Since the entire charge distribution averages to zero between the two boundary lines we may fit Neumann conditions without having the static solution approach infinite values at infinity. It may also be seen directly that ar/ax may equal zero for x = 0 and x = h, by differentiating (7.2.35). The final expression for Gk is therefore Gk(rlro)
=
R(x
+ xolY -
+4
2: (
.
Yo)
+ R(x -
xolY - Yo)
+ (~~i) eik111-1101
1rYx) (1rYxo) {e- v(n/h) '-k'ly-Yol cos cos - 2
... =1
h
h
V
-
y
- (kh/1r)2
(~) «"...
/h) IY-lIol }
(7.2.36)
This series converges quite rapidly. Other cases, where the static Green's function turns out to be a closed expression, may be worked out from the results of Chap. 10. The method of images is restricted to boundaries which are composed of straight lines in two dimensions or planes in three dimensions. There is one exception to this rule. In the case of the Laplace equation (with Dirichlet conditions) the method of images may be applied to the circle in two dimensions and to the sphere in three dimensions. This limitation on the image method may be expected from an elementary knowledge of geometrical optics, for it is well known that the only mirror for which the image of a point sour ce is itself a point is the plane mirror. Of course this does not mean that the image method cannot be applied to other shapes, rather that it can be applied only approximately. We therefore turn to a more general representation of Green's functions, by eigenfunctions. Expansion of Green's Function in Eigenfunctions. The method of eigenfunctions discussed in Chap. 6 is limited only by the ease with which the requisite eigenfunctions can be determined. Since precise solutions are available for only the separable coordinate systems, the expansion of Green 's functions in eigenfunctions is practical for only these cases. Let the eigenfunctions be Vtn and the corresponding eigenvalues k~, that is, (7.2.37) Here n represents all the required indices defining the particular Vtn under discussion . Moreover, as was shown in the preceding chapter, the functions Vtn form an orthonormal set : (7.2.38)
§7.2]
Green's Functions for Steady Waves
821
where the region of integration R is bounded by the surface upon which Y;n satisfies homogeneous boundary conditions. The Green 's function Gk(rlro) satisfies the same conditions. In addition it is assumed that the functions Y;n form a complete set so that it is possible to expand Gk(rlro) in a series of Y;n: Gk(rlro) =
LAmY;m(r) n
Introducing this expansion into the partial differential equation satisfied byGk ,
we find that
l:A m(k 2 - k;")Y;m(r) = -411'o(r - ro)
Employing Eq . (7.2.38), we multiply both sides of the above equation by If;n(r) and integrate over the volume R . Then A n
so that
= 4~n(ro) k~ _ k2
Gk (rlr) = 411' 0
2: n
If;n(ro)Y;n(r) k2 _ k 2 n
(7.2.39)
the desired expansion. An example of such an expansion was given in Eq. (7.1.10). One unexpected feature of Eq. (7.2.39) is its unsymmetrical dependence on rand ro for complex Y;n in face of the proof given earlier that Gk must depend symmetrically on these variables. This is, of course, not a real dilemma. The solution lies in recognizing that, since the scalar Helmholtz equation does not involve any complex numbers explicitly, If;n is also a solution of (7.2.38) and therefore will be included as one of the orthonormal set Y;n. We have here a simple case of a degeneracy, for to one eigenvalue k~ there belong two eigenfunctions, If;n as well as Y;n. Thus included in the sum (7.2.39) there will be the term If;m(ro)Y;m(r) k;" - k 2
and also the term
If;m(r) Y;m (ro) k;" - k 2
so that actually (7.2.39) is symmetric and real. Another matter of interest is the behavior of Gk as k ~ k«. We see that, as a function of k, Gk is analytic except for simple poles at k = ± k«, with residues +211'lf;n(r')Y;n(r) jk n. Thus if it should happen that a Green's fun ction is known in a closed form, the eigenfunctions Y;n and the eigenvalues k« may be found by investigating Gk at its poles. . The singularities have a simple physical interpretation, for they are just the infinities which occur when a nondissipative vibrating system is driven at one of its resonant frequencies. To make this correspondence
822
Green's Functions
[cH.7
more explicit, we recall that the partial differential equation satisfied by the velocity potential set up by a point source at ro, with angular frequency w, is
But 1/; = e-u"tGk , k = we. Hence if k = k n , the system is being driven at one of its resonant frequencies and gives an infinite response if the system has no friction. There is one situation for which the response will not be infinite. This occurs if the source space-dependence is orthogonal to l/;n. For if 1/; satisfies then 1/; =
f
p(h
dV o = 411"
LNm~~~r~2
o dV 1/;m(r)
(7.2.40)
m
The nth term vanishes if Nn(r)p(r) dV = O. Then the nth term in series (7.2.39) is missing, and k can equal k; without Gk becoming infinite. The Green's function for such problems (we shall use here the term "modified Green's function" and the notation rkJ satisfies
where by m ~ n we mean that all terms for which k; = ± k« are to be left out. We have already discussed one such case for the Green's function for the Laplace equation when the boundary conditions were homogeneous Neumann. In that case one of the eigenvalues was k = 0, corresponding to a constant for an eigenfunction. The k for the Laplace equation was also zero, and we found it advisable to use the modified Green's function roo We shall now give some examples of the application of (7.2.39). These are fairly simple when completely enclosed regions are under discussion, for when the eigenfunctions and the corresponding eigenvalues are known, it is only necessary to normalize the 1/;'s in order to be able to fill in the formula. An example of the expansion under these circumstances is given in Eq. (7.1.10). Expansions for the Infinite Domain. We therefore turn to other types of regions, of which the simplest is unbounded and infinite. We showed earlier in this section [Eq. (7.2.18)] that the two-dimensional Green's function for this case, for a source (as opposed to a sink) is
Green's Functions for Steady Waves
§7.2]
823
7l"'iH o(k lr - rol). A possible complete orthonormal set m which to expand this is furnished by the plane wave [l j(27l')]eiK .r
where K . r = K,» + Kyy where K", and K; may assume any numerical" value. To obtain a complete set it is necessary, according to the Fourier integral theorem, to have the range in K extend from - 00 to + 00 following a route in the complex planes of K", and K; joining these two points. Ky Plane
Ky =+/k 2_K 2x
Fig. 7.7
Contour C for integration of Eq . (7.2.42).
Since K", and K; are continuous variables (see page 762 for the discretecontinuous transition), the sum in (7.2.39) must be replaced by an integral : g~(R)
f '" f'"
. = t7l'Ho(kR) =:;;:1 _., d.K;
iK R
_., dK y K2e _• k 2
(7.2.42)
This representation was utilized in Eq . (7.2.31) (K 2 = K; + K;) . Again it is important to note that the integral is undefined unless the path of integration about the poles of the integrand is specified. The K y path of integration is given in Fig. 7.7. It is chosen so as to lead to an outgoing wave from the source point R = O. The K; integrations may then be performed to yield, for Y > 0,
Let K", = k cos
(~
+ tjJ)
where tjJ = tan- 1 (YjX)
7l'Ho(kR) =
t': e »:
ik R
C08"
d~
The contour of integration for ~ must, of course, be such as to yield a convergent integral. In view of the original limits, it must go from -i 00 to +i 00. Convergence is obtained by running the contour somewhat to the left of the imaginary axis in the upper half plane and to the
824
Green'« Functions
[cH.7
right of it in the lower half plane as is illustrated in Fig . 7.8. Our final result is the well-known integral representation for the Hankel function [see Eq. (5.3.69)] : H o(kR) Polar Coordinates.
=
(71"1) -
/,,/2- i .. i k R e cos " dlJ -../2+i ..
(7.2.43)
Equation (7.2.42) is the proper representation of the Green's function Yk(r) for t'iPlone a two-dimensional infinite region, to be employed in problems for which rectangular coordinates are most appropriate. Let us now apply the general formula (7.2.39) employing the eigenfunctions appropriate to polar coordinates. These are
One must also normalize. The normalization factor for the ep dependenceis I/Y271". The normalization factor N m for the radial dependence is obtained from the equation appropriate to continuous eigenvalues (in this case k) :
Fig. 7.8 Contour B for integral represcntation of Ho(kR).
limlim {N;' Ak-+O,R-+ ..
(k+t>k jk-Ak
dk , (rJm(kr)Jm(k'r)rdr} = I jo
The value of the indefinite integral over r is (see formulas at end of Chap. 11)
f
' ) d = k'rJm(kr)Jm_l(k'r) - krJm(kr')Jm+t(kr) J m (k r )Jm (k r r r k2 _ k'2
Since R is large, the asymptotic behavior, Jm(x) ~ y2/7I"x cos [x p(m + -!)], given in Eq. (5.3.68) may be used . We find that N'; is independent of m and is equal to 0. Hence the normalized eigenfunctions are (7.2.44) yk/271" e-«:m(kr) Substituting (7.2.44) into (7.2.39) yields
.
i7l"H o(kR ) =
(;71")
2:
eim(4)-
4> o)
f-.. . Jm(~~)~mk~ro)
K dK
m=-ao
where the contour of integration is still to be specified. Actual evaluation demonstrates that, for outgoing waves, the required contour is just the one illustrated in Fig . 7.7 with the poles at ± k. The above expansion
Green's Functions for Steady Waves
§7.21. may be rewritten
80
as to involve only positive m, as follows :
L 00
i7rH o(kR) =
(2~)
825
.I.
Em COS [ m ( '1'
_
.I. )]
'1'0
f
_
00 00
J m(Kr)Jm(Kro) K dK K2 _ k 2
m=O
(7.2.45) It is possible to evaluate the K integral by methods of function theory. However, the procedure is just a difficult way to do a simple calculation. It is preferable to derive expansion (7.2.45) by another procedure, which may be extended to other coordinate systems and boundary surfaces. A General Technique. The method is just that employed in Sec. 7.1 to establish a connection between the surface Green's function G(x,yl~) [Eq. (7.1.8)] and the volume Green's function [Eq . (7.1.10)]. We expand the volume Green's function in terms of a complete set of functions involving all but one of the coordinates (in the present case there are only two coordinates, rand 1jJ) with coefficients which are undetermined functions of the uninvolved coordinate. Thus let
L(2~) 00
gk(R) =
(7.2.46)
eim(,;-,po)Pm(rlro)
Comparing with (7.2.45) we note that pm (r Iro)
=
f
J m(Kr)Jm(KrO) K dK K2 _ k2
00
_
00
Introduce (7.2.46) into the equation for gk(rlro): 'V 2gk + k 2gk = -47ro(r - ro) In polar coordinates this becomes
! ~ [r i}gk] + J:2 i} 2gk + k2 k = r i}r
i}r
r i}1jJ2
_ 47ro(r - ro) o(IjJ g r
-
IjJQ)
(7 2 47) ' .
[The right-hand side of this equation involves the expression of the 0 function in polar coordinates. This expression must satisfy the requirement that t5(r - ro) must vanish unless r = ro and IjJ = ljJo and must integrate to unity over all space, f f oCr - ro)r dr dljJ = 1. It is easy to verify that these requirements are satisfied.] Inserting expansion (7.2.46) into Eq. (7.2.47) one obtains
47rt5(r - ro) o(IjJ r
-
ljJo)
826
Green's Functions
[cH.7
Multiply both sides of this equation by e-in~ and integrate over cP from The integration on the left-hand side involves the orthogonal properties of the set eim~. We obtain
o to 2?r.
!!!.
r dr
(r
~Pm) + (k2 dr
2) _ m pm = _ 411"o(r - ro) r2 r
(7.2.48)
We see that Pm(rlro) is a one-dimensional Green's function for a SturmLiouville operator (d/dr)r(d /dr) + r[k2 - (m 2/r 2)] [see Eq. (6.3.12)]. The solution of the inhomogeneous linear second-order differential equation £(if;) = v is given in Eq. (5.2.19) as if;
= Y2
!Z
VYl dz A(Yl,Y2)
+
Yl
f
z
VY2 dz A(Yl,Y2)
where z is the independent variable, Yl and Y2 are the two independent solutions of the homogeneous equation
!!. z dy + Z (k2 dz
dz
2) _ m Y Z2
=
0
and A(Yl,Y2) is the Wronskian A(Yl,Y2) = IYl Y2
Y:I Y2
The function v is the inhomogeneous term, in this case -411"o(r - ro) /r. The limits of integration in the expression for if; depend upon the particular choice of independent functions u, and Y2, and the boundary conditions on pm. We shall take the limits (this is permitted since we have not chosen Yl and Y2) to be less than z( =r) in the first integral and greater than z in the second. Thus
For r < ro, the first integral vanishes, while for r integral vanishes. Hence P _ m -
-411" { Yl(r)Y2(rO); rOA(Yl,Y2) Y2(r)Yl(rO);
r::; ro r ~ ro
> ro, the second (7.2.49)
The Wronskian is evaluated at roo Boundary conditions determine which of the solutions of the homogeneous equation are to be employed. In the case under discussion the solutions are the Bessel functions J m(kr), Nm(kr) or any linear combination. The boundary conditions are (1) pm is to be finite at r = 0, since the only singularity of gk occurs at r = ro, and (2) the point r = ro must be a source, since gk(R) has been taken to be a diverging wave. Hence Yl = J m(kr) and Y2 = Hm(kr).
Green's Functions for Steady Waves
§7.2]
827
Finally we must evaluate tl(Yl,Y2) at r = roo It is useful to use the relation Eq. (5.2.3) , giving the space dependence of the Wronskian tl(z) = tl(zo)[f(zo)/f(z)]
where the differential equation satisfied by Yl (or Y2) is
~ 0~~) + qy
=
0
In the case under discussion f = r, so that tl(Yl,Y2) = constant/r. To determine the constant, one may employ the first terms of either the power series about the origin (r = 0) or the asymptotic series about r = 00, since the relation tl(Yl,Y2) = constant/r must be satisfied for each term of the power or asymptotic series for tl(Yl,Y2). To illustrate, we utilize Jm(kr) - - ----t V2/7rkr cos [kr - in"(m + kr->
Hm(kr)
-m
00
-~
kr->
V2 /7rkr ei1kr-!T,m+! )]
00
The Wronskian is asymptotically
(7r~r)
l-
cos [kr -
in"(m
+ ~) )ei[kr-!T(m+m + k sin
[kr - in"(m
This equals 2i/7rT so that tl(Yl,Y2) = 2i/7rT. results,
+ ~))eilkr-!T(m+m}
Finally, collecting all our
- 2 2 ' {Jm(kr)Hm(kr o); r ~ ro p.; - 7r '/, J m(kro) H« (kr) ; r ~ To
(72 0) . .5
Note that pm is the value of the integral occurring in (7.2.26). Introducing (7.2.50) into the expansion for gk(k) = i7rHo(kR), an expansion for H o(kR) is obtained : Ho(kR) =
~
Ho(kR) = mi..:.
~ eim(~-~o) -L..
{Jm(kr)Hm(kro) ; r ~ ro Jm(kro)Hm(kT); r ~ ro
cos[m(q, - q,0)]
{ J m(kr)Hm(kro) ; r::; To Jm(kTO)Hm(kr); r ~ TO
(7.2.51) o We have given the derivation of (7.2.51) in detail because it will serve as the prototype for the calculation of expansions of other Green's functions. These expansions are of considerable use, as the following calculat ion will show. We shall derive the expansion of ei b (a plane wave traveling from left to right) in polar coordinates and then the integral representation of the Bessel function J m [described in Eq. (5.3.65)). We note that Ho(kR) represents a wave traveling from the Em
828
Green's Functions
[cH.7
source at ro. To obtain a plane wave traveling from left to right it is necessary to place the source at - 00, i.e., let ro -7 00 and cfio -711'. Then R = .yr2
-
+ r3 ro'-+
2rro cos (cfi - cfio)
------t
ro
00
(1 + .:.. cos ro
cfi) = ro
+X
q,o' = rr
Hence
Ho(kR)
------t
ro--.+
00
I2 '\J1I'kro
e i1k(ro+z)-lrJ
q,o =rr 00
\ ' Em( -1)m
~
cos(mcfi) J m(kr)
m=O
L
I 2 eilkro-jr(m+iH '\J1I'kro
00
or
i kz
e
=
rn Emi
(7.2.52)
cos(mcfi) Jm(kr)
m=O
the desired expansion. This series has been given in Chap. 5 in another form [see Eq. (5.3.65)]. Finally, by employing the orthogonality properties of the set cos(mcfi) it is possible to derive an integral representation for J m(kr). Multiply both sides of Eq. (7.2.52) by cos(vcfi) , and integrate from 0 to 11'. Then Jv(kr)
= i-v 11'
rr
)0
e ikrc•• .p
cos(vcfi) dcfi
(7.2.53)
This relation was derived in another fashion in Chap. 5 [see Eq. (5.3.65)]. A General Formula. Let us now turn to the problem of deriving the expansion of the Green's function for any of the generalized coordinate systems for which the scalar Helmholtz equation is separable. We need to review some of the results of the discussion of separation (see pages 655 et seq.). If h , ~2, and ~a are three orthogonal, generalized coordinates, with scale factors hl, h2, and h a, then the Laplacian is 3
v
2.1. _ 'I' -
\'
u; a
~ Sin a~n
(In a~na1/;)
(7.2.54)
n=l
The quantities In are functions of ~n only (that is, It is a function of ~l only) ; S is the Stackel determinant [Eq. (5.1.25)] whose elements nm are fun ctions of ~n only (that is, lm is a function h only); M n is the minor of S which multiplies nl in the expansion of S [Eq . (5.1.26)]. M 1 is a function of ~2 and ~a but not of ~l. The scalar Helmholtz equation separates in coordinates h , ~2, ~a, so that 1/; = Xl(~l)X2(~2)Xa(~a)
where
In~n d~n [In ~~nn ] +
3
2: nm(~n)k;'
m=l
=0
(7.2.55)
§7.2]
Green's Functions for Steady Waves
829
where k~ = k 2 and k~ and ki are two separation constants. The factors fn and the elements of S are given in the table at the end of Chap. 5. We shall also need the Robertson condition (5.1.32), h 1h2h a = S fdda. To see how we expand a Green's function in these general coordinates it is well to return to the derivation of Eq. (7.2.51), the expansion of a Green's function in polar coordinates. There the cP factors turned out to be eigenfunctions, independent of the constant k in the Helmholtz equation. The r factors, on the other hand, depended on k and on the eigenvalues m of the cP factors and for these and other reasons could not be made into eigenfunctions. The function G was then expanded into an eigenfunction series in the cP factors. The r factors for each term in the series then satisfied an inhomogeneous equation which could be solved in terms of the two solutions of the homogeneous equation, and the expansion was achieved. We try the same procedure with the three general coordinates ~l, ~2 , ~a. Of the three separation constants, k 1 = k, k 2 , k a, the first one, k, is fixed in value by the Helmholtz equation which we are solving. The other two, k 2 and k a, are available to become eigenvalues for a set of twodimensional eigenfunctions, in terms of which we are to expand the Green's function G. Usually the choice of which two of the three coordinate factors are to be eigenfunctions is an obvious one. In spherical coordinates, for instance, two of the three coordinates, cP and {}, are angles , with finite range of values and simple boundary conditi ons (periodicity and finiteness) which may be imposed to obtain eigenfunctions. In other cases (such as the circular cylinder coordinates, r, cP, z) only one of the coordinates (cP for the circular cylinder) has a finite range of values, and one of the other two (z, for instance) must produce a set of eigenfunctions for an infinite domain, having a continuous range of eigenvalues for one of the separation constants k 2 or k a• Suppose we find that the ~2 and ~a fa ctors may be made into eigenfunctions, with corresponding pairs of eigenvalues for k 2 and k a. We order these eigenfunctions in some manner, with respect to the allowed values of k 2 and k a; for instance, the lowest value of k 2 may be labeled k 20 , the next k 21 , and the mth k s«, whereas the allowed values of k a are k ao, k a1, . . • , k an • • • ; the eigenfunction corresponding to k 2m , k an being X2m(~2)Xan(~a) though X 2 may also depend on nand X a on m. To simplify the notation we label the pairs of integers (m,n) by a single letter, p or q, and express the eigenfunction product by a single letter W. Then the pth eigenfunction for the coordinates ~2, ~ a is W p(~2,b) with eigenvalues k 2p , k ap • In what follows we shall assume that both ~2 and b have finite ranges of values, so that both k 2p and k ap have a discrete set of values and the eigenfunction expansion is a series over p (m and n) . The extension to cases where one or both of the eigenvalue sequences is a continuum, so that the expansion is an integral (like a Fourier integral,
Green's Functions
830
[CR. 7
instead of a Fourier series), is one which is not difficult in any particular case. Therefore we assume the existence of a complete set of eigenfunctions Wq(b,h) (we chose ~2, and ~3 as examples, any pair for which the necessary conditions are satisfied) satisfying the orthonormal condition (7.2.56)
where p is a weight function. (See page 781 for a discussion of weight functions in one dimension.) We shall assume that this set of eigenfunctions exists for arbitrary k for the range in ~2 and h in the domain of interest. The function W q may depend upon k, of course . Then in analogy to (7.2.46), we write
Gk(rlr')
=
LX1q(~1IWBq(~~,~~)Wq(~2,~3)
(7.2.57)
q
where the functions X q and B; are to be determined. In generalized coordinates the equation determining Gk is 3
~ Mn ~
L.t 8fn a~n n=l
[f
aG] + k'KJ =
n
a~n
-4,r
O(~l - ~Do(b - ~~)O(~3 - ~~) h 1h2h 3
1
(7.2.58)
Note that the representation chosen for oCr - r') as given by the coefficient of -411" is such as to vanish unless all three coordinat es ~1, ~2, ~3 equal the corresponding primed coordinates, respectively, and so that its integral over all of space is unity. We now pro ceed to introduce series (7.2.57) into Eq. (7.2.58). We require the result of applying ('V2 + kD to W q(~2, ~3). Only two of the terms in the sum in (7.2.58) involve derivatives of W q . Since W q is a product of X 2(b) and X 3(~3), solutions of Eqs. (7.2.55) , it follows that
2: z: a~n 3
[fn
n=2
aa~q]
=
-
3
3
n= 2
m=l
2: ~n {2: ~nm(~n)k~q}
Wq
where the separation constants have been given the additional label q to indicate their correspondence to function W q. The sum over n may be simplified by utilizing the properties of the determinant 8 [see Eq . (5.1.27)].
LM n~nm = 801m n
Then
~ Mn~
L.t
n=2.3
8fn a~n
[f aw a~n
q
n
]
-kiW q +
~l
[2: k;"q~lm] m
Wq
§7.2]
Green's Functions for Steady Waves
831
Hence, upon substitution of the series (7.2.57) into (7.2.58) one obtains
LBg(~,~~)Wq(b,~3) ~1 D1 d~1
d~~qJ +
[h
L 3
k;'iP1m(h )X 1q}
m=l
q
= -411" o(h -
~~)o(b - WO(~3 - ~~)
h 1h2h3
By employing the orthonormality condition of (7.2.56) and the Robertson condition (5.1.32) it is found that
Bq(~~,~~) = ;1~~~,W)~tit»);t~~)
(7.2 .59)
and d [I Tr1 dh
3
dX1 q] 1 dh
\ ' k2 ] + [l..i mq1m
X 1q -__ (411") /l 0(~1 _ ~l')
(7.2.60)
m=l
Thus, as in the example in polar coordinates discussed above, X 1q is a one-dimensional Green 's fun ction. Following the procedure employed in solving (7.2.48), it is possible to express X 1q in terms of two independent solutions (Yl and Y2) of the homogeneous equation
We obtain
X (~I~/) = _ 1q
1
1
411"
!:J.(Y1q,Y2q)/1(~;)
{ Y1q(~1)Y2q(W;
(7.2.61)
Y1g(~~)Y2q(~1) ;
where !:J. is the Wronskian evaluated at ~/1' As in the discussion on page 826, which solutions Y1 and Y2 are used depends upon the boundary conditions of the problem. The form of the result for the Wronskian is !:J.(Y1,Y2) = constant//l
(7.2.62)
so that the factor !:J.(Y1,Y2)/l is a constant. The expansion for Gk(rlr /) is then Gk(rlr
/)
= -411" (h~~3)
p(~~,W
Lwq(~~,~)
W
q(b,~3) .
q
!:J.( 1 ) Y1q,Y2q
{Y1q((~;»Y2g((~D) ; Y1q ~1 Y2q
.(x!xo) has simple poles at A = An with residues -471"Y;n(x),yn(XO)' These very singularities must be present in the closed form (7.2.68) . Hence by examining (7.2.68).it is (in principle) possible to obtain the eigenvalues An and, in addition, the corresponding eigenfunctions ,yn already normalized.
Green's Functions for Steady Waves
§7.2]
833
Let us clarify the suggested procedure by means of a simple example. Let (7.2.65) be and the boundary conditions if; = 0 at x = a, x = b (a < b). Then the appropriate Yl is sin [ 0 (x - a)] and Y2 = sin [V}; (x - b)]. The value of I::>.(Yl,Y2) is
o
[0 (x - a)] cos [0 (x - b)] - 0 cos [0 (x - a)] sin [0 (x
sin
- b)]
=
0
sin
[0 (b -
a)]
Therefore G>.(xlxo)
=
o
-47r ' sin [VA (b - a)]
sin
[0 (x
- a)] sin
[0 (Xo
sin
[0 (Xo
- a)] sin
[0
1
- b)]; X S Xo (x - b)] ; x :2: Xo
The eigenvalues occur at the zeros of sin [0 (b - a)] , so that one obtains the familiar result vx.; = [n7r/(b - a)J. The residue here is -[87r/(b - a)]( -I)n sin [n7r(x - a) /(b - a)] sin [n7r(xo - b)/(b - a)] ,
or -[87r/(b - a)] sin [n7r(b - x) /(b - a)] sin [n7r(b - xo)/(b - a)] : Hence if;n(x)if;n(xo) is [2/(b - a)] sin [n7r(b - x)/(b - a)] sin [7rn(b - xo)/(b - a)) so that the normalized eigenfun ction is .1. 'l'n
=
I 2 . [n7r(b - X)] "Vb - a sin b- a
These eigenfun ctions satisfy orthogonality condition (7.2.66) with r = 1 and form an orthonormal complete set. We thus see that, if any two solutions whose combination satisfies the proper boundary conditions (actually u, satisfies those which exist at one boundary point, Y2 at the other) can be obtained, and if the evaluation of the Wronskian can be obtained, then the normalized orthogonal eigenfun ctions and eigenvalues may be found. Comparing this method with the more usual methods discussed in Chap. 6, we find that the amount of labor is the same for .all. However, the method just described also yields the normalization which often involves a difficult integral in the other methods. We shall 'have occasion to employ this method for more complex functions in the problems for this chapter and for the determination of eigenfunctions and their normalizations which arise in problems of two and higher dimensions. We shall also encounter the same procedures under a somewhat different guise in Sec. 11.1.
834
Green's Functions
[cH.7
7.3 Green's Function for the Scalar Wave Equation The Green's function for the scalar Helmholtz equation, just discussed in Sec. 7.2, is particularly useful in solving inhomogeneous problems, i .e. problems which arise whenever sources are present within the volume or on the bounding surface. The Green's fun ction for the scalar wave equation must perform a similar function ; thus it should be possible to solve the scalar wave equation, with sources present, in terms of a Green's function. To obtain some notion as to the equation this function must satisfy let us consider a typical inhomogeneous problem. Let Vt' satisfy V 2•1• 't'
-
1
-
c2
a1/t = -41rq(r t) at2 ,
-
(7.3.1)
The function q(r,t) describes the source density, grving not only the distribution of sources in space but also the time dependence of the sources at each point in space . In addition to Eq. (7.3.1) it is necessary to state boundary and initial conditions in order to obtain a unique solution (7.3 .1). The condition on the boundary surface may be either Dirichlet or Neumann or a linear combination of both. The conditions in time dimension must be Cauchy (see page 685, Chap. 6) . Hence it is necessary to specify the value of Vt and (aVtj at) at t = to for every point of the region under consideration. Let these values be Vto(r) and vo(r), respectively. Inspection of (7.3.1) suggests that the equation determining the Green's function G(r,tlro,to) is 1 a2G V2G - - 2 - 2 = -41ro(r - ro)o(t - to) c at
(7.3.2)
We see that the source is an impulse at t = to, located at r = roo G then gives the description of the effect of this impulse as it propagates away from r = ro in the course of time. As in the scalar Helmholtz case, G satisfies the homogeneous form of the boundary conditions satisfied by Vt on the boundary. For initial conditions, it seems reasonable to assume that G and aGjat should be zero for t < to; that is if an impulse occurs at to, no effects of the impulse should be present at an earlier time . It should not be thought that this cause-and-effect relation, employed here, is obvious. The unidirectionality of the flow of time is apparent for macroscopic events, but it is not clear that one can extrapolate this experience to microscopic phenomena. Indeed the equations of motion in mechanics and the Maxwell equations, both of which may lead to a wave equation, do not have any asymmetry in time. It may thus be possible, for microscopic events, for" effects" to propagate backward in
835
Green's Funclionfor the Scalar Wave Equation
§7.3]
time; theories have been formulated in recent years which employ such solutions of the wave equation. It would take us too far afield, however , to discuss how such solutions can still lead to a cause-effect time relation for macroscopic events. For the present we shall be primarily concerned with the initial conditions G(r,tlro,to) and aGlat zero for t < to, though the existence of other possibilities should not be forgotten. The Reciprocity Relation. The directionality in time imposed by the Cauchy conditions, as noted above, means that the generalization of the reciprocity relation Gk(rlro) = Gk(rolr) to include time is not G(r,tlro,to) = G(rotolr,t) . Indeed if t > to, the second of these is zero. In order to obtain a reciprocity relation it is necessary to reverse the direction of the flow of time, so that the recipro city relation becomes G(r,tlro,to)
= G(ro, - tolr,-t)
(7.3.3)
To interpret (7.3.3) it is convenient to place to = 0. Then G(r,tlro,O) = G(ro,Olr, -t) . We see that the effect, at r at a time t later than an impulse started at ro, equals the effect, at ro at a time 0, of an impulse started at r at a time -t, that is t earlier. To prove (7.3.3) let us write the equations satisfied by both of the Green's fun ctions : -47ro(r - ro)o(t - to)
Multiplying the first of these by G(r, -tlrl, -h) and the second by G(r,tlro,to) , subtracting, and integrating over the region under investigation and over time t from - ao to t' where t' > to and t' > h , then
f~
. f dt
dV {G(r,tlro,to)V2(i(r,-tlrl,-tl) - G(r,-tlrl,-tl)V2G(r,tlro,t o)
a G(r, + C21 G(r,tlro,to) iJt2 2
1 a2 G(r,tlro,to) } tlrl ' - h) - C2 G(r, - tlrl' - t l ) iJt2
= 47r{G(ro,-tolrl,-t l) - G(rl,tllro,t o)I (7.3.4) The left-hand side of the above equation may be transformed by use of Green's theorem and by the identity
Green's Functions
836
[cH.7
We obtain for the left-hand side
f~
00
+ !c2
dt
f as-
f
- G(r, -tlrl, -t l) grad G(r,tlro,to)] l) dV [G(r, tlro, t) aG(r, -tlrl' 0 at -t - G(r, -tl rl, -t I ) aG(r,tlro,to) at
[G(r,tlro,to) grad G(r, -tlrl, -t l )
r
~_
The first of these integrals vanishes, for both Green's functions satisfy the same homogeneous boundary conditions on S. The second also vanishes, as we shall now see. At the lower limit both G(r,- 00 Iro,to) and its time derivative vanish in virtue of the causality condition. At the time t = t', G(r,-t'lrl,-tl) and its time derivative vanish, since -t' is earlier than -tl • Thus the left-hand side of (7.3.4) is zero, yielding reciprocity theorem (7.3.3). We shall demonstrate that it is possible to express the solution (including initial conditions) of the inhomogeneous problem for the scalar wave equation in terms of known inhomogeneities in the Green's function. We shall need Eq . (7.3.1): 2.1,(
VO'Y
also
V 2rt( fP
. tl t) _ r, ro, 0
)
1 a21J; _
ro,to - C2 at3 -
!c2 a2G(r,tlro,to) at3
=
This last equation may be obtained from (7.3.2) by the use of the reciprocity relation. As is usual, multiply the first equation by G and the second by y; and subtract. Integrate over the volume of interest and over to from 0 to tr. By the symbol tt we shall mean t + E where E is arbitrarily small. This limit is employed in order to avoid ending the integration exactly at the peak of a delta function. When employing the final formulas, it is important to keep in mind the fact that the limit is t+ rather than just t. One obtains
Again employing Green's theorem, etc ., we obtain
(t+
Jo dto
¢ dS o ' (G grads '" -
'" grads G)
+ 4?r
f It+ f + C21
dt«
[aG ay; Jt+ dVo ato'" - G ato 0 dYoq(ro,to)G = 4?r1J;(r,t)
§7.3]
837
Green's Funciionfor the Scalar Wave Equation
The integrand in the first integral is specified by boundary conditions. In the second integral, the integrand vanishes when t = r: is introduced by virtue of the initial condit ions on G. The remaining limit involves only initial conditions. Hence, 4mj1(r,t) = 411"
It+ dto f dVoG(r,t!ro,to)q(ro,to) + It+ dt« ¢ as, . (G grad, t/; -
- bf zv, [G~}._o
t/; grads G)
t/;o(ro) - Gt._ovo(r o) ]
(7.3.5)
where t/;o(r o) and vo(ro) are the initial values of t/; and at/;/at. Equation (7.3.5) gives the complete solution of the inhomogeneous problem including the satisfaction of initial conditions. The surface integrals, as in the Helmholtz case, must be carefully defined. As in that case we shall take a surface value to be the limit of the value of the function as the surface is approached from the interior. The first two integrals on the right side of the above Eq. (7.3.5) are much the same sort as those appearing in the analogous equation for the case of the Helmholtz equation. The first represents the effect of sources; the second the effect of the boundary conditions on the space boundaries. The last term involves the initial conditions. We may interpret it by asking what sort of source q is needed in order to start the function t/; at t = 0 in the manner desired. We may expect that this will require an impulsive type force at a t ime t = 0+. From (7.3.5) we can show that the source term required to duplicate t he initial conditions is (1/ c2)[t/;o(r o) 0' (to) + vo(ro)o(t o)] where by o'(t o) we mean the deri vative of the 0 fun ction. It has the property (b j(x)o'(x ) dx }a
= {-1'(O); ~f 0;
If
x = 0 x = 0
~s with.in i?terval IS
(a,b )
outside interval (a,b)
The physical significance of these terms may be understood. A term of type voo(to) is required to represent an impulsive force, which gives each point of the medium an initial velocity vo(ro). To obtain an initial displacement, an impulse delivered at to = 0 must be allowedjto develop for a short time until the required displacement is achieved. At this time a .second impulse is applied to reduce the velocity to zero but leave the displacement unchanged. It may be seen that the first term .t/; (ro,to) 0' (to ) has this form if it is written
J.!-.~
{t/; (ro,to) [O(t o +
E~
;: oCto - E)]}
Green's Functions
838
[CR. 7
Form of the Green's Function. Knowledge of G is necessary to make (7.3.5) usable. As in the case of the scalar Helmholtz equation we shall first find G for the infinite domain. Let us call this function g. The method employed in the scalar Helmholtz case involves assessing the relative strength of the singularities in the functions V 2g and iJ 2g /iJt 2 in the equation
It may be argued that V2g is the more singular, since it involves the second derivative of a three-dimensional 5function 5(r - r') = 5(x - x') . · 5(y - y')5(z - z') . Such an argument is not very satisfying. However, for the moment, let us assume it to be true. We shall return to the above equation later and derive the result we shall obtain in a more rigorous manner. Integrating both sides of the equation over a small spherical volume surrounding the point r = ro, that is, R = 0, and neglecting the time derivative term, one obtains as in the previous section g ------7 5(t - to) / R
(7.3.6)
R--->O
As before we now proceed to find a solution of the homogeneous equation satisfying this condition, for it is clear that g satisfies the equation V2g -
1 iJ2g
C2
iJt 2
= 0 ; Rand t - to not equal to zero
At R = 0 condition (7.3 .6) must be employed. Since we are dealing with point sources in an infinite medium g is a function of R rather than of rand ro separately. Hence
~2 iJ~ (R2 :~) - ~ (~:~) =
0
(7.3 .7)
or
The solutions of this equation are g
= h[(R /c)
- (t - to)]
+ k(R/c) + (t -
to)]
R
where hand k are any functions. Comparing with condition (7.3.6) we see that two possibilities (or any linear combination of these) occur, 5[(R/c) - (t - to)l/R or 5[(R/c) + (t - to)l/R . The second of these must be eliminated, for it does not satisfy the condition imposed earlier, which requires that the effect of an impulse at a time to be felt at a distance R away at a time t > to. Therefore - t g = 5[(R/c) -R (t - to)l., R,t 0
>0
(7.3.8)
§7.3]
Green's Funetionfor the Scalar Wave Equation
839
representing a spherical shell about the source, expanding with a radial velocity c. We may now make an a posteriori check of our initial assumption, that the singularity of V 2g was greater than that for iJ2g/at 2 • This is indicated by the presence of the l /R factor, but to prove it requires a rather nice balancing of infinities. We shall therefore stop to put (7.3.8) on a more firm footing and only then return to discuss the deductions which follow from this formula. Using spherical coordinates for Il(R) = Il(r - fo) and defining
= t - to
T
it is immediately possible to retrace the steps leading to (7.3.7) and obtain the more general equation, valid also for Rand T equal to zero,
The numerical factor 2 enters because the variable R can never be negative.
Hence
fo
00
Il(R) dR
= t.
To .proceed further it is desirable to employ the relation Il(R) /R
=
(7.3.9)
-1l/(R)
To demonstrate this multiply Il(R) / R by a differentiable function j(R) and integrate over R . Let j(R) = j(O) + l' (O)R + 1"(0) (R2/2!) + .. .. Then
00 1_ 00 j(RkIl{R)
dR
=
j(O)
1-"'",
Ile:) dR
+ 1'(0) 1-"'00 Il(R) dR + 1'i~)
1_
00
00
RIl(R) dR
+
The first of these terms is an integral over an odd function, so that it has a Cauchy principal value of zero; the second one gives 1'(0) ; the third and all higher terms give zero. Hence
0000 1_ j(R~(R)
1_
00
dR
= 1'(0) =
00
1_
00
1'(R)Il(R) dR
= -
00
j(R)Il/(R) dR
This equation may also be derived more dire ctly from the definition of derivative as follows: 1l/(R)
= lim e--e0
[1l(R
+ E)
- Il(R - E)] = lim 2E ....0
t
[1l(R
+ E)
-.R
_ Il(R - E)]
R
Il(R) ;:: ---n:-
Green's Functions
840
[cB.7
Returning to the equation, we may now write (}2
aR2 (Rg) -
1
(}2
/
C2 aT 2 (Rg) =
20 (R) O(T)
It is clearly appropriate to introduce the variables ~
= R - CT;
1/
= R
+ CT
(7.3.10)
The meaning of o/(R) O(T) in the new variables must also be determined. To do this note that f_O
Hence the summation (7.3.29) reduces to a sum over zero-order Bessel functions. The absence of any angular dependence is not surprising in view of the circular symmetry of the initiating pulse (7.3.28). The response at any subsequent time t and at a position r is given by 1/;(r,t) = A
l cos (kopct)N5~ o(kopr) p
(7.3.31)
§7.3]
Green's Funclionfor the Scalar Wave Equation
853
Equation (7.3.31) is exact. Note that the set N op cos(kopct) Jo(kopr) describes the free radial vibrations of the membrane. Generally the response to an initial impulse may be expressed in terms of a superposition of free vibrations, each mode vibrating with its own frequency. This is to be contrasted to the response of the system to a steady driving force of a given frequency. In that case, the response has the same frequency as the driving force and the space dependence involves a superposition of the 1/;n(r)'s, all of them vibrating with the frequency of the driving force. Let us consider the response back at the starting point, r = 0. Then (7.3.31) becomes 1/;(O,t) = A
Lcos(kopct) Ngp p
We introduce the approximate value of the zeros : 1/;(O,t)
~A
L
cos [(2 P :
3) 7!-;] N~p
(7.3.32)
p
When will the original pulse refocus at r = O? On first consideration we might think this would occur when t = 2a/c, the time for the pulse to go to the edge of the membrane and back . This is, however, not the case. As may be seen from the asymptotic behavior of J o(z) ~ V2/rrz . . cos(z - {-7r) , a phase change of 11'/4 occurs in passing from the region r ~ to r ~ a. This is characteristic of propagation in two dimensions. No such phase change occurs in either one or three dimensions. Because of this phase shift, it is necessary for two traversals from the center out to the edge to occur before a final phase shift of 11' occurs and the pulse is refocused . Hence we may expect that, when ct = 4a, the pulse will reform itself at r = 0. This may be readily verified by sub-
°
stitution in (7.3.32) for 1/;(0,4a) ~ - A
LNgp'
(The initial pulse
p
1/;(0,0) is A ~N5p ') We should like to emphasize again that this phenomenon occurs only in two dimensions; in one and three dimensions it does not occur. The pulse from the center of a sphere of radius a re-forms at the center at a time t = 2a/c. There is one final point which also shows the striking difference in wave propagation in two as compared with one or three dimensions. In the latter the" initial pulse re-forms exactly at the proper time. In two dimensions this is not so because of the wake developed as the wave progresses. This may be seen in the present instance as follows. Expression (7.3.32) is approximate, for the approximate values of the roots of the Bessel function J 0 were utilized. If the precise values of the
[cH.7
Green's Functions
854
roots had been employed, there would have been no value of ct at which the phase (kopct) would be exactly the same for all p. In other words, there would be no value of ct for which the free vibration initiated by the pulse would have all returned to their initial phase. Thus the free vibrations would never interfere in the proper fashion to re-form the initial situation exactly. As another example of the construction of Green's function for the scalar wave equation let us derive expression (7.3.8), the infinite space Green's function, by direct utilization of the superposition method. In that case so that g(R ,r) = - I
~R
f-_.
f-
ei(kR-wT) dw = -1- __ eiCol[(RIC)-T) dw ~R
It should be noted that we have carefully chosen the relative sign between the factor kR and wt to be such that ei(kR- CoIT)/ R represents a wave diverging from the source as time progresses, i.e., as r increases. This is the manner in which we satisfy the causality principle. We now make use of the integral representation for the 0 function, Eq. (7.3.22), to obtain
g(R,r) = o[(R/c) - r]/R
Klein-Gordon Equation. The Green's function for the time-dependent Klein-Gordon equation satisfies the equation (7.3.33) It is easy to verify that the Green's function for the Klein-Gordon equation may be employed in much the same way as the Green's function for the scalar wave equation. For example, the reciprocity condition (7.3.3) and the general solution (7.3.5) apply as well here . There are, however, important physical differences between the two. These may be best illustrated by considering the Klein-Gordon Green's function for the infinite domain, thus obtaining the analogue of Eq . (7.3.8) . The function g(r,tlro,to) may be obtained by superposition of the solutions obtained for a simple harmonic time dependence e-iCol(t-to) rather than the impulsive one given by oCt - to). The necessary superposition is given by Eq. (7.3.22). The individual solutions may be then given by g(RIVw 2 - C2Jc2) , where
§7.3]
Green's Funciion for the Scalar Wave Equation
855
The solution of this equation is g =
exp [i
y(w/c)2 R
K2
R]
; R = [r - rol
(7.3.34)
In the limit w/c» K Eq. (7.3.34) becomes g = ei(.,/c) R/R as it should. the opposite case w/c « K
For
) e-KR/R
g (W / C)«K
grvmg a characteristically "damped" space dependence. This is, of course, not related to any dissipation. From the one-dimensional mechanical analogue (Chap. 2, pages 138 et seq.), a string embedded in an elastic medium, we see that it is a consequence of the stiffness of the medium. w Plane
Contour
W=+CK
Branch Line
Fig. 7.14
Contour for integral of Eq. (7.3.37) for R
> ct.
Employing (7.3.22) we may now write as the solution of (7.3.33) valid for an infinite medium 1 g(r,tlro,to) = 27r-R
f'"_ '" exp i[Y(W/C)2 -
K2 R - wr] dw
(7.3.35)
where r = t - to. Function g is a function of Rand r only, as expected. We must now specify the path of integration. Before doing so it is convenient, for convergence questions, to introduce the function h(R,r) such that (7.3.36) iJh(R,r)/iJR = Rg(r,tlro,to) Hence _ 1 h(R,r) - -2.
m
f"
exp i[Y(W/C)2 - K2 R - wr] Y (W/C)2 - K 2 dw _ '"
(7.3.37)
. Green's Functions .
856
[CR. 7
The integrand has branch points at w = ± CK. The relation of the path of integration relative to these branch points is determined by the causality condition. We choose the path and branch line shown in Fig. w Plane 7.14 . First note that h = 0 if R > cr, as the causality postulate would demand for this case. In the limit of large w the exponent in (7.3.37) approachesiw[(Rlc) - r] = Branch Line iwl(R lc) - r]. The path of integration may then be closed in the upper half of the w plane without changing the value of the integral. Fig. 7.16 Contour for integral of Eq . Since the integrand has no singulari(7.3.37) for R < ct. ties in the upper half plane , the integral is zero. N ow consider h for R < Cr . The contour is then deformed to the one shown in Fig. 7.15. It may now be reduced to a more familiar form . We introduce a new variable 1J, such that r = Ivr2 - (Rlc)21 cosh 1J c
Then h(R,r) = 2 . 1r'/,
f"
Finally let (x - 1J) = h(R,r)
=
-i..i
ee +~.-i
2c 1r
i~;
f
and let
w=
CK
cosh x
exp [-iKClvr2 - (Rlc)21 cosh (x - 1J)] dx
then
-i.. - i "
+l ..-
exp [-iKClvr2 - (Ric) 21 cos
i.,
~] d~
This is just the integral representation of the Bessel function of zero order [see Eq. (5.3.65)] so that h(R,r) =
vr 2
-CJo[KC
-
(Rlc)2] ; R
Combining this with the expression for cr h(R,r) =
-cJO[KC
vr 2
-
to. In other words the development backward in time of a source placed at ro at a time to. The reciprocity condition now reads
G gives
G(r,tlro,to)
=
G(ro,to/r,t)
(7.4.6)
Function G describes development as time increases, leading from the initial source to the final distribution. Function G describes the same process in reverse time order, beginning with the final distribution and going backward in time to the initial source. The question of functions and their adjoints will be discussed later in this chapter. The proof of (7.4.4) or (7.4.6) follows the pattern developed in the preceding section. The two equations to be considered are
ata G(r,tlro,t o)
= -41To(r - ro)o.(t - to)
+ a ~G(r,-tlrl,-tl)
-47To(r - rl)o(t - tl)
V 2G(r,t/ro,to)
V2G(r,-tlrl,-tl)
2
a2
Green's Function for Diffusion
§7.4)
859
Multiply the first of these by G(r, -tlrl' -t l) and the second by G(r,tlro,to), subtract, and integrate over the region of interest and over t from - ao to tt. Then using Green's theorem one obtains
f~: dt
f
{G(r, -tlrl' - tl) grad[G(r,tlro,to) -G(r,tlro,to) grad[G(r - tlrl' - tIl} • dS -a 2
f r: { ' dV Jo
+G(r,tlro,to)
G(r, -tlrl, -tl)
ata [G(r,tlro,t o»)
~ [G(r,-t1rl,-t l))} dt = 41T[G(rl,t 1Iro,t o) - G(ro,-tolrl,-tl»)
The first of the integrals vanishes by virtue of the homogeneous boundary conditions satisfied by G. In the second, the time integration may be performed to obtain [G(r, - tlrl' - tl)G(r,tlro,t o) )::~:
At the lower limit the second of the two factors vanishes because of (7.4.2). At the upper limit the first factor vanishes again because of (7.4.2), and it is recognized that we have tacitly assumed in all of this that t, is within the region of integration. The reciprocity condition now follows immediately. We may also obtain the equations satisfied by G and G as functions of to. For example, from (7.4.6)
vgG + a2(aG/ato) = -41To(r - ro)o(t - to) vlf] - a 2(aG /ato) = -41To(r - ro)o(t - to)
(7.4.7)
Inhomogeneous Boundary Conditions. We shall now obtain the solution of the inhomogeneous diffusion equation, with inhomogeneous boundary conditions and given initial conditions, in terms of G. The equation to be solved is (7.4.8) where P, the source function, is a known function of the space and time coordinates. Multiply this equation by G and the first .of Eqs. (7.4.7) by if;; subtract the two equations; integrate over space and over time from 0 to tr :
It+ dto f dV
0
[if;VgG -
GV~if;] + a2
f
It+ dto[if; (:~) + G (:~) ] = 41T It+ dt o f dV opG - 47Tif;(r,t)
dV 0
860
Green's Functions
[cH.7
We may apply Green's theorem to the first of these integrals. In the case of the second, the time integration may be performed. Note that G(r,tlro,t+) = O. Finally 1f;(r,t) =
It+ dto f
1 r + 47r)0
dto
f
dVo p(ro,to)G(r,tlro,t o)
dSo• [G grads 1f; - 1f; grad, G) +
2/
:11"
dVo[!f;G]to=o
(7.4.9)
G is chosen so as to satisfy homogeneous boundary conditions corresponding to the boundary conditions satisfied by 1/;. For example, if 1/; satisfies homogeneous or inhomogeneous Dirichlet condition, G is chosen to satisfy homogeneous Dirichlet conditions. The first two terms of (7.4.9) represent the familiar effects of volume sources and boundary conditions, while the third term includes the effects of the initial value 1/;0 of 1/;. If the initial value of aN at should be given, it is necessary to consider the equation satisfied by fJ1f;jat rather than that satisfied by 1/;. Let II = a1/;jat. Then from (7.4.8) we obtain
'v 211
-
a2(all jat) = -411"(apjat)
an equation of the same form as (7.4.8) and to which the same analysis may be applied. As a consequence, either type of initial condition may be discussed by means of (7.4.9). As we saw in Chap. 6, we should not specify both initial value and slope for the diffusion equation. Green's Function for Infinite Domain. We now go on to construct specific examples of Green's functions for this case. As usual the Green's function g(R,r) , R = [r - rol, r = t - to for the infinite medium- is the first to be discussed . It is possible to derive the expression for one, two, or three dimensions simultaneously. Let g be a one-, two-, or threedimensional Fourier integral: g(R,r) =
(2~)n /
eip.R'Y(p,r)
zv,
where n is 1, 2, or 3 depending on the number of dimensions and the dimensionality of the integration variable dV p is the same . Since
and we finally obtain an equation for 'Y: a2(d'Y jdr)
with a solution
+ p 'Y
'Y = (411"ja
2
2)e-(p'
= 411"o(r) /a ')Tu(r)
Green's Function for Diffusion
§7.4]
861
where we have picked that solution which conforms with the causality requirement. Hence
I
=
g(R,r)
= L2;)"na 2 ] u(r) [
'
or
[~J (21T)na u(r)
g(R r)
2
eip'Re-(p 2/a' )T dV
1-.. .
p
eiP.R.e-(p.2/a2 )_
II_
«>
(iJG) dtoif!(O,t o) iJxo z.=o dt o
cos(wto) e- a ' z ' / 4 ( t -
t o) [
dt o
(t - to)'
«>
]
A more convenient integration variable is given by ~2
Then
r y:;;: Jo
if!(x,t) = 2T 0
ee
= [a 2x 2/4(t - to)}
d~ e-~' cos w (t
_ a
2x 2 )
4e
=
~ Re {l«> d~ eiHH",(a'z·/4E' )}
=
~; Re{ei",e--yr;:,ax l«> d~ e-H-(-yr;;:;ax /2~)1'}
We can show that this integral is a constant, independent of x. J(OI) =
fo
«>
We set
d~ e-IHa'/~)I'
The integral J may be written in another form by the substitution 'II = 0I2/~:
J(OI)
0 =
Hence
=
l«> (;:) d'll e-IHa'/~)J'
l«> d~ (1 - ;:) e-IHa'/~)J'
By differentiating the first form for J we find J'(OI)
=
2l«> d~ [ ~ - (~2) ](2;) l«> d~ [1 - (;:)]
= 401
rlHa'IW
e-IHa'IE)!'=
0
Green's Functions
864
[CR. '[
Therefore J is independent of a and is equal to its value for a = 0, which is just t -V;;:. Hence
"'(x,t)
= Re[To ei CT, the contour may be closed by a semicircle in the upper half plane. The integral is then zero, since there are no singularities within the contour. When R < cr, the contour is deformed so as to extend along the negative imaginary axis. We must then evaluate an integral which is very similar to the integral involved in the calculation of the Green's function for the Klein-Gordon equation. We obtain (see table of Laplace transforms at the end of Chap. 11)
.: (:i) e-!a2c'T Jo[ia2c yR2 -
c2r2]u(cr - R)
Consider now the contribution coming from the integral involving e-W - T :
~ e-!a'c'T ( exp[ipR 2c
Jc
+i
yp2 - (a 4c 2/4) cr] dp Y p2 - -Ha4c 2)
This integral is zero when (R + cr) > 0 (recall that in one dimension R can be negative) but is not zero when (R + cr) < O. Then we obtain
- (:i) e-!a'c'T Jo[ia2c yR2 -
c2r2][l - u(R
+ cr)]
Combining these two expressions yields gl(R,r) = 21l'ce-!a 2c'TJ o [a~c yR2 - c2r2J u(cr -IRI)
(7.4.25)
The reader should be able to verify that this result tends to the correct limiting forms given by Eqs. (7.4.10) and (7.3.16) as c - 00 or as a - 0, respectively.
868
Green's Functions
[cu. 7
We may now obtain the three-dimensional g from the differential equation (7.4.24) :
g3(R,r) =
~ e-ia' c'< {o(cr -
+
a
R)
2cR
2 yR2 - c2r 2
J 1[-!a 2c yR2 - c2r 2]u(cr - R)}
(7.4.26)
We shall obtain the Green's function for two-dimensional problems by integrating g3(r,r) over the z component of R rather than by direct consideration of (7.4.23). Let R2 = + p2. Then
e
g2(R,r) = or
g2(R,r) =
1-.. . d~ g3(R,r)
ia c 2c e- ' '< u(cr - p) {I
yc2r2 _ p2
+ 2 sinh
2c 2[a
4
yc 2r2 - p2]} (7.4.27)
Here we have employed the formula
t: I }o
2
1(2z
sin e) de = 2 sinh z z
where I 1 (x ) = -iJ 1 (i x) (see tables at end of Chaps. 10 and 11). The three-dimensional case exhibits the physical phenomena which occur as a consequence of the inclusion of a velocity of propagation into the diffusion equation or the inclusion of a dissipative term into the wave equation. Both terms in (7.4.26) vanish when R > cr, as is to be expected whenever effects propagate with a finite velocity. The first term is a reproduction of the initial pulse, reduced, however, by two factors. The first, l/R, is the geometrical factor which appeared in the solution of the simple wave equation. The second is the fa ctor e- ia' c' < which tells us that this part of the wave, generated by the point source, decays with time as it moves through the medium. The second term in (7.4.26) constitutes the wake. For sufficiently long times Cr» R, it is the term which yields the usual diffusion approximation. These differences may be exhibited in another fashion. Let us solve the one-dimensional initial-value problem. From (7.4.20) we find 2
a If; = 4?r
f
dXo[1/tg1] to-O
12 + 4?rc
f
[alf;
a
g1 dxo gl et o - If; ato ] to~O
(7.4.28)
§7.5]
Green's Function in Abstract Operator Form
869
where 1/!o(xo) and vo(xo) are the initial values of 1/; and a1/;/at, respectively. Note that this becomes d'Alembert's solution, Eq. (7.3.18), if a - O. The first term is the same as that given by d' Alembert except for the decay in time as given by e- i a ' c' l • The second term is new and represents the effect of diffusion . The third term reduces to the d' Alembert term when a - O.
7.5 Green's Function in Abstract Operator Form So far our discussion has been limited to a particular type of partial differential equation. The space operator has been V 2, and the time operators have been absent or a/at or a2jat 2 in the Helmholtz, diffusion, and wave equations, respectively. In the present section we shall generalize these considerations so that they apply to any operator, permitting the application of the theory to any of the equations of physics as long as they are linear. Our plan will be to emphasize the important elements in the previous discussion and then to see how they are most appropriately generalized. It is natural that we shall have to be somewhat abstract. For example, instead of writing out a specific form for the homogeneous equation to be considered, we shall state it in operator form: a1/;
=0
(7.5.1)
where a operates on the coordinates giving the dependence of 1/;. For example, in the diffusion equation a = V 2 - (a 2 )( il j ilt) and is a function of rand t. Another linear type, the integral equation, is mentioned in Chap. 2 (see page 180) and will be discussed more fully in Chap. 8. Such an operator a is a
= 1
--&.
Jb K(x,xo)
.. . dxo
and the equation a1/; = 0 reads 1/;(x) -
lab K(x,xo)1/;(xo) dxo =
0
The variables may include more than just space and time dependence. In transport problems (Sec. 2.4) the distribution function f depends not only on rand t but also on the momentum p and energy E. In the same notation, the equation for the Green's function G is aG(xlxo)
=
-41l'o(x - xo)
(7.5.2)
where x is a generalized vector representing all the independent variables which are relevant; a operates on x. For the wave equation x = axX + ally a.z + alt, where a, etc ., are unit, mutually orthogonal vectors.
+
Green's Functions
870
[CR. 7
Then lJ(x - xo) becomes the product of the IJ functions for each coordinate; e.g., for the wave equation lJ(x - xo)
= lJ(x - xo)lJ(y - yo)lJ(z - zo)lJ(t - to)
Generalization of Green's Theorem, Adjoint Operators. The most important mathematical tool employed in the analysis of the pre ceding sections was Green's theorem ; our first task will be to generalize it . In differential form Green's theorem states that UV2V - VV2U = V· (uVv - vVu) A generalization of this equation in terms of a which immediately suggests itself is uG.v - vG.u = V . P(u,v) (7.5.3) where P is a generalized vector in terms of the same unit vectors as that describing x while V is the corresponding gradient operator. Hence V · P = (OP;e/ax) + (apy/oy) + (ap. /az) + (apt/at) + . . ' . For example, in the case of the wave equation, G. = (1/c 2)(o2j ot 2), we find from Eq. (7.5.3) that
V2 -
U[V2 - !c2~] v - V [V2 - !~] u = ~ [u av . at2 c2 at2 ox ox . + ~ [u av
,
ay
ay
_ v au] ay
+ i- [u ov _ az
oz
- v ou] ax
v ou] _ ! 2 ;~ [u av _ v ou] oz c at at at
Here P = uVv - vVu, where V is the general gradient operator. Relation (7.5.3) is not satisfied by all operators a. For example, in the case of the diffusion equation for one dimension, G. = (a2jax 2) - a 2 ( a/ ot), we find uav - vG.u = :x [ u
~: -
2 v ::] - a [ u
:~ - v ~~]
The first pair of terms on the left side is in the proper form . However, the second pair cannot be written as the time derivative of a function of u and v. We must therefore generalize Green's theorem beyond (7.5.3) : uG.v - vau = V · P(u,v)
(7.5.4)
a
a
where is an operator which is called the adjoint of a. When = a, that is, when (7.5.3) applies, the operator G. is said to be self-adjoint. In the case of the diffusion equation, = (a 2/ax 2) + a 2(a/ at). Definition (7.5.4) is just a generalization of a definition of the adjoint operator employed in Chap. 5 (see page 526). In Eq. (5.2.10) was defined by
a
a
v(z)G.[y(z)] - y(z)a[v(z)) = (d/dz)P(v,y) where P(v ,y) is the bilinear concomitant. dimensional problems of Eq. (7.5.4) .
This is the statement for one-
Green's Function in Abstract Operator Form
§7.5]
871
Looking back at the manner in which Green's theorem is utilized, we see that we shall be interested in the solutions of the equation (7.5.5)
&{t = 0
and the corresponding Green's function for the adjoint operator &G(xlxo) = -41ro(x - xo)
(7.5.6)
Equation (7.5.5) is known as the adjoint of (7.5.1), the equation involving el, while {t is referred to as the adjoint of if;. In the case of the onedimensional diffusion equation, &{t = 0 reads iJ2{t iJx2
+
2 iJ{t a iJt
=0
We see that {t satisfies the diffusion equation with the time variable reversed. Hence if if;(t) is a solution of (7.5.1), then {t(t) = if;( -t) is a solution of (7.5.5). Once the generalization of Green's theorem is available, it becomes possible to solve the inhomogeneous problem elif; = -41rp(x)
(7.5.7)
with inhomogeneous boundary condit ions. Since the adjoint operator & is involved in Green's theorem (7.5.4), it is clear that we must compare (7.5.7) and (7.5.6). Multiply the latter by if;(x) and the former by G(xlxo), and subtract: G(xlxo)elif;(~) - if;(x)&G(xlxo) = 41rif;(x)o(x - xo) - 41rp(x)G(xlxo)
Employ (7.5.4), and integrate over a volume in x space (which includes the entire range of each component of x which is of physical interest) . For example, in the case of the wave equation, it includes integration over time from 0 to tt and an integration over x, Y, and z coordinates within the surface upon which boundary conditions are to be satisfied. Then if;(xo) = Jp(x)G(xlxo) dv + (1/41r)JV. P[G(xlxo) , if;(x)] dv if;(xo) = jp(x)G(xlxo) dv + (l/41r).fn . P[G(x'lxo), if;(x')] dS (7.5.8)
or
where n is an outward-pointing unit vector orthogonal to the surface S bounding the volume in x-space. In the scalar wave equation this term is -1
41r
~to+ dt 0
f
n· [GVif; - if;VG] dS - - 1 2 411'c
f
O
ilG]t=t + dv [iJif; G- - if;iJt iJt 1=0
Effect of Boundary Conditions. To proceed any further it becomes necessary to consider the boundary conditions satisfied by if;. Consider the case in which if; satisfies homogeneous boundary conditions on S;
872
[CR. 7
Green's Functions
that is, there are no sources of the field if; on the surface S. By the principle of superposition it must then be possible to obtain the solution as an integral over the volume source distribution p(x) multiplied by the effect due to a source at x. This involves us in two considerations. In the first place, we must relate G and G. As we shall see, this will lead to a generalized reciprocity condition. We shall postpone the proof of this theorem for a short time. In the second place, in order to obtain a solution of the proper form it is necessary for the surface term in (7.5.8) to vanish. The Green's function G and if; must satisfy homogeneous boundary conditions which must be so correlated that .'in· P [G(x'lxo), if;(x')] dS
=
0
(7.5.9)
In the simplest case we have considered, the scalar Helmholtz equation, the surface term vanishes if the Green's function and the fun ction if; satisfy the same homogeneous boundary condition. In the scalar wave equation, we employed initial values for (aif;/at) and if;, the Cau chy condit ions; for the Green's function we employed the causality condition (see page 834). It is also possible (as we pointed out earlier, in Sec. 7.2) to determine the proper boundary conditions if; must satisfy. For example, in the Helmholtz equation, the surface term involves the surface values of Vt and aif;/an. Placing both of them equal to zero is manifestly improper, for in that event the surface integral automatically vanishes, the boundary condition on G is left arbitrary, and the solution of the inhomogeneous equation becomes nonunique. Since the solution is in fact unique, the initial assumption about the surface values of if; and aif;/ an is incorrect and we are led to relax the boundary conditions to be either homogeneous Dirichlet or homogeneous Neumann or some linear combination of the two. Then in order for the surface term to vanish, G (in the case of the Helmholtz equation) must satisfy the same boundary condition as if;. In a similar fashion, the examination of Eq. (7.5.9) will lead to a determination of the proper boundary conditions for if; and the corresponding ones for G. Having determined in an implicit fashion the boundary conditions on if; and G(xlxo), we may now turn to the reciprocity condition. We compare the equations satisfied by G and G: aG(xlxo) = -47ro(x - xo);
aG(xlxl) = -411"o(x - Xl)
Multiply the first of these by G and the second by G, subtract, and integrate over the relevant volume in X spa ce. Employing the generalized Green's theorem (7.5.4), we obtain
§7.5]
Green's Function in Abstract Operator Form
873
In order that the solution of the inhomogeneous source problem with homogeneous boundary conditions be expressible in terms of G and not G, it is necessary for a simple algebraic relation between them to exist, which in turn requires the surface term in the above equation to vanish. Comparing this surface term with (7.5.9), we see that G(xlxo) satisfies the same conditions on S as 1/;, a result which may be expected from our intuitive idea of G and its relation to 1/;. We finally obtain
G(xlxo) = G(xolx)
(7.5.10)
In words, the left-hand side of the equation describes the effect at x of a point source at Xo, the propagation being governed by the operator a and the boundary conditions. On the right-hand side, the source is placed at x; the effect is measured at Xo, the propagation from x to Xo now and the corresponding boundary condition for G. being governed by If G is not the same function as G, a directionality in x space must exist, for reversing the direction of propagation changes the consequent observations. This irreversibility must be apparent in the operator a or in the boundary conditions. For example, the operator a for the diffusion equation V 2 - a 2 (a/ at ) is not invariant against a change in the sense of time, i.e., against the substitution of - t for +t. The operator for the wave equation a = V 2 - (1/c 2 ) (a 2/ ot 2 ) is self-adjoint (a = (1), so that a directionality, for example in the time coordinate, cannot arise from it. However, a directionality can perfectly well arise from the boundary conditions imposed. For example, the application of the causality condition imposes a definite asymmetry with respect to past and future. As a consequence, the reciprocity principle for the Green's function for the wave equation for this initial condition reads
a
so that
G(r,tlro,t o) = G(ro, -tolr, -t) G(r,tlrot o) = G(r, - tiro, - to)
(7.5.11)
W-e see that G describes the propagation from a source point ro to one at r with, however, the sense of time reversed, so that the event at t occurs at some time earlier than the impulse causing it at a time to (note that t < to). For example, in the case of the Green's function for the infinite domain
G(r,tlro,t o) = (1/ R) o[ (R/c)
+
(t - to) 1; R = [r - rol
At a given R, an effect is felt at a time t = to - R/c, that is, at a time R/c earlier than the initiation of the motion at to. For this reason G is often referred to as the advanced potential while G = (l /R)o[R/c (t - to)] is the retarded potential. Both are solutions of the source problem with differing initial conditions.
874
Green's Functions
[cB.7
Because of the effect of boundary conditions it is useful to generalize the idea of adjoint. We introduce two terms: adjoint boundary conditions and adjoint problem. An adjoint problem will be satisfied by {; if {; is a solution of
arid if it satisfies adjoint boundary conditions. latter by the requirement n . P[,p,{;] = 0;
We shall define the
on the boundary surface
(7.5.12)
Hence if ,p satisfies a certain boundary condition, (; will satisfy a corresponding boundary condition which we shall call the adjoint boundary condition. A problem is considered self-adjoint when = d and the boundary conditions for,p and"" are the same. For self-adjoint problems
a
G(xlxo) = G(xlxo).
More on Adjoint Differential Operators. Let us now become more' definite and consider some operators and their adjoints. As a first example consider the one-dimensional situation. Here we shall generally be interested in second-order operators, so that we may specialize to d 2v dv (7.5.13) dv=p-+q-+rv dz 2 dz The adjoint is [see Eqs. (5.2.10) et seq.] _ d2 d du = dz 2 (pu) - dz (qu)
+ Ttl
(7.5.14)
The bilinear concomitant P(u,v) (7.5.15) Under what conditions will (i be self-adjoint? Upon placing we find that dp/dz must equal q. Under these circumstances,
a=
(i
dV)
d ( p- +rv dv=dz dz
The equation dv = 0 is just the Sturm-Liouville one discussed in Sec. 6.3. We see that it corresponds to the only linear self-adjoint operator containing at most second-order differential operators.
~
Green's Function in Abstract Operator Form
§7.5] When
875
dp/dz = q, the bilinear concomitant Pis P=p ( udv- - vdU) -
dz
The requirement n-
dz
P(I/;,~) = 0 is just that 0 =
p (I/;
~: - .jJ ~~)._
p (1ft °a.jJ
- .jJ aal/;) ,where b and a are the points at which boundary z z .=1> conditions are satisfied. If p is finite 'at the end points, then possible boundary conditions are Dirichlet, I/; = 0 at a and b; Neumann, dl/;/ dz = 0 at a and b; or mixed, al/;/az = IN at a and b. All of these are self-adjoint boundary conditions, for 1f; must satisfy the same boundary conditions as 1/;. Periodic boundary conditions 1ft(a) = I/;(b) and (dl/;/dz)._a = (dif;/dz).,..,J, are also self-adjoint. Another type of boundary condition occurs when p has a zero at either a or b. In that event P is zero at the point only if the functions I/; and .jJ are bounded. Again this boundary condition is self-adjoint. We have, of course, considered these very same conditions in Chap. 6. For all of them, the Green's function must be symmetric. Expressions (7.5.13) to (7.5.15) may be generalized to include operators involving higher order differentials and more than one dimension. Consider first the operator (7.5.16) Anyone-dimensional operator is, of course, a linear combination of operators of type an. The adjoint is (7.5.17) The bilinear concomitant is
P
n-2V) 2(PU») 3V) Uv = U(dn-1V) _ (d(PU») (d n 2 (d (d n_ p dz n-I dz dz - + dz 2 dz n - 3
( , )
(_1)n-1
(dn~:~~») v
(7.5.18)
In several dimensions, the most general differential operator would be of the form
.) axaax b 1
where
XI,X2, .
2
an • ••
ax k ; •
a
+ b + .. . + k =
n
(7.5.19)
, z, are generalized coordinates. The adjoint &. is (7.5.20)
876
[CR. 7
Green's Functions
The bilinear concomitant P(u,v) is P(uv) = ,
al [ T.U ( r: axi
a(n-I)v
)
laxt . . . ax:
a(pu) ( a(n-2)v --aXI axi 2ax 2 . .
+ . . . (_)a aa(PU)) ( aa +( -l) 2 [( aXi
a(n-a)V
a~-I
+ ... +( -l)n
k
a•
I
8
a(n-a-2)V ) axl2-2 . .. aX~
. ... () _ b-I (aa+b-I(pu)) (aln-a-b)~)J aXaaXb-1 .. . aXk I
[ ( an- .(pU) ) (ak-Iv)
aX~a~ . ..
+ ...
)J
2
aX~aX2
-
ax~
ala-I)(pu) ( a(n-a)v axa-l axb. .. ax k
I
(aa(PU)) (
aX~
•••
+ _
)
-
)
.
aX~-1
2
•
k 2v) • (an-.+I(PU) ) (a ax~ax~'" ax. ax:-~
(ax~a~n~l~p~)aX~-I) vJ
(_1)k-1
(7.5.21)
where an is the unit vector corresponding to the coordinate x n • As a simple example, consider the operator a 2v
a2v O. Show that the proper Green's function is (eikRIR) + (eikR'jR') , where k = wlc, and R2 = (x - XO)2 (R') 2 = (z - XO)2
+
(y - YO)2 - YO)2
+ (y r 2 = x 2 + y2 + Z2 »
+
(z - ZO) 2
+ (z + ZO) 2
Show that, when a 2, the asymptotic expression for if; is if; ~ (Va 2/r)eikT-iwI[J 1(ka sin tJ)lka sin tJ] Use this result to discuss the Fraunhofer diffraction of waves from a circular orifice. 7.4 The inner surface of a sphere of radius a is kept at potential if;a(tJ,l{J) , where tJ and l{J are the angle coordinates of a spherical system concentric with the sphere. Show that the Green's function appropriate for this problem is ! [r2 + r3 - 2rro cos 0]-; - [(rrola)2 + a 2 - 2rro cos 0]-; where 0 is the angle between r the radius vector to the observation point and ro the radius vector to the source point [cos 0 = cos tJ cos tJo + sin tJ sin tJo cos (l{J - l{Jo)]. Show that the interior potential is .I'(r o ) = !!:.... 't' "l{J 411"
[1 _(:.)2] Jo(Z" 0(" Jo d
a
l{J
if;a(tJo,l{JO) sin tJo dtJo [a2 + r 2 - 2ar cos OF
Find a series expansion of if;.in terms of powers of ria useful for points near the origin. 7.6 Show that the Green's fun ction in spherical coordinates for the Laplace equation is
l.
~ (n-m)! m m {(rnlr(j+l); r:::;To R= ~Em(n+m)!cos[m(l{J·-l{JO)]Pn(COStJ)Pn(COStJO) (To/rn+l); r;:::ro n,m
and that for the Helmholtz equation is eikR
If:
~
= ik ~ Em(2n
- m)' + 1) (n (n + m) i cos[m(l{J -
l{Jo)]P;:O(cos tJ) .
n,m
T :::; To _0 ) {jn(kr)hn(kro) ; . P ::' (cos vo . I n(kro)hn(kr) ; r;::: To
Green's Functions
888
[cH.7
where in and hn are the spherical Bessel functions (see Prob. 5.20 and the tables at the end of Chap. 11). 7.6 A solution of the Helmholtz equation, originally of the form if;o(r) =
~Amn
cos(m
+ am)p;:'(cos TJ)jn(kr)
extending throughout all space, is perturbed by the presence of a sphere of radius a with center at the origin , at the surface of which if; must now satisfy the boundary condition - (iJif;jiJr) = 1/if;;
r = a
Show that the new if; solution of the equation V 2if; + k 2if; = 0 outside the sphere, satisfying the above boundary condition at r = a and the condition at r - 00 that if; must equal if;o plus an outgoing wave, is a solution of the following integral equation:
=
if;(r)
if;o(r)
+ 4~ ¢ if;(ro) [a~o G(rlro) -
J o; r ~ a
1/G(rlro) dA
where G is the second series of Prob. 7.5 and the integration is over the surface of the sphere. 7.7 A wire of radius b is immersed in an oil bath of infinite volume. The heat-diffusion coefficient of both oil and wire is a. Both are originally at zero temperature. A pulse of current is sent through the wire, heating it momentarily up to temperature To. Show that the temperature a distance r from the wire axis a time t later is T = (TO) e-r2/4G21 (b e-1I2/4a2IJo(iyr)YdY 2a 2t Jo 2a 2t
By use of the series expansion and asymptotic expression for J 0, compute · T for the two limiting cases, one where 2a 2tj r is much smaller than b, the other where it is much larger. 7.8 Determine the one-dimensional Green's function Gk(rlro) for the Bessel differential operator
!!:... (r dGk) dr
dr
+ v-a,
= -li(r - ro)' r a '
-
where Gk ( alro) = O. Show that Gk is singular whenever k = kr , where Jo(kna) = O. From the behavior of Gk at this singularity determine the normalization integral
foG rJ~(knr)
dr
7.9 Show that, in cylindrical coordinates,
L
(2 - liom) cos[m(p - po)]
m
~o
00
Jm(Ap)JmC'Apo)
ei '\l' k
V
2
- X2 !z-
k2 -
zol
A2
AdA
CH.
Problems
7]
889
7.10 Let u = Ex and v = - Ell, where E is a two-dimensional electric field. Show that the equations satisfied by u and v may be summarized as follows :
(u)
y) a/ ax ( a/ay -a/a a/ax v
= 0
Define a Green's dyadic
satisfying the equation
a/ ax -iJ/a y) a/ax ® = -4d(x - xo)o(y - Yo) ( a/ay
(10 01)
Show that
®= (
a/ax a/ay) G -a/ay a/ax
where G is the Green's function for the two-dimensional Laplace equation . Discuss the meaning of ®, and obtain the solution in terms of ® of the inhomogeneous form of the equations for u and v. 7.11 Let if; satisfy the following equation : (d 2if; /dx 2 )
+ k 2if;
= 0; 0
~
x ~ l
and the boundary condition if;(0) = 0 and if;(l) = ft/!'(l) where f is a complex constant. Show that the eigenfunctions are sin(knx) where tan(k n ) = fk n • Show that the adjoint solution satisfies the same equation as if; but with the boundary conditions ,jt(0) = 0, ,jt(l) = N'(l) . Show that ,jtn = 1f;n. Show that
Discuss the normalization for if;n, and verify by examining the Green's function for the problem, which may be obtained in closed form . 7.12 A self-adjoint operator \! may be broken up into two selfadjoint parts \!r and \!p, where \!r operates on variable r only and \!p operates on p only : Let the orthogonal and normalized eigenfunctions of \!p be 'Pn(P) :
Show that the Green's function GA(r,plro,po) which satisfies the equation [,c(r,p) - A]GA =
-
o(r - To)O(p - po)
Green's Functions
890 is given by (h. =
[cH.7
Lg~-~.(rlro)pn(p)Pn(po) n
7.13
Xn)]g~-~n =
[\!r - (X -
where
- o(r - ro)
If
f G~(rlro) + (4~) f GO(rlrl)G~(rllro)
G~(rlro) = Go(rlro) and
Go(rJro) =
= -471"o(r - rs)
X)G~
(\! Show that
-
(4~) G~(rlrl)GO(rllro) dV
1
dV 1
7.14 If G is the Green's function for the scalar Helmholtz equation for the semi-infinite region x 0, satisfying mixed boundary conditions
iN/ax =
FI/;; at
x
= 0
show that
(aaxG) -
_ - (axa + F) (eRikR -
FG -
where R is [r - rol and R'
ikR eR' ')
=
[r + rol . By integration show that
G
= (eikR/R)
+T
Determine T.
Table of Green's Functions General Properties. equation
The Green's function
G~(rlro)
satisfies the
£(G) - XG = -471"o(r - ro)
with certain homogeneous boundary conditions on the boundary surface S . Its adjoint [see Eq. (7.5.4)] O~(rlro) satisfies the equation ;£(0) - XO
= -471"o(r - ro)
satisfying adjoint boundary conditions [see Eq. (7.5.9)] on the boundary surface S: The reciprocity principle is that
If £ is Hermitian (if its adjoint equals its conjugate), then (h. is also Hermitian. In this case the eigenvalues Xn for £,
Table of Green's Functions
CH.7]
891
are real, the set of eigenfunctions 1/;n is mutually orthogonal, and G (r lr ) = 4 A
0
7r
\ ' y;,,(ro)1/;n(r) An)
L.t Nn(A n
where N'; = fl1/;,,12 dv. If £ is not Hermitian, then the eigenfunctions ln of the Hermitian conjugate equation £*(In) - Jlnln
= 0; .c* =:£.;
P.n
= Xn
may not equal the eigenfunctions 1/;n and neither set may be mutually orthogonal. The double set 1/;, I then constitutes a biorthogonal set, however, and
and the adjoint G is not necessarily equal to the conjugate (j: [see Eq. (7.5.44); s, = fln1/;n dv] . Green's Function for the Helmholtz Equation. G is a solution of V2Gk(rlro) + k2G k(rlro) = -4d(r - ro) satisfying homogeneous boundary conditions on some surface S . Then the reciprocity relation is Gk(rlro) = Gk(rolr), since the equation is selfadjoint. If 1/; is a solution of (V2 + k 2 )1/; = -47rp, having value 1/;o(r on the surface S and having outward-pointing normal gradient No(r-) = (iJ1/;/iJn)s on S, then within and on S B
1/;(r) = f p(ro)Gk(rlro) dvo
tf.. + 47r1 'f [Gk(rlrg)N o(r~)
)
a
- 1/;o(~) iJno Gk(rl~)] dA o
where the first integral is over the volume enclosed by S and the second is a normal outflow integral over all of S. The normal gradients are taken in the outward direction, away from the interior where 1/; is measured. If the surface S is at infinity and if outgoing waves are specified [causality condition, Eq. (7.2.17)], then G takes on the simple form gk(r lr o) for the infinite domain: gk(rlro)
= eikR/R; = i7rH~l)(kP);
=
(~:) eikl"'-"'.I;
3 dimensions; R2 = (x - XO)2 + (y - YoF + (z - ZO)2 2 dimensions; P ? = (x - XO)2 + (y - YoF" 1 dimension
The Green's function for the Poisson equation V21/; = -47rp is Go(rlro) , for A = O. The corresponding forms for the infinite domain are go(rlro) = (l /R); = - 2 In R;
3 dimensions 2 dimensions
892
[cH.7
Green's Functions
When surface S coincides, all or in part, with one of the set of separable coordinate surfaces discussed in Sec. 5.1, we ca n expand G in series of separated solutions. Suppose the boundary conditions (finiteness, periodicity, or homogeneous conditions on the boundary) are such that two of the factors can be eigenfunctions, say the ~2 and ~a factors. The b factor must also satisfy homogeneous conditions at the surface, which we assume corresponds to the surfaces ~l = a, b = b, b a. The coordinates have scale factors hl, h 2, h a; a Stackel determinant S with elements ilmn(~m); and minors M m = as/ailml [see Eqs. (5.1.25) et seq.]. The Helmholtz equation
n
and eigenfunction solutions satisfying appropriate boundary conditions are chosen for X a and X 2 : Wq(h,~a) = ~ ••(h)XI'.(~a);
v, P. = 0, 1,2, . . .
These are orthogonal with respect to a density function so that
p
(often
p
= h 2h a)
and the W's const it ut e a complete set for the coordinates ~2, ~a within the surface S. Two independent solutions, Ylq(b) and Y2q(b) , are chosen for the b factor, each corresponding to the separation constants of W q and arranged so that Yl satisfies the required boundary condition at b = a and Y2 the required condition at b = b(b a) . Then
where the scale factors have functions of the primed coordinates and where ~ is the Wronskian for the two ~~ solutions: ~ = ~(Ylq,Y2q)
=
YlqY~q
- Y~qY2q = (constant/h), a function of ~;
893
Table of Green's Funclions
CH.7]
Expansion of the Green's function for the infinite domain, of the sort generalized above, is given for two-dimensional polar coordinates in Eqs. (7.2.51) and (11.2.23), for rectangular coordinates in Eq. (11.2.11), for parabolic coordinates in Eq. (11.2.67) , and for elliptic coordinates in Eq. (11.2.93). The expansions for three-dimensional systems for rectangular coordinates are in Eq. (11.3.10), for spherical coordinates in Eq. (11.3.44) , and for spheroidal coordinates in Eq. (11.3.91). Similar expansions for vector solutions are given in Eqs. (13.3.15) and (13.3.79). Green's Function for the Wave Equation. G is a solution of 1 13 2
V 2G(r ,tlr o,to) -
C2 at2 G(r,tlro,to) =
-41l"0(r - ro)oCt - to)
satisfying homogeneous boundary conditions on surface S and obeying the" causalit y " requirement that G and aGlat = everywhere for t to. The reciprocity relation is then
°
G(r,tlro,to) = G(ro, -tolr, -t)
If if;(r,t) is a solution of V2if; - (1/c 2)(a2if; /W) = -41l"p(r ,t) having value if;.(r·) and outward normal gradient N.(f') on the surface S and having initial value if;o(r) and initial time derivative vo(r) = aif;latt=o within S at t = 0, then for t 0 and within and on S if;(r,t) =
ft+ dt« f dV oG(r,tlro,to)p(ro,to)
+ ~ ~t+ dto ¢dA o [G(r,t1ro,to)N.(ro) . - 4;C2
f
dV o [
- if;(ro) 13:0 G(r,tlro ,to)]
(~~\=o if;o(ro)
- Gto=ovo(ro)
l
E
~ 0+
The closed forms for the Green's function for the infinite domain are g(r,tlro,to)
= (l IR)o[(Rlc) - (t - to)]; for 3 dimensions ; R2 = (x - xo)2 =
- YO)2
to');;-2-----;P=2]U[(t - to) - (Pic)]; for 2 dimensions; P ? = (x - XO)2
= 2c1l"U[(t - to) -
where
+ (y
+ (z -
ZO)2
[2c/v,...,c2"(tc------:-
u(x) = 0 ;
x
0;
(Ix - xol/c) ]; u(x) = 1;
+ (y
- YO)2
for 1 dimension x 0;
t(x) = u'(x)
J-.. . o(x)f(x + a) dx = f(a) ; J-.. . o'(x)f(x + a) dx = -f'(a) The Green's function for the wave equation is related to the Green's function for the Helmholtz equation by the Fourier integral relationship
894
Green's Functions G(r,tlro,t o) = (c/27r)
f-.. .
[cH.7
Gk(r!ro)e-ikc(t-t,) dk
= 4?rc 2u(t - to)
L(:)
1f;n(ro)1/;(r) sin[wn(t - to)]
n
where 1/;n is the eigenfunction solution of V 21/; + k~1/;n = 0 within Sand where W n = knc. The contour for the integration over k is just above the real axis . Green's Function for the Diffusion Equation. G is a solution of V2Ga(r ,tlro,to) - a 2 (aj at)Ga(r,tlro,t o)
=
-411"o(r - ro)o(t - to)
satisfying homogeneous boundary conditions on surface S and obeying the causality requirement that G be zero for t to. The adjoint function 't(r,tlro,t o) = Ga(r, -tiro, -to) satisfies the adjoint equation V2G + a 2 (aGj at) = -4?ro(r - ro)o(t - to). The reciprocity relationship is Ga(r,tlro,to) = Ga(ro,tolr,t) = Ga(ro, - tolr, -t) If 1/;(r) is a solution of V21/; - a 2 (a1/;j at) = -411"p(r) , having value 1/;.(r') and outward normal gradient N.(r') = a1/;jan on surface Sand having initial value 1/;o(r) within S at t = 0, then for t > 0 within and on S
It+< dto f dV o p(ro,t) G(r,tlro,to) + 4~ It+< dto f dAo [ G(r,tlr~,to) N.(r'O) -
1/;(r,t) =
+ ::
f
1/;8(r~) a~o G(r,tlr~,to) ]
dV 0 1/;o(ro) G(r,tlro;O) ;
E
~ 0+
The form of the Green 's function for the infinite domain, for n dimension~, IS
411" ( ga(r,tlro,t o) = 2" a
)ne-
_a~ 2 v 1I"r
a ' R' / 4T
u(r)
where r = t - to and R = [r - rol. The Green's function for the diffusion equation is related to the eigenfunctions 1/;n for the related Helmholtz equation, (V2 + k~)1/;n = 0 for the domain within S, by the equation
411" \ ' e-(kn'/a')(t-I,) Ga(r,tlro,to) = a2 u(t - to) Nn 1f;n(ro) 1/;n(r)
Lt n
CR.
7]
Bibliography
895
Bibliography The literature on Green 's functions and their applications is rather spotty. Satisfactory accounts of various aspects of the theory : . Bateman, H. : "Partial Differential Equations of Mathematical Physics," Chap. 2, Cambridge, New York, 1932. Carslaw, H . S. : "Mathematical Theory of the Conduction of Heat in Solids," Macmillan & Co., Ltd., London, 1921, reprint Dover, New York, 1945. Courant, R., and D. Hilbert: "Methoden der Mathematischen Physik," Vol. 1, Springer, Berlin, 1937. Kellogg, O. D. : "Foundations of Potential Theory," Springer, Berlin, 1939, _ reprint, Ungar, New York, 1944. Murnaghan; F. D., "Introduction to Applied Mathematics," Wiley, New York, 1948. Riemann-Weber : "Differential- und Integralgleichungen der Mechanik und Physik," Vieweg, Brunswick, 1935. Sommerfeld, A. : "Partial Differential Equations in Physics," Academic Press, New York, 1949. Webster, A. G.: "Partial Differential Equations of Mathematical Physics," Chap. 5, Stechert, New York, 1933.
CHAPTER
8
I ntegral Equations
In the preceding chapters, we have relied mainly upon the differential equation to describe the propagation of a field y.,. Boundary condit ions have to be specified in addition, for the differential equation describes only the local behavior of y." relating y., at a point r to y., at r + dr. Starting from a given point r, the differential equation permits the construction of the many possible solutions in a stepwise fashion . The boundary conditions are then invoked to choose the solution which is appropriate to the physical situation of interest. Inasmuch as the boundary values are a determining feature, it would be useful to formulate the determining equation for y., in such a manner as to include the boundary conditions explicitly. Such a formulation must relate y.,(r) not only to the values of y., at neighboring points but to its values at all points in the region including the boundary points. The integral equation is an equation of this form . Since it contains the boundary conditions, it represents the entire physics of the problem in a very compact form and, as we shall see in many instances, a more convenient form than the more conventional differential equation. This is not the only reason for studying integral equations. We have already seen, for example, in the discussion of diffusion and transport phenomena, that there are many situations which cannot be represented in terms of differential equations. In other words, there are problems in which the behavior of y., at r depends on the values of y., at some distance from r, not just on the values at neighboring points. In the first section of this chapter we shall display some of the integral equations which arise in physics and shall discuss their classification into various types, each having different properties and techniques for solution. Then after a discussion of general mathematical properties 'of these types, we shall devote the rest of this chapter to discussing techniques of solution.
.
8.1 Integral Equations of Physics, Their Classification We consider first an example from transport theory. Here , as the consequence of a collision, a particle which was originally traveling in a 896
Integral Equations of Physics
§8.1]
897
given direction with a certain energy, as specified by its momentum po, may acquire a new momentum P which is very different both in direction and in size from the original Po. More explicitly, let P(plpo) dp dt be the probability that a particle having a momentum Po be scattered into momentum between p and p + dp in a time dt. If the original distribution function !(r,po,t) dpo gives the relative number of particles having a momentum between Po and Po + dpo at a point r, then the collisions which occur in a time dt contribute the following to !(r,p,t) : [fP(plpo) fer ,Po,t) dpol dt We see immediately that the value of !(r,p,t) at p is related to all the values of !(r,po,t) for all values of Pe consistent with conservation of momentum and energy. To complete the picture let us obtain the complete equation for! by considering the other changes in !(r,p,t) in a time dt. The above term gives the number of particles scattered into the volume of configuration space at rand p. A number of particles leave this region because of scattering or because of their complete disappearance by absorption. Let the probability per unit time for scattering or absorption out of p be PT(p). If there is no absorption, PT(p) = fP(polp) dp. The number of particles leaving in a time dt is P~(p)
!(r,p,t) dt
Finally, even if there are no collisions, a change in! occurs simply because the particles are moving. The particles at r were, at a time dt earlier, at r - (p/m) dt. Thus fer , p, t
+ dt)
=
![r - (p/m) dt, p, tl - PT(p)!(r,p,t) dt + [fP(plpo)!(r,po,t) dpol dt
The equation states that the particles at r at a time t + dt consist (1) of those which arrived there as a consequence of their motion (2) less those which scattered or absorbed out of the momentum range dp (3) plus those which were scattered into the range dp by collision. By expansion we finally obtain a differential-integral equation (see Sees. 2.4 and 12.2):
:{ = -
(!.V)! -
P T! +
f
P(plpo)!(r,po,t) dpo
(8.1.1)
Under steady-state conditions, ! is independent of t and
(~ . V)! =
-Pd
+
f
P(p!po)!(r,po) dpo ; st eady state
(8.1.2)
We emphasize again the dependence of ! on the complet e fun ctional dependence of! on p rather than just on the relation of! for neighboring
898
Integral Equations
[cH.8
values of p. In Sec. 2.4 the probabilities P T and P are directly related to the cross sections and the above equation is converted by direct integration from a differential-integral equation into an integral equation. We shall again discuss transport equations in Chap. 12. Example from Acoustics. It should not be thought that this type of equation occurs only for transport problems, where the collision process % is a rather obvious source of discontinuous changes in the momentum p. We may, for example, consider a case in acoustics. As we shall see, differentialRigid Plate integral equations arise whenever two Membrane \--+--_._ _.x_ - systems with distributed mass or other relevant parameters are coupled. For example, consider the vibrations of a membrane set in a rigid plate as illustrated in Fig .8.1. The vibrations of the membrane give rise tosound waves which Fig. 8.1 Radiation from a memin turn react back upon the membrane, brane in a rigid plate. influencing its vibration, etc. Suppose that the displacement of the membrane is given by t/;(y,z) ; the corresponding velocity in the x direction is iN /iJt = -iwt/;, assuming simple harmonic time dependence. The resulting velocity potential in the medium to the right is, from Eq. (7.2.10), cp(x,y,z)
=
-
Lf
Gk(x,y,zIO,yo,zo)vn(yo,zo)
as,
where Gk is a Green's function satisfying the condition (aGk /an) = 0 ; at x =
°
where k = w/c, c = velocity of propagation of sound and where V n is the normal component of the velocity, i .e., the velocity in the negative x direction. Hen ce V n = iwt/;. The function Gk is provided by the method of images (see page 812) : Gk(x,y ,zlxo,yo,zo) = (eikR/R) + (eikR'/R') (RF = (x - XO)2 + (y - YO)2 + (z - ZO ) 2 (R'F = (x + xo)2 + (y - yoF + (z - zoF
The sound thus generated in the region x > 0 gives rise to a pressure which now becomes a forcing term in equation of vibration of the membrane. The pressure is related to the velocity potential by the equation p = po(acp/at) = -iwpocp
where Po is the mean density of the medium in which the Round propagates. The equation of motion of the membrane is
§8.1]
Integral Equations of Physics v~
where
+ K2if; =
-(piT); T = tension ;
K
899
= wlV ;
JL =
V = VTlp. mass / area
Substituting for p we obtain
In this equation we again see that the behavior of if; depends not only upon the value of if; at a point on the membrane and neighboring points but also on its values at every point on the membrane in the manner dictated by the integral on the right. The above equation is a differential-integral equation, but by employing the Green's function for the membrane it may be reduced to an integral equation. It should be clear from the above example that an integral equation will result when an impulse at one point in a medium may be transmitted to a point some distance away through another medium ' coupled to the first . The equation for describing the vibrations of the first medium will contain a term arising from the propagation in the second medium. This term will involve the values of if; at all points at which the two media are in contact; the integral in the case discussed above is just such a term. Radiation problems, in which the reactive effects of the emitted radiation back on the source cannot be neglected, will naturally lead to integral equations. Solution of the integral equation will give rise to precise evaluation of radiation resistance or, more general, radiation impedance. We shall consider such problems in more detail in Chaps. 11 and 13. An Example from Wave Mechanics. As a final example let us turn to quantum mechanics. The Schroedinger equation must be written as an integral equation whenever the potential energy is velocity dependent. Let the potential energy V be V(r,hVI i )
where we have already replaced the momentum operator by (hl i)V. The Schroedinger equation in differential form is V2if;
+ 2h",: {E
- V[r,(hli)V]!if; = 0
If the dependence of V on V is not that of a simple polynomial, this equation is not one of finite order. To obtain the equivalent integral equation, introduce the Fourier transform of if;:
if;(r) = -1(27rh)i
foo_
00
r,o(p)e{i/,,)p·r dp
900
[CH. 8
Integral Equations
Substituting in the Schroedinger equation, multiplying through by [1/ (21rh)lje-(i/fl)q.r, and integrating on r yields
(i~) \p(q) ;+
1-.. .
\p(p)V(p - q, p) dp = E\p(q)
V(p - q p) = -1-
,
(21rh)f
J"
_..
e i(p-q).r/A
(8.1.4)
VCr p) dV
'
This integral equation determining \p(q) was given earlier in Sec. 2.6. The meaning of the integral can be most easily seen if we consider a scattering problem in which the scattering is caused by the potential V. If a plane wave of amplitude \p(p) is incident upon the region where the potential exists, then it is scattered. In other words , some of the incident wave is diverted into other directions, possibly with loss of momenta. We now ask for the contribution from a variety of plane waves with different momenta, to a given momentum q upon scattering by V . This is given by the integral term. Here is a correspondence with transport phenomena, discussed earlier in this chapter, which may be employed to obtain a graphic understanding of quantum mechanics. We have devoted some space to discussing problems which require integral operators. We have also pointed out that even those problems which are expressible in terms of a differential equation may be reformulated in terms of an integral equation. Several examples of this sort will now be given. Boundary Conditions and Integral Equations. The integral equation formulation is of particular advantage in the case of boundary-value problems associated with partial differential equations. In the example which follows, a second-order partial differential equation in two dimensions will be restated as an integral equation in only one dimension. A reduction in the dimensionality is, of course, extremely worth while from the point of view of obtaining both exact and approximate solutions. Consider the Helmholtz equation V 2if;
+ k if; = 2
0 Place a barrier along the negative x axis as shown in Fig. 8.2. A plane wave eik ' r traveling in. the k direction is in cident upon the barrier. We shall be interested in the effect of the barrier on this wave in the case that the solution if; satisfies the boundary condition aif;/ ay = 0 on the barrier. This solution must satisfy the following conditions at large distance from the origin. In the lower half plane, y < 0, if; - - - - ? 2 cos(kyY) ei le. ::; + a diverging cylindrical wave r.......
(8.1.5)
00
In the upper half plane, y
> 0, there will be
no reflected wave , so that
if; - - - - ? eik ' r + a diverging cylindrical wave r-+
00
(8.1.6)
§8.1]
Integral Equations of Physics
901
Because of the asymmetry of these boundary conditions we shall have to treat y; for y > 0 and y < 0 somewhat differently, so that it will be necessary to verify that 1/; and (iN/ay) are continuous at the interface of these two regions, i.e., for y = 0, x > O. From the theory of Green's functions given in Chap. 7 we have the general equation for y ~ 0 :
1/;(r) = 2 cos(kyY) eik•x
+
L¢[
Gk(rlro) :to - y;
aG~~~ro) ] as,
The path of integration is shown in Fig. 8.2. The proper function Gk is chosen by the requirement that for large r the proper boundary conditions be satisfied. It is clear that Gk must be a diverging source, in which
,
I
Barrier\.
ti" , ''',I< "'"
??
1-. -
-
- -
-
x
--'"-
Contour C
Fig. 8.2 Diffraction from a half-plane barrier at y x < o. Contour for integral equation.
=
0,
event the integral around the large semicircle vanishes. The integral along the x axis is then simplified by choosing [aGk(rlxo,Yo)/ayo] = at Yo = 0. Then
°
+ 41 )0i " Gk(rlxo,O) (aay;) dx«; y < ° (8.1.7) Yo Gk(rlro) = 7ri[H o(kR) + Ho(kR')] xo)2 + (y - Yo) z; R' = yf'7(x---xo""'):::-z-:-+--;-(y----,-+-y-07"""::)
1/;(r) = 2 cos(kyY) eik•x
7r
R =
vi (x
-
yo-O-
Z
°employing the same G one finds (8.1.8) y;(r) = - -4 fo" Gk(rlxo,O) (a1/;) -a dx«; y > ° Yo The boundary. condition (8.1.6) for the region y > °is clearly satisfied. In the region y
>
k
1
7r
0
"0=0+
It is now necessary to introduce continuity conditions. At y = 0, the function y;(xo,O) computed from (8.1.7) and from (8.1.8) should agree . Hence , for x > 0, 2eik• x =
-...!.- )0(., Gk(x,Olxo,O) [(a1/;) ayo 47r
Yo=O+
+ (ay;) ayo
Yo=O-
Jdxo
Integral Equations
902
[cB.8
Moreover the slopes must be continuous; i .e.,
(::0)
Hence
1"'
2eik.", = - o~ 1
0
UTr
or
2eik.", = - i
yo= o+
=
(:t)
yo=o -
Gk(x,Olxo,O) (Of) ~ uYo
t : H~l)[klx -
dx«;
xol] (of)
)0
x
>0
Yo- O
oyo
dxo
(8.1.9)
yo= o
This is an integral equation for (of/oyo)yo=o . Once (of/oyo) yo=o is known it can be substituted into (8.1.7) and (8.1.8) to obtain the diffracted as well as reflected wave. One needs another equation for x < O. Note that the integral equation is one-dimensional. It includes all the boundary conditions. Other boundary conditions on the barrier or on the aperture (i.e., other than continuity on the aperture) would have led to another integral equation. Equations for Eigenfunctions. Another type of integral equation may be obtained from the Schroedinger equation : [ V2
or
+ 2h7 (E
- V)] f = 0
+
[V k2 2)E; k = (2m/h U 2
2
Rewriting (8.1.10) as (V2 is given by
+k
2)f
U]f
=
0
(8.1.10)
= (2m/h 2 ) V
= Uf, we see that a solution of (8.1.10) . (8.1.11)
where Gk has been chosen so as to ensure satisfaction of boundary conditions for f . Equation (8.1.11) is an integral equation for f . It differs from (8.1.9) in that f appears both within and outside the integral. To obtain a better understanding of (8.1.11) it is convenient to turn to a one-dimensional example. We consider the Sturm-Liouville problem discussed in Sec. 6.3. The unknown function f satisfies
t [p ~~] +
[q(z)
+ Xr(z)]f
=
0
(8.1.12)
To perform the reduction from Eq. (8.1.10) to Eq. (8.1.11) here, the Green's function is introduced, which satisfies d [ P dG(zIZo)] dz dz
+ qG(zlzo)
= - 8(z - zo)
(8.1.13)
We now must add the boundary conditions to be satisfied by G and f. For definiteness one assumes that feZ) and f(O) are known, i .e., Dirichlet
Integral Equations of Physics
§8.1]
903
conditions on if;. The corresponding conditions on G are G(Ol zo) = 0 = G(llzo). Then, transposing (Arif;) in (8.1.12) to the right-hand side of the equation and considering it to be an inhomogeneous term, one finds if;(z) = A
II
G(zlzo)r(zo)if;(zo) dz«
+ [if;(O)P(O) dGJ::1z) 1.-0 - [if;(l)P(l)
(dG~::lz») 1.-1
(8.1.14)
This is an integral equation for if;. The boundary conditions on if; enter explicitly, indicating again how the integral equation includes all the pertinent data on the problem. There are no additional conditions on if; to be satisfied. If the boundary condit ions satisfied by if; are homogeneous Dirichlet, if;(0) = 0, if;(l) = 0, then the integral equation for if; becomes (8.1.15) Eigenfunctions and Their Integral Equations. Let us illustrate this last equation by giving the integral equation satisfied by the classical orthogonal functions:
d~: + Aif; =
(a)
if;(z) = A G(zlzo) =
if;(l)
= if;(0) = 0
fol G(zlzo)if;(zo) dz«
.!. { z(l l
- zo); z zo(l - z); z
< Zo > Zo
sin (n7rz/l) ; A = (n7r/l) 2; n integer
Solution :
~ [(1
(b)
0;
if;(z)
- Z2) : ] =
G(zlzo) =
A
+ Aif; = 0 ;
if; finite at z
fl G(zlzo)if;(zo) dzo -
i f~l if;(zo)dzo zo)]; z < Zo z)]; z > Zo
i { In [(1 + z) /(1 In [(1
+ zo) /(1
-
Solution: Legendre polynomials Pn(z) ; A = (n)(n (c) if;(z)
= ±1
+
1);
n integer
~ :z [z~~] + [A - ::] if; = 0; \II finite at z = 0, ec = At : G(zlzo)if;(zo)zo dz o; G(zlzo) = (21) {«z//zo»n ; z < Zo 10 n Zo Z n; Z > Zo
Solution :
0
Bessel functions J n(
z)
904
Integral Equations
~~ + [13 2 -
(d)
a 2z2]f = 0;
-ddzf2 + [A 2
or
[cH.8
f( 00 ), f( - 00) finite
a - a 2z 2]f = O',
A
= 13 2
+a
J-.. . G(z!zo)f(zo) dz« eiaz. f e-a~' d~ eiaz.' J." e-a~' d~; G(zlzo) =!: - .. ~ { eiaz.' J~' . e-a~' d~ eiaz' J."z. e-a~' d~; f(z) = A
z
7r
Hermite function s e- iaz'H n (0
Solution: (e)
~ tz [Z2
:J + [
-132
+ 2za
A = 2(n
z);
J
f = 0;
Solutions :
. 1 0
Go(zlzo)zof(Zo) dz«; A = 2a;
e-/lzL~(2I3z)
< Zo
z
> Zo
+ 1)a;
n integer
f(O) , f( 00) finite
We may identify A with either 2a or a 2 - 13 2• f(z) = A
z
In the first case, e-/llz-z.1
G(zlzo) = - 213zz o
where L n(2I3z) are the
Laguerre polynomials ; (aim - 1 = n;
n integer
In the second case an equivalent integral equation is
(f)
1.!!. [ Z2dfJ + [A Z2 dz dz f(z) = A
- a 2 + 2aJ f = O' z'
fo" G(zl zo)z~f(zo) dzo
G(zlzo) = e-a(z+z.)
f
l
zo
.
( ..
e2a~ -
~
2
e2a~
J. p
d~ ;
z < Zo
d~; z > Zo
Types of Integral Equations; Fredholm Equations. We may now proceed to classify the integral equations already discussed and to generalize somewhat. Turning to Eq. (8.1.14), we see that it may be written f(z) = A
lab K(zlzo)f(zo) dz o + cp(z)
(8.1.16)
where in th~t case K(zlzo) = r(zo)G(zlzo) and cp(z) is a known function ; f satisfies boundary conditions. An integral equation for f of the form of (8.1.16) is called an inhomogeneous Fredholm equation of the second kind. The quantity K(zlzo) is called the a and b are the fixed points at which
§8.1]
Integral Equations of Physics
905
kernel of the integral equation. A kernel is symmetric if K(zlzo) = K(zolz). In (8.1.14) the kernel is not symmetric if r(zo) ~ 1. A homogeneous Fredholm equation of the second kind may be obtained . by omitting Ip(z) :
(8.1.17) Equation (8.1.15) and the cases (a) to (e) tabulated immediately below it are examples of homogeneous Fredholm equations of the second kind . In cases (a), (b), (d), the kernel is symmetric. Cases (e) and (f) and the kernel in (8.1.14) are examples of a polar kernel: K(zlzo) = G(zlzo)r(zo); where G(zlzo) = G(zolz)
(8.1.18)
The kernels in everyone of the cases are definite in the region of-interest, i.e., in the range 0 :::; z :::; l for case (a), 0 :::; z < 00 for case (c), etc . A positive definit e kernel satisfies the inequality
fab dzo fab dz[K(zlzo),p(z)1f(zo)] >
0
where K is presumed real and 1f is arbitrary. For a negative definite kernel the above integral is always less than zero. In either case the kernel is definite. In the event that neither inequality is true for an arbitrary 1f, the kernel is said to be indefinite. Equation (8.1.9) is an example of a Fredholm equation of the first kind which has the general form
= fa bK (z[zo)1f(zo) dzo
Ip(z)
(8.1.19)
Here 1f is the unknown function , Ip is known. Volterra Equations. Fredholm integral equations (8.1.16), (8.1.17), and (8.1.19) involve definite integrals. When the limits are variable, the corresponding equations are Volterra equations . An inhomogeneous Volterra equation of the second kind is, corresponding to (8.1.16), 1f(z)
=
fa' K(zlzo)1f(zo) dzo + Ip(z)
In the homogeneous variety corresponding to (8.1.19), is Ip(z)
Ip
= O.
(8.1.20)
A Volterra equation of the fir st kind,
= fa" K(zlzo)1f(zo) dzo
(8.1.21)
The Volterra equation, when convenient, may be considered as a special case of the corresponding Fredholm equation where the kernel employed in the Fredholm equation is M(zlzo) = {OK(zIZo); ;
Zo Zo
z
(8.1.22)
906
Integral Equations
[CR. 8
In the preceding chapter, we have met Green's functions which satisfy Eq. (8.1.22). It will be recalled that, in discussing time dependent problems, it turned out that G(r,tlro,to) = 0 for t < to, as a consequence of the principle of causality, which requires that an event at a time to cannot cause any effects at a time t earlier than to. We may expect that an integral equation involving G as the kernel will be of the Volterra type. To illustrate the origin of the Volterra equation, consider the motion of a simple harmonic oscillator: (d2if; /dtg)
+ k 2if; =
0
Define a Green's function G(tlto) for the impulse function oCt - to) by d2G(tlto) /dt~
=
-oCt - to)
and
G(tlto) = OJ
if t
< to
Multiplying the first of these equations by G and the second by if;, subtracting the two, and integrating on to from to = 0 to to = t+ (where t+ signifies taking the limit of the integral as to ~ t from the side for which to > t), we have
-
fo'+ dto {G(t1to) ~:t d2~~~lto) if;} + k 2fo'+ G(tlto)if;(to) dto = if;(t) or if;o [dG d~t\to) ] o
- G(tIO)vo 1,=0
+ k 2 10(I+ G(tlto)if;(to) dto =
if;(t)
(8.1.23)
where if;o and Vo are the initial values of the displacement if; and the velocity dif;/dt. Equation (8.1.23) is an inhomogeneous Volterra equation of the second kind. The initial conditions on if; are explicitly contained in it . From the example it is clear that Volterra equations will result whenever there is a preferred direction for the independent variable; in the above example this is one of increasing time. Another case of a similar kind occurs in transport theory whenever collisions are made with massive scattering centers. Then the energy of the scattered particles cannot be greater than their energy prior to the scattering. As a consequence there is a degradation in energy giving a preferred direction to the energy variable. As an example of this sort of equation consider a beam of X rays traversing a material in the direction of increasing x. We shall neglect the change in direction in scattering, presuming that the X rays all propagate directly forward after scattering. In ' passing through a thickness dx , the number of X rays of a given wavelength are depleted by absorption and scattering out of that wavelength range and are increased by scattering from those X rays whose energy (energy 0: 1/;\) is greater (in other words, whose wavelength is shorter) . Hence if
§8.2]
General Properties of Integral Equations
907
!(X ,x) ax is the relative number of X rays with wavelength between X and A + dX, then
a!~~x) =
-p.f(X,x)
+
lA
P(XIXo)!(Xo,x) dXo
where p, represents the absorption coefficient and P(XIXo) dX is the probability, per unit thickness, that an X ray having a wavelength Xo is scattered into a wavelength region dX at X. This is an integral-differential equation; to reduce it to a pure integral equation let !(X,x) =
fo
00
e- p"if;(X,p) dp
Then if;(X,p) satisfies the homogeneous Volterra equation of the second
.
~~ :
(p, -
p)if;(X,p)
=
loA P(XIXo)if;(Xo,p) dXo
8.2 General Properties of Integral Equations In discussing the general properties of integral equations it will be convenient to utilize the results obtained for operator equations in abstract vector spa ce. As we shall now see, the Fredholm integral equation is just a transcription of an operator equation to ordinary space. Consider the inhomogeneous equation in vector space: ~' e
=
Xe
+f
(8.2.1)
Since the Green's fun ction which occurs so frequently in the formulation of integral equations is intimately connected with the inverse operator (see page 882), it will be profitable to rewrite (8.2.1) as e =
X~-l •
e
+ g;
g =
~-l • f
(8.2.2)
To transcribe this equation to ordinary space let us expand the vectors in terms of the eigenvectors of the operator z, e(zo). Let
e = f e(zo)if;(zo) dz o g ~-l
•
e(zo)
~-l
•
= f e(zo)lp(zo) dzo
fe(Zl)K(Zllzo) dZ l e = fe(zo) dzofK(zolzl)if;(Zl) dZ l =
(8.2.3)
Introducing these definitions into (8.2.2) yields an inhomogeneous Fredholm equation of the second kind: if;(zo) = XfK(zolzl) if;(Zl) de;
+ lp(zo)
The limits of integration are included in the definition of K .
(8.2.4)
908
Integral Equations
[cH.8
The integral equation is thus often equivalent to a differential equation "handled in reverse." Instead of the differential operator, equivalent to m:, which is studied, it is the integral operator, symbolized by m:- 1, which is examined. The Fredholm equation of the first kind - 0 if K is positive definite) . A variational principal for A,
914
Integral Equations
[CR. 8
obtained from Eq. (8.2.15), is A = min
I!
I
!{;(z)1/;(z) dz 1fi(z)K(zlzo)1/;(zo) dz dzo
(8.2.24)
Kernels and Green's Functions for the Inhomogeneous Equation. We turn next to the inhomogeneous equation 1/;(z) = >'IK(zlzo)1/;(zo) dzo + x(z)
(8.2.25)
This is to be solved in terms of a Green's function. We shall choose one to correspond to the (~h of (8.2.16). In abstract vector space Eq. (8.2.25) reads e = >,~-l. e + q The solution for e is obtained as follows: , (1 Hence
>,~-l)
e =
• e = q or (~- >.) . e = . q; ® = (~ - A)-l
~.
q
(%~)
This solution is not convenient, inasmuch as it involves the product of two operators. However, a little rearrangement of factors circumvents this difficulty. Write e so that
=
[®>.(~
e
=
- >.)] • q + >.% • q q + >'®>" q
(8.2.26)
This is the solution of Eq. (8.2.25) in terms of %. Expressed in ordinary space it reads (8.2.27) 1/;(z) = x(z) + AIG>.(zlzo)x(zo) dzo The Green's function given here is referred to, in the theory of integral equations, as the solving kernel . To obtain Eq. (8.2.27) directly without the intermediary of vector space it is necessary to formulate the integral equation for G>.. This may be obtained from the defining equation (8.2.16) for ®>., as follows: (~
or or
~-l(~
- >')®>. = - >.)% = ®>. =
3 ~-l
~-l
=
®>. _
>,~-l®>.
+ A~-l®>.
(8.2.28)
Translated into ordinary space this becomes an integral equation for G>.: G>.(zlzo) = K(zlzo)
+ >.IK(zlzl)G>.(Zl!ZO) de,
(8.2.29)
From this integral equation it is apparent that Go(zlzo) = K(zlzo)
Combining Eqs. (8.2.28) and the integral equation for 1/;, (8.2.25), it is possible, by employing the symmetry of G>. and K, to derive solution
§8.21
General Properties of Integral Equations
915
(8.2.27). However, it should be emphasized that this symmetry involves the possibility of interchanging limits of integration. This is, of course, contained in the definition of symmetry. It will be useful, for later discussion, to obtain an integral equation for K in terms of GA' Again turning to vector space, note t~at (~
-
A)~-l
= 3 -
~A-I
or
~-l
=
@ -
A@~--'l
In ordinary space this is K(zlzo)
=
GA(zlzo) - AfGA(zlzl)K(Zllzo) de,
(8.2.30)
The difference between this integral equation and the one for GA, Eq . (8.2.29), is only apparent. In Eq . (8.2.30) let z and Zo be interchanged, so that K(zolz) = GA(zolz) - AfGA(zolzl)K(zrlz) dz; Employing the symmetry properties of K and GA we are led to Eq . (8.2.29) from (8.2.30). This relation between K and GA led Volterra to introduce the term reciprocal functions to describe K and -GA' Expansions corresponding to Eqs. (8.2.18) and (8.2.19) for % and ~-l may be obtained for GA and K (see discussion on page 908) : (8.2.31) and
(8.2.32) m
To give the analogues for Eqs. (8.2.17), (8.2.20), and (8.2.22), it is first necessary to obtain the transcription of ~-p to ordinary space . Corresponding to the expression ~-p • e we have f K p(zlzo)1f(zo) dz o
where it now becomes necessary to find Kp(zlzo) . In ordinary space this is written as
Consider first
~-l
•
e.
~-2 •
e;
f K (zlzo)1f(zo) dz o
The effect of ~-2 on e may be obtained as
~- l • (~-l •
e) or
ffK(zlzl)K(Zllzo)1f(zo) dz o dZ I Hence K 2(zlzo) = fK(zlzl)K(Zllzo) dZ I The effect of ~-3 on e may be obtained by operating with ~-l on hence K 3(z!zo) = fK(zlzl)K2(Zllzo) dZ l It is clear from the general relation ~-(p+q) = ~-p~-q that Kp+q(zlzo) = fKp(zlzl)Kq(Zllzo) dZ l = fKq(zlzl)Kp(Zllzo) dZ l
(8.2.33)
916
Integral Equations
[CH. 8
We may now write the equivalents of Eqs, (8.2.17), (8.2.20), and (8.2.22). The analogue of (8.2.17) is ee
Gx(zlzo)
=
L Kn+1(zlzo)A n
(8.2.34)
n=O
The expansion for K; is the analogue of that for ~-p as given in (8.2.20): Kp(zlzo) =
1¥-m(Z~~m(Zo)
(8.2.35)
m
The statements as to the scalar values of @x and ~-p may be converted into statements about G» and K p once the equivalent process to that of obtaining the scalar value I@I is known. This consists of placing z = Zo in the kernel, i.e., obtaining the diagonal element of the matrix representing @x, and then integrating over z, the equivalent of summing over these diagonal elements. Hence .
while and
f f f
Gx(zlz) dz =
1 1 1
Am
~X
(8.2.36)
m
Kp(zlz) dz =
1 Xf:. =
c,
(8.2.37)
m
Gx(zlz) dz =
Cn+1xn
(8.2.38)
n
Semidefinite and Indefinite Kernels. In many cases the kernels are not definite and the corresponding operators not Hermitian, so that the discussion above does not apply. Even after iteration, for some cases, the final operator is only semidefinite rather than definite, so that the above theorems are still not appropriate. What may be said about kernels which are not definite? First of all it is no longer necessarily true that the eigenvalues are real. The eigenvalues again are in some cases finite in number. In the case of the Volterra equation, for example, there are no eigenvalues; no solutions of the homogeneous equation (see page 920). As an example consider the following simple, nondefinite kernel and the corresponding Fredholm integral equation: ¥-(z)
=
X
fo
1
(z - 2zo)¥-(Zo) dz«
It is clear that ¥-(z) is a linear function of z : ¥-(z) =
O:Z
+ {3
§8.2]
General Properties of Integral Equations
917
The constants may be determined by introduction of the lin~ar function into the integral equation az
+ {3
= A fol (z - 2z o) (azo + (3) dz o
Equating like powers of z, one obtains a = A(-!a
+ (3);
(3 = 2A( -ia - -!(3)
These are a pair of homogeneous linear simultaneous equations for a and {3. For a nonzero solution the determinant of the coefficients must vanish. Hence 1 - -!A 1
1
%A
-AI +A
=0
There are two roots : Al = !( -3
+i
y15);
A2 = XI = -!(3
+ i Vl5)
The corresponding solutions are 1/;1
;= Z -
-! + (l /AI);
1/;2 = z - -! + (1/A2)
In this example there are only two eigenvalues and there are only two • solutions; the two eigenvalues are complex conjugates. Because the eigenvalues and eigenfunctions may be finite in number, it may be, of course, no longer possible to expand an arbitrary function in terms of these eigenfunctions. The statement of completeness must therefore be made only with respect to a certain class of functions. For example, the eigenfunctions for the problem considered just above can be employed to express any linear function of z, More important, the eigenfunctions are sufficiently complete to permit the expansion of the kernel K(zlzo) and the Green's function Gx(zlzo). Hence it stili remains possible to solve this inhomogeneous Fredholm equation of the second kind in terms of its eigenfunctions. The necessary formalism for this situation has already been considered in Chap. 7. The eigenvalue problem in abstract vector space is &.e
= xe
(8.2.39)
We define the Hermitian adjoint eigenvalue problem [see Eq. (7.5.40)]: &* . f = Xf
(8.2.40)
The solutions of (8.2.39) are orthogonal to the solution of (8.2.40), so that we may set (8.2.41) This relation may be employed to evaluate the coefficients in the expansion of those vectors which may be expressed in terms of 'en [see Eq. (7.5.43)] :
918
Integral Equations g =
Lgnen;
[cH.8
(8.2.42)
gn = (f:. g)
n
Expansions of the Green's operator are also available :
e, = @x
where
\' emf:'
Lt 'Am -
(8.2.43)
'A
m
=
(~
-
'A)-1
The Spur of an operator, when expanded is defined by
III
terms of em and
f:',
(8.2.44) The expansion factors (Spurs) 1~-II, I~-pl, I@xl and the expansion of in terms of I~-pi are given in Eqs. (8.2.19) to (8.2.22), respectively. If there are only a finite number of eigenvalues, relations must exist between the Spurs of various powers of 1~-II . If there are, say, q eigenvalues, it is possible to express these eigenvalues in terms of the Spurs of the first q powers of 1~-II. Hence 1~-(q+l)1 may be expressed in terms of the scalars 1~-II, 1~-21, . . . , I~-ql. The variational problem satisfied by the eigenvector solutions of Eqs. (8.2.39) and (8.2.40) is
I@xl
'A = stationary value of (f* • ~ • e) j(f* • e)
(8.2.45)
By varying f* in the above equation, we find [see Eq. (6.3.74)] that 0'A = 0 yields Eq. (8.2.39) , while by varying e, we find that 0'A = 0 yields Eq. (8.2.40) for f. Equation (8.2.45) is similar to (8.2.14). A variational problem similar to (8.2.15): 'A
= stationary value of
(f* • e)j(f* • ~-l • e)
(8.2.46)
The case of the skew-Hermitian operator (corresponding to an antisymmetric real kernel) deserves special attention. In that case ~* = -~ . Moreover, if ~
(~*~)
then
.e
=
xe
. e = -'A 2e
From the fact that ~*~ is definite, that is, (e* • ~*~ • e) ~ 0, it follows that -'A 2 ~ 0 or that 'A is a pure imaginary. A second point of interest concerns the solutions of the adjoint eigenvalue problem. Whenever ~ is skew-Hermitian, it follows that en = fn; for if ~
then
~*
• en = 'Ane n • en = - ~ . en = - 'Ane n
Turning to orthogonality relation (8.2.41), we see that the en vectors for skew-Hermitian operators are mutually orthogonal, as in the case of solu-
§8.21
General Properties of Integral Equations
919
.
tions of eigenvalue problems involving Hermitian operators. ' The expansion of the Green's operator can be carried through for the latter case, and hence the formulas for positive definite operators may be taken over bodily. Kernel not]Real or Definite. Let us now consider the application of Eqs. (8.2.39) to (8.2.46) to integral equations. Because of the greater complication of the present case we shall illustrate some of the results with the specific integral equation with the kernel (z - 2z o) discussed above. The eigenvalue equation with a kernel which is not real, positive definite, (8.2.'17) 1/;(z) = AIK(zlzo)1/;(zo) dz« has nonzero solutions for only special values of A; call them Am. Corresponding to these values Am there are corresponding 1/;'s, 1/;m, the eigenfunctions. The Am's are not necessarily infinite in number ; they are not generally real. The eigenfunction set, besides being not necessarily complete, is not mutually orthogonal. For this reason, we must introduce the Hermitian adjoint problem [corresponding to (8.2.40)] cp(z)
=
XIK*(zolz)cp(zo) dz o
(8.2.48)
As noted earlier, the eigenvalues of the adjoint problem are complex conjugates of those of (8.2.47) . Moreover, I i{Jp1/;q dz = Opq
(8.2.49)
where CPP and 1/;q are individual eigenfunctions. The eigenfunctions and eigenvalues for the kernel z - 2z o are given above (page 916). The solutions CPP satisfy the adjoint problem : cp(z)
= Jl 10 (zo - 2z)cp(zo) dz« 1
(This kernel K* was obtained by exchanging z and Zo in K and taking the complex conjugate of the result.) Again note that cp(z) is linear in z, so that cp = az + b. Then
-(a Hence
+ 2b)Jl = a; I
Jl ;- 1 1f.u
I
~.u
(ia
+ ib)Jl
= b
2.u I = 0 - 1
This equation, determining .u, is precisely the same as that determining A. Indeed, we may place JlI
However,
CPl
= Xl;
Jl2
= X2 = Al
and 1/;1 are not complex conjugates of each other, for
920
Integral Equations
[cH.8
which is to be compared with the earlier expressions for Y;I and Y;2. now very easy to show that, in agreement with Eq. (8.2.49),
10
1
ihY;2 dz
It is
=0
The fun ctions (()i and Y;i have not as yet been normalized in the manner determined by (8.2.49) . The normalization integral is e l l Jo Zo
where k(zlzo) and h(zlzo) are continuous but do not necessarily have equal values at z = zoo The iterated kernel K 2(zlzo) is given by the integral K 2(zlzo)
f
=
K(zlzl)K(Zllzo) dZ I
Introducing the definition of K we can evaluate K 2 : K 2(zlzo) =
fa
to
+t
h(zlzl)k(Zllzo) dZ I
h(zlzl)h(Zllzo) dZ I
+ f.b k(zlzl)h(Zllzo) dz 1;
=
f' k(zlzl)k(Zllzo) dZ
faz h(zlzl)k(Zllzo) dZ + I
z;::: Zo
1
+ i:(b k(zlzl)h(Zllzo) dz1 ;
Z
=:;
Zo
The function K 2(zlzo) is continuous for z ~ Zo, for by placing z = Zo, we find that the expressions valid for z ;::: Zo and z =:; Zo join on continuously. A singular kernel of the type K(zlzo) = H(zlzo)/[z - zol"; IH(zlzo)1
=:; M
(8.2.51)
may be reduced to a nonsingular one by a sufficient number of iterations if a < 1, that is, if the kernel is quadratically integrable. Let us compute a few of these:
-lb I
K 2 (Iz Zo) -
"
H(zlzl)H(Zllzo) d I - Zo I" Zl ZlI"Zl
Z -
923
General Properties of Integral Equations
§8.2]
We have IK2(zlzo) I :::; M2
J{ba IZ -
Idt I ZI "ZI - Zo a
:::; Iz - ZOI2"
l 11 - drrlalr"l :::; Iz - ZOI2"-1 j_ 11 - drrl"lr"l fJ
M2
1
M2
ee
'Y
ee
where 13 = (b - zo) /(z - zo) and 'Y = (a - zo)/(z - zo). The value of the final integral may be easily expressed in terms of beta functions. These are finite as long as a is not a positive integer or zero. We assume this to be true. Then
where Co is the value of the integral. It is clear that K 2 is bounded as long as 2a - 1 :::; 0, that is, as long as a :::; i . If this condition is not satisfied, we may continue the iterative process. However, as we shall now see, the iterative process will yield a bounded kernel only if a < 1. The basis of the proof lies in the fact that IKp(zlzo) I
Hence
IK3(zlzo)1 :::;
s
f
!K(zlzl) IIK p- 1(Zll zo)I dZ I
lab IK(zlzl) IIK 2(ZI!ZO) I de,
< COM3
-
<
r
CoM 3 zol3a-2
- Iz -
_..,..,-d_Z-=-1_----;-;;----;"
b _
Ja Iz - zll"lzl - ZOI2"
j'"_
ee
dr 11 - rl"lrI 2"
1
so that where C1 is a new constant. Generally (8.2.52)
The nth iterate will be bounded if na - (n - 1) a
<
(n - l/n)
< 0, that
is, if -(8.2.53)
Hence for a given a < 1 an n can be found such that K; 'is bounded. The Green's function for the Laplace equation gives rise to kernels whose singularities are very much like the ones just being considered, that is, similar to (8.2.51) . The three-dimensional Green's function is proportional to (8.2.54) The two-dimensional Green's function is proportional to the logarithm
924
Integral Equations
[CR. 8
of the similar two-dimensional quantity. By considering the more singular kernel [(x - XO)2 + (y - YO)2j-! the two- and three-dimensional cases may be considered together. sider the three-dimensional case. IK (r lr o) I ~
Let
M/ lr -
Con-
rol
Then K 2(rlro)
f
~
M2 .
v[(x - :l'd 2 + (y - YIP
~I~I~I
+ (z -
Z,)2J(XI - XO)2
+ (y -
YO)2
+ (z -
zoP]
We may reduce this case to that of (8.2.51) by employing the inequality (x - XI)2
+ (y
- YI)2
+ (ZI
- ZI)2 ;::: 3[(x - XI)2(y - YI)2(Z - ZI)2]!
Hence
~
(CM2/3 )[(x - xo)(y - Yo)(z - zo)]!
Therefore K 2 is bounded. A similar proof holds for the two-dimensional case. Green's functions appropriate to the Helmholtz equation do not have just the branch point or pole type of singularity as in Eq. (8.2.54) but contain an essential singularity as Ir - rol ~ 00 [see integral equation (8.1.9)]. Kernels of this type cannot be made nonsingular by iteration. We shall call such kernels intrinsically singular. The major difference between an intrinsically singular kernel and a bounded kernel, which is of interest to us, is the nature of the eigenvalue spectrum associated with the homogeneous Fredholm equation of the second kind. It may be shown that for bounded kernels the eigenvalues are denumerable even if there are an infinite number of eigenvalues. This is not necessarily true for intrinsically singular kernels. In that case the eigenvalue spectrum may be continuous; i.e., for a range of A, all values of A have nonzero solutions Y;>.. We may understand this difference as follows. If K; is quadratically integrable, then there exist meaningful expansions of K n in terms of a denumerable set of eigenfunctions, expansions which converge in the mean (see page 739). For example, it is possible to expand such a kernel in a double Fourier series. This is, however, not possible for a kernel which is not quadratically integrable. Generally speaking, in addition to a Fourier series, a Fourier integral over a properly chosen path of
§8.3]
Solution of Fredholm Equations of the First Kind
925
integration (avoiding the singularity) is required to represent a function which is not quadratically integrable. Indeed we have seen in our discussions of Green's functions that eigenfunction expansions for Green's functions (for both Laplace and Helmholtz equations) in the infinite domain must involve Fourier-type integrals, corresponding to the fact that these Green's functions are not quadratically integrable. We shall close this section with an example of an integral equation with an intrinsically singular kernel and a continuous eigenvalue spectrum. The equation is 1/;(z)
=
X
1-.. .
e-lz-zol1/;(zo) dzo
(8.2.55)
The singularity of the kernel occurs at the infinite limits. This may be reduced to a differential equation by differentiating twice. Then (d 21/;/dz 2 )
+ (2X -
I)1/; = 0
or
1/;>. = Aeyl-2>'z
+ Be- V l -
2>.z
However, the integral in (8.2.55) exists only if Reh/I - 2X] < 1. All values of X which satisfy this restriction are eigenvalues of (8.2.55). All corresponding 1/;>. are eigenfunction solutions. The eigenvalue spectrum is clearly continuous. Another example is given by case (c) following (8.1.15).
8.3 Solution of Fredholm Equations of the First Kind We shall limit ourselves to a discussion of those cases in which an exact solution may be obtained. Approximate techniques will be discussed in Chap. 9. It should be emphasized that approximate methods for the solution of physical problems are most conveniently based on an integral equation formulation. Thus although we shall not be able to solve a great many integral equations exactly, the approximate treatment of such equations will occupy a special position and will be discussed in detail in Chap. 9. The general methods to be described here are similar to those employed in the solution of differential equations discussed in Chap. 5. The principal feature of the procedure is the expansion of the unknown function in terms of a complete set of functions. This expansion would be given in the form of a sum or integral over the set, with undetermined coefficients. Upon introduction of the expansion into the differential equation a relation among the unknown coefficients could be determined. In other words, the differential equation is transformed into an equation or set of equations determining the unknown coefficients. The complete set is chosen, if possible, so that the new equations are readily solvable. For example, if the expansion is a power series 1/; = ~anZn+., the transformed equation is a difference equation for the coefficients an. For
926
Integral Equations
[CH. 8
certain types discussed in Chap. 5, this difference equation involved only two different values of n and could be easily solved . Series Solutions for Fredholm Equations. We shall now apply this method to Fredholm integral equations of the first kind, for which it is particularly appropriate. This integral equation, as given in Eq. (8.1.19) , is . y'2; V, where V is the transform of e-I",I. This may be easily obtained : V(k) = -1-
f"
y'2; - ..
e- I",lei k2: dk
where V(k) is analytic for the region 11m 1-
vz;;: >.V =
or
V2J;
V(k) = 1
kl < 1.
(k2 - (2)' - 1)l!(l
+ k2
We then obtain
+ k2)
The zeros occur at k = ± k« where k o = .y2>. - 1. These zeros are simple. Hence the solutions of the homogeneous equation are
x=
(8.5.24)
e±ik,,,,
These are solutions only if ± k o is within the domain of regularity of V, 11m kol < 1, which corresponds to the requirement that Im >. < 2 Re >.. These solutions may be verified directly by substitution in the original integral equation for x or by reducing the latter to a differential equation by differentiating the entire equation twice. We then obtain
We consider next the solution of the inhomogeneous equation as given by the first two terms of Eq. (8.5.17). For this purpose, cf>+ and cf>_ are required : . cf>+ =
~ ~ .. ea"'e
e., =
~
1°
y'2; -..
ikz
dx = -
e-a"'ei kz dx =
vz;;: (~ + ik) ; A
-yI2; (ik - Oi)
1m k >
; 1m k
<
a
-Oi
Integral Equations
968
The solution (8.5.17) is valid for only
f
a
[CR. 8
< 1. Then
ee + i.o" (1 + k2 )e- ikz _ .. +i.o" (k 2 - k3)(k _ ia) dk + in" (1 + k 2 )e- ikz } 2 + _.. + in" (k - k5) (k + ia) dk
A { t/;(x) = 21ri -
.f
(8.5 .25)
where we may add any solution of the homogeneous equation, i.e., any linear combination of the two solutions given in Eq . (8.5.24). The integration limits satisfy the conditions 'T~ < 1, -Ii > -1 as shown in Fig. 8.3. Consider two cases, x > 0, and x < O. In the first case we can add on a semicircle in the lower half plane to the contour of each integral and so obtain a closed contour. The integrals may then be
t
Im(k)
k Plane
---------------.---
t
'- Contour
for First Integral
I"
To
Re(k)-
Contour for Sec.and Integral
" - },
\
1
1
·-ko
_ _ _ _ _ _.
Fig. 8.3
. ---1.
_
Contours for integration of Eq. (8.5.25).
evaluated by an application of Cauchy's integral formula. The second integral vanishes since all the singularities of its integrand occur outside the contour. The first term may be readily evaluated to yield t/;
+
= A
a2 - 1 (k 2+ l)e(k 2+ l) e {a2+ k5 e + 2k0o(ko - ia) + .;;-;-::0,--;-;-----:,-:-....,. 2k o(ko + ia) {aa +- k\ e + k o V_k~a++1 k3 COS[ k oX - tan-1(ka)]} ; i koz
i k oZ
az
}
2
A
=
az
2
2
0
x> 0
0
For x < 0, the semicircle now runs in the upper half plane , so that only the second integral contributes: t/;-
=
A
{~2; ~2 e- a z + k a 0
o
.J-a+ +1 k3 COS[k 2
oX
+ t~n-l(ka)]} ;
x
TO ; the second for Im k < TI. The solution of the homogeneous equation may be obtained from (8.5.29) by replacing q,+ by 8+, q,_ by S_ and finally by placing S_ = -8+ so that if; = _1_ J. [S+(k) -
y'2; 'f
y'2; AV(k)S+(-k)] e- ikx dk 1 - 21l"A 2V(lC) V( -k)
where the contour is within the region in which V(k) and V( -k) are
Fourier Transforms and Integral Equations
§8.5]
971
analytic. By reversing the direction of integration on the second term if; may be written as J,. + .y2; AV( -k)e 2ikz] . if; = y!211' r S+(k) 1 _ 211'A2V(k) V( -k) e->kz dk (8.5.30)
1
[1
Since S+ and 1 + .y2; AV ( - k) are both analytic in the region within the contour, we may replace their product by one analytic function of k. Hence the homogeneous solutions are given by if; =
l Arox·-Ie-ikr"
(8.5.31)
T,'
where it has been assumed that 1
1 - 211'A 2V(k)V(-k)
The coefficients An are to be determined by initial or boundary conditions, though ratios between some of them must be adjusted to be consistent with the original integral equation. An Example. To illustrate, consider the integral equation if;(x) = Aeal"l
+ A J-.. . e-I"+".lif;(xo) dx«
From the results obtained in the discussion of integral equation (8.5.22) we have V(k) = vf27;/(1 + k 2) ; 11m kl < 1 ip+ = -A/.y2; (a + ik); Imk> a ip_ = A/.y2; (ik - a); Im k < -a Consider the solutions then at the roots of 1 given by k k
of the homogeneous equations first. We look - 211'A 2V(k) V( - k) = 0 which for this case are
= ±ko; = ±ik l
;
k o = y!2A - 1 k l = y!2A + 1
From (8.5.31) we may write the solution of the homogeneous equation as if; = ale- ik." + a2e ik oz + b1e-k,,, + b2e-k,,, It is now necessary to evaluate the ratios between coefficients of this expression by direct substitution in the integral equation. One obtains if;(x)
=
A y!211' [alV (ko)eik oz
+ a2V ( -
ko)e- ik . " + bl V( -ikl)ek,,,
+ b2V(ikl)e-
k ,,, ]
Equating coefficients of like exponentials yields the independent pair of simultaneous equations al - A y'2; V( -k o) a2 = 0;
A y'2; V(k o) al - a2
=0
972
Inlegral Equations
[cH.8
and b1
-
A vz;;: V(ik 1 ) b2 = 0;
A y211" V( -ik 1) b 1
-
b2 = 0
The requirement for nonzero solutions, that the determinant of these equations vanish, leads to the equation employed above to determine ko and k 1 and is therefore automatically satisfied. The ratio of the coefficients may now be found. For the ko solutions it is adal = A vz;;: V(k o) = 1 Hence if; = cos kox is a solution of the homogeneous equation. Similarly cosh k1x is another independent solution. Turning now to the inhomogeneous equation (8.5.29), we see that there is no essential difficulty in evaluating the integrals by applying Cauchy's integral formula with a procedure completely analogous to that employed in example (8.5.22). Applications of the Laplace Transform. As may be predicted from the above discussion, the Laplace transform may be most gainfully employed for integral equations whose kernels permit the application of the faltung theorem for the Laplace transformation:
.e [j"
v(x - xo)f(xo) dXo] = V(p)F(p)
where V(p) and F(p) are the Laplace transforms of v(x) and f(x) , respectively. This suggests that we should consider Volterra integral equations f(x) = cp(x)
+ f; v(x
- xo)f(xo) dx«; z
>0
(8.5.32)
Examples of Volterra equations arising from the solution of vibration problems and from energy absorption of X rays in matter have been discussed on pages 905 to 907. For the present let us consider the general situation. Take the Laplace transform of both sides of (8.5.32). Assume that the transform of cp is analytic for Re p > TO and that the region of analyticity of the transform of v has at least a strip, parallel to the imaginary axis of p, in common with the band for cpo In that strip, F(p) = 4>(p)
+
V(p)F(p)
Solving for F(p), we have F(p) = 4>(p) j[l - V(p)]
(8.5.33)
Hence a particular solution of Eq. (8.5.32) is obtained by inverting the Laplace transform [Eq. (4.8.32)] 1 f(x) = - . 211"1,
ji..
+T' [ 4>() ~( -i"'+T' 1 p)
1
ePZ dp;
Re x> 0
(8.5.34)
Fourier Transforms and Integral Equations
§8.5]
973
There are no nonzero solutions of the homogeneous Volterra equation, so that formula (8.5.34) gives the unique solution of Eq. (8.5.33). As an example of this, we shall consider the Volterra integral equation resulting from the differential equation (d 21/t/dt2) + k 21/t = 0 initial conditions at t = 0, 1/t = 1/to, ()1/t/ iJt = Vo. equation is given in Eq. (8.1.23):
1/t(t) = 1/to [ iJG~:lto)
1.=0 - voG(tIO) + k 2 fot+ GUlto)1/t(lo) dto
The Green's function G is determined by ()2G/iJt 2 = -o(t - to);
G(tlto) = 0;
The equivalent integral
t
~Im(p)
/
< to
if t
/'
p Plane
ik
/'
The equations may be readily solved :
G(tlto) = (to - t)u(t - to) Re(p)-
where u(t) is the unit function (see page 840). Substituting this result for G, the integral equation becomes
1/t(t) = 1/to
+ vot + k 2 fot+ (to . 1/t(t o) dto;
t) .
Contour
>0
t
This is now in proper form to apply the preceding discussion :
Fig. 8.4 Contour Laplace transform .
tp(t) = 1/to + vot; cI>(p) = (1/to/p) + (VO/p2); v(t - to) = k 2(to - t); V(p) = _k 2/p2;
for
inversion
of
Re p > 0 Re p > 0
Substituting in solution (8.5.34) one obtains
1/t(t)
1 = 2~
11"/,
ji . + OO
T '
- , 00 +TO
[
Vo 2++Pk1/t20] P
ep:r;
dp;
TO
>0
We may evaluate this integral by adding on an infinite semicircle, as indicated in Fig. 8.4, and then applying Cauchy's integral formula. We, of course, obtain the familiar result
1/t = 1/to cos(kt)
+
(vo/k) sin(kt)
Volterra Integral Equation, Limits (x , 00). The Laplace transform may also be applied to integral equations of the following form: .
1/t(x) = tp(x) +
L" v(x -
xo)1/t(xo) dx«
(8.5.35)
Integral Equations
974
[cH.8
Integral equations of this type occur in transport problems where x may be the energy after a collision and Xo the energy before collision (see Sees. 2.4 'and 12.2) . For collisions with fixed systems with no internal degree of freedom, Xo ;::: x; that is, the collisions always result in a loss of energy of the incident particle. To solve Eq. (8.5.35) by application of the Laplace transform it is necessary to develop a faltung theorem for the form
L..
vex - xo)if;(xo) dxo
We start from the faltung theorem for Fourier transforms : g:
{J-.... g(x -
xo)ep(xo) dXo} =
y'2;Gj(k)~j(k)
Now let g(x) = v_ex), that is, equal vex) for x < 0 and equal zero for > 0 ; similarly let ep(x) = if;+(x), that is, if;(x) for x > 0 and zero for x < O. Then the above equation becomes x
To convert the Fourier into a Laplace transform, we recall that F , (p) = y'2; [F+(ip) ]j . Hence
.e
{L" vex -
xo)if;(xo) dXo} = y'2; [V_(ip)]tl'!'+(p)],
We can express [y'2; V _(ip)]j in terms of a Laplace transform:
fO-.. v(x)e-
[-\/271" V _(ip)]j =
If we therefore let v( -x)
Pz
[y'2; V _(ip)]j
dx;
=
t : v( -x)e
Jo
Pz
dx
= w(x), then
. [y'2; V -(ip)lj = W,( -p) Finally (8.5.36)
We may now return to integral equation (8.5.35) . Taking the Laplace transform of both sides (we shall dispense with subscript l from here on, since we shall be dealing with Laplace transforms only) , one obtains
+
'!'(p) = ~(p) W( -p)'!'(p) '!'(p) = ~(p) /[l - W( -p)]
or Finally . .
1
if;(x) = - . 271"'/.
fi.. +" [ - i"+T'
~()
p 1 - W(-p)
] ePZ dp
(8.5.37)
(8.5.38)
§8.5]
Fourier Transforms and Integral Equations
975
is a particular solution of integral equation (8.5.35). It should be emphasized that, for solution (8.5.37) or (8.5.38) to be meaningful, it is necessary for the regions of analyticity of W( - p) and ~(p) to overlap. As has been mentioned before , if this occurs for only certain range of the parameters in either ~ or W, it may be possible to extend the range by analytic continuation. As an example for this case let cp(x)
= C; v(x) = Ae a"'; A, a real and positive
so that (8.4.35) is if;(x) = C
+A
tOO ea("'-""lif;(xo) dx«
We now apply the Laplace transform [though before we carry through the analysis we should point out that for this particular v(x), the integral equation may be reduced to a first-order differential equation which can be easily solved]. To fill in formula (8.5.38), ~(p) and W( -p) are required: ~(p) = C/p ; Re p > 0 W( -p) = 00 eP"'v( -x) dx = A 00 e(p-a)", dx
fo
= a
fo
A /(a - p);
Re p
O.
if;(x)
= -1 . 21l't
fi
{C(
)}
00 +TO p - a eP"'dp -iOO+TO p[p - (a - A)]
where 0 < TO < a. We again close the contour by adding a semicircle extending around the left-hand half plane of p as in Fig . 8.4 above. Then the integral for if; may be evaluated by the Cauchy integral formula, there being simple poles at p = 0 and at p = a - A (if TO is taken greater than a - A) . This arbitrariness concerning the residue at a - A, which may be included or not as desired, corresponds to the arbitrariness of the complementary function, a solution of the homogeneous equation, which in this case is proportional to the difference of two particular solutions of the inhomogeneous linear equation. We obtain CA if;(x) = -c« -- - - - e(a-A)", a-A a-A
The first term is Ii particular solution which represents the "steady state" induced by the "source" term C if a < A . The second term indicates that the solution of the homogeneous equation, obtained by placing C = 0, is proportional to e(a-A)",. We note the homogeneous term again filling the familiar role of the transient.
976
Integral Equations
[cH.8
. Mellin Transform. We commence this subsection by recalling the definition and inversion formula as given in Chap. 4 and Prob. (4.48) . The Mellin transform of f(x) is given by F(s)
=
fa" f(x)xa-l dx
If this should not exist, then it is often possible to introduce the " halfplane" transforms equivalent to those introduced in Eourier transform theory: (8.5.39) F_ exists for Re s > 0' 0 whereas F+ exists for Re s < 0'1. In the event that F(s) does not exist, 0'0 > 0'1, while the opposite inequality holds if f does exist. The inversion formula may accordingly be written f(x)
OO+ C1O ' -_(S)] - ds + - i 00 +C10' X·
21l'2
where
O'~
[F
= - 1 . {ji
>
0'0
and
O'~
< 0'1.
ji
00 +C11' +C11'
- i 00
[F-+(s) - 1ds} X·
(8.5.40)
The faltung theorem is
~ {~ .. if;(xo)v (:0) (:~o) ]}
= V(s)'l'(s)
(8.5.41)
suggesting that the Mellin transform may be gainfully employed in solving integral equations of the following type : if;(x)
=
~(x) + ~ 00 v (:0) if;(xo) (~o)
(8.5.42)
The analysis leading to the solution of this equation is so completely similar to the corresponding Fourier integral treatment [leading to solution (8.5.17)] that we give only the results here :
o'
1 {jiOO+cr [ _ ] ds if;(x) = - . -- 2m -iOO+C10' 1 - V X·
+
ji
OO + C1,'
[
-+- ] -ds
-ioo+cr,' 1 - V ,x'
+ ¢ [1
~ V] ~:}
(8.5.43)
where the contour integral on S is within the region O'~ < Re p < O'~ , within which S must be analytic. The solution X of the homogeneous counterpart of (8.5.42) is given just by the contour integral. If the zeros of 1 - V occur at Sr, and if these zeros are of order t, then (8.5.44)
where B r t are arbitrary constants.
§8.5]
Fourier Transforms and Integral Equations
977
As an example of the use of the Mellin transform, we set
0
110
This integral may be evaluated by use of the Cauchy integral formula. For aX > 1, the contour is closed by a semicircle in the right-hand half plane. In that event, the only singularity of the integrand occurs at So for which 1 - Cr(so) = O. Then
"'= C(ax)80A "'l(SO) .,
(8.5.46)
ax>1
where "'l(SO) is the logarithmic derivative of the gamma function at So. For aX < 0 the singularities occur at all the negative roots of 1 - cr(s) = 0, so that
.
'" A,\,
1
'" = - C ~ (ax)" "'l(ST); aX < T=l
1
(.8.5.47)
[Note that these results for", are not solutions of the homogeneous equation as given in Eq. (8.5.45) .] The series for'" in the range (ax) < 1
978
Integral Equations
[cH.8
converges very well, since s, is a sequence of negative numbers whose absolute value increases with r, The Method of Wiener and Hopf. It is possible to extend the class of integral equations which can be solved through the use of Fourier transforms so as to include the following type: r/I(x) = A
fo
00
v(x - xo)r/I(xo) dxo
(8.5.48)
as well as its inhomogeneous counterparts of both the first and second kind. It is important to realize that the above equation is presumed to hold for all real values of x, both positive and negative. To make this point more obvious in the writing of the integral equation, let us introduce the functions r/I+ and r/I- with the usual definitions : r/I+(x) = r/I(x); r/I-(x) = r/I(x);
x x
>0 p., while for 'It_the band is for 1m k < TO . These are illustrated in Fig. 8.5, where it is seen that there is a strip p. < 1m k < TO wherein all the relevant transforms, V, 'It +, and 'It _, are regular. This result is, as we shall see, fundamental in the Wiener-Hopf technique. The transform of expression (8.5.49) may now be taken, employing the faltung theorem : 'It + + 'It _ = .yz;;: AV'It+ 'It +(1 - .yz;;: AV) + ''It_ = 0 (8.5.51) or It is clear that some added information must be brought to bear on this equation before 'It+ and 'It_ can be independently determined. This is provided by the method of factorization as applied to the function (1 - .yz;;: AV). This quantity is regular in the strip - T l < 1m k < TO . We now seek to break it up into factors T+ and T_ such that
(8.5.52)
980
I ntegral Equations
[CR. 8
These factors are to be regular and free of zeros in the half planes Im k > J1. and 1m k < TO, respectively. It is usual in addition to require that T+ and T_ have algebraic growth, as compared with exponential growth. That this is possible is shown by Wiener and Hopf in their original memoir. In any given problem this factorization must be carried out explicitly. Assuming Eq. (8.5.52) , we may rewrite the equation for 'It+ and 'It_, (8.5.51) , as follows : (8.5.53) The left-hand side of this equation is regular in the region Irn k > J1., while the left-hand side is regular for Im k < TO . Sin ce they have a common region of regularity TO > 1m k > J1., in which they are equal, we may assert that - 'It_ T_ is the analytic continuation of 'It+ T+ in the lower half plane. Hence 'It+ T+ is regular throughout the entire complex plane and is therefore an entire function which we shall call P(k) . This whole discussion, together with Eq. (8.5.52) , is, of course, not definite enough to determine the form of 'It+ T+; this reveals itself through its behavior for large k. Note that T+ has already been chosen to have algebraic growth ; i.e. it behaves like a polynomial for large k. The behavior of 'It+ for large k is determined by the behavior of f+(x) as x ---+ 0+. The condition that f+(x) be integrable at the origin, necessary for the existence of 'It+, leads to the asymptotic dependence
'It+( k)
------+
Ikl-> co
0
We therefore observe that P(k) is a polynomial of degree less than T+ (since P / T+ approaches zero for large Ikl). This fixes the form of P(k); the undetermined constants may be obtained by substitution into the original equation (8.5.48) . Equation (8.5.53) may now be solved for 'It+ and 'It_: (8.5.54) The inversion formula is
f+( x) = _ ~ V 211'
f.
+iT [P(k) ] e- ik % dk ; J1. - .. +iT T+(k)
< T < TO
(8.5.55)
From f+ one may determine f- by substitution in (8.5.50) or more directly
f-(x)
= -
1.
V~1I'
f"- ..
+iT [P(k) ] e- ik % dk ; +iT T_(k)
~< < T
TO
(8.5.56)
This completes the solution of the Wiener-Hopf integral equation (8.5.48) . We shall now illustrate this discussion with some examples.
§8.5]
Fourier Transforms and Integral Equations
Illustrations of the Method. the integral equation 1/;(x)
The transform of
981
A very simple example is furnished by
= A fo
00
e-I"- ,,.I1/;(xo)
dxo
y(k) is
e-I"I,
2 V(k) = y'2; (1
+ k 2) ;
=
TO
Tl
= 1
The quantity which must be factored into T+ and T_ of Eq. (8.5.52) is
= (1 - .V !2=AV) ,"ITr
[k
2 -
(2A -
k2 + 1
l)J = T+ T_
The factorization may be performed by inspection :
-
T+ = [k 2
so that
-
[k
J[
2
J
(2A .- -1) 1 ----. k+t k-t (2A - l)JI(k + i); T_ = (k - i)
1 - V21rAV
=
-
The first of these, T+, is clearly regular and free of zeros for Im k > JJ., where JJ. is less than TO = 1, while T_ is regular and free of zeros for Im k < TO, as long as Re A > O. Hence
+ i)
P(k) = '¥+(k)(k2 - (2A - l)JI(k
= -(k - i)'¥_(k)
The function P(k) is determined from the condition that it must be regular in the finite complex plane of k, while '1'+(k) ~ 0 as Ikl ~ From this it follows that P(k) for this example must be a constant C. It cannot increase as rapidly as k, for this would imply '1'+(k) ~ 1. It cannot decrease more rapidly than a constant, for this would imply a singularity (pole or branch point) in the finite complex plane. We may now solve for,!,+ and '1'_: IX) .
'1'+ = C(k and
1/;+ = C
+ i)/(k2
f
-
00
'1'_ = -C/(k - i)
- (2A - 1)];
+': {(k
OO+ tT
+ i) /(k2
- (2A - l)]}e- ib dx
Since , in this expression, x > 0, we may close the contour with a semicircle in the lower half plane. Employing Cauchy's formula one obtains 1/; = -'-
s. {[vu-=I + iJ V2A - 1
e- iv'2>..-lx
+
[vu-=I - .iJ
ei v'2>.. - l X}
+
,I2A - 1 [sin(V2A - 1 x)/(V2A - I)]} ; x > 0 (8.5.57)
where D is a new constant.
In the same manner 1/;- may be evaluated :
+
41rt
= D{cos(V2A - 1 x)
1/;- = De"; x
0, I/; = 1/;+ satisfies the differential equation
1/;" For x
< 0, I/; = 1/;- = X
+
(2X - 1)1/; = 0;
x
>
°
1/;_ is given by
fo" e-I"- ,,orl/;+(xo) dxo
or
1/;- = Xe"
fo" e-"OJ/;+(xo) dx«
This is obviously in the form given by (8.5.58) . The solution given in Eqs. (8.5.57.) and (8.5.58) is that one which is continuous and has a continuous slope at x = 0. This factorization is not so mysterious as it might seem. In the first place it is not necessarily unique, since the requirements on P = '1'+ T+ = -'1'_ T_ and on the asymptotic form of '1' are not completely rigid . It turns out, however, that the interrelations between P and the T's and the '1"s are such that the final solutions come out the same, no matter which choice is made . In. many cases the factorization is unique. It is in the example just given. For instance, we might have tried T+ = l[k 2
(2X - I)l!(k
-
+ i)}(k
- (3);
T_ = (k - i ) (k - (3)
If T+ is to have no zeros in the range Im k > p. < I, we must have Im {3 < p. < I ; but in this case T_ would be zero at k = {3, which would be in the region where T_ is not supposed to have zeros. Or we might try
T+ = [(k 2
-
2X
+ I)j(k + i)(k -
(3)];
T_ = (k - i) j(k - (3)
but this would put a pole in an undesirable region of the k plane. Consequently the only choice which keeps the zeros and poles of T+ below i and those for T_ above i is the one given , if we are to restrict our choice to functions which go to infinity with a finite power of k as Ikl---+ 00 . As a second example, we turn to a problem considered by A. E . Heins : x - X (.,
1/;( ) -
I/;(xo) dxo xo)]
)0 cosh[j(x -
The Fourier transform of sech[i(x - xo)] may be readily obtained by contour integration. The function to be factored is X7r 27r XV = 1 - cosh(7rk)
_ j-
1Let cos(7ra)
=
V
cosh(7rk) - X7r cosh (7rk)
X7r where lal < j; then
cosh(7rk) - cos(7ra) = 2 sin[~(a
+ ik)] sin[~(a -
ik)]
Fourier Transforms and Integral Equations
§8.5]
983
These factors may in turn be expressed directly in terms of their zeros by means of their infinite product representations [Eq. (4.3.8)] or equivalently in terms of r functions from the relation [Eq. (4.5.33)] r(z)r(l - z) = 7f' csc(7f'z) so that
sin[7f'
(a ~ ik)] = r(ia + iik)ro. _ ia _ iik)
and similarly sin[7f'
(a ~ ik)]
=
r(ia _ iik)ro. _ ia + iik) ik)r(i + ik)
cosh(7f'k) = r(i -
Also Consequently [1 _ y'2; AVl =
ik)r(i + i~) r (a ~ tk) r ( a ~ tk) r a ~ tk) r 27f'~(i -
.
(1 _
.
(1 _a ~ tk)
From this we can write down , with some arbitrariness, T
_
+ -
ik)(a + ik)ex(k) . r (a ~ ik) r a ~ ik)' 7f'r(i -
_ r(l
(1 _
+~)r(l- y) eX(k ) r(i
T_ -
+ ik)
The function x(k) is determined by the requirement that T+ and T_ be of algebraic growth for large values of k. To examine the behavior for large k we employ Stirling's theorem :
i] In z
In[~(z)l----? [z z---+ 00
Then
- z +
ik In 2 +
In( T+) ---+ X -
Ikl---+ 00
i In(27f')
In(ik) + .. .
In order that T+ behave like a polynomial for large choose X = ik In 2, in which case we have T+---+
Ikl---+ 00
Ikl,
we must
ik
Of course this holds only where T+ is regular. We may now determine P(k) . Since 'l'+ ---+ 0, it follows from the
Ikl---+ 00
regularity of P(k) = -'l'_ T_ = 'l'+ T+ that P(k) is a constant which we shall call C, which thus determines function 'l'+ : 'l'
2- _ 2- [
cosh (7f'k) ] + - T+ - T_ cosh(7f'k) - ros(7f'a)
984
Integral Equations
[cH.8
The latter form will prove to be more useful in the present. problem. The function 'hex) is C
1/;+(x)
=
f.
+ iT
cosh (1I"k) dk . - cos(1I"a )] 'r_
.
vz;;: _.. + iT e-'kx [cosh(1I"k)
For x > 0, we may close the cont our in the lower half plane where 'r_ is regular. The poles of the integrand occur then only at the zeros of [cosh(1I"k) - cos(1I"a)] which are at -ik = -2n ± a, n = 0, 1, 2, . . Therefore 1/;+(x)
=
. L
, [cot(1I"a)] ~ { e - (2n+al'" e-( 2n-al",} C 11" 'r_[ - (2n + a) i] - 'r_[ - (2n - a)tl n=O
where C' is a new constant.
.
Substituting for 'r_, 1/;+ becomes
= C' [cot(1I"a)] \ ' {rei + a + 2n) (2e"')-(2n+al 1/;-t
11"
Lt
n=O
r(I
+ a + n)
n!
_ rei - a + 2n) (2e"')-