Analytic functions and distributions in physics and engineering 0471733342, 9780471733348

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ANALYTIC FUNCTIONS AND DISTRIBUTIONS IN PHYSICS AND ENGINEERING

ANALYTIC FUNCTIONS AND DISTRIBUTIONS IN PHYSICS AND ENGINEERING BERNARD W. ROOS John Jay Hopkins Laboratory For Pure and Applied Science Gulf General Atomic, Inc. San Diego, California

Jolin Wiley &: SOlIS, Inc., New York • London • Sydney • Toronto

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Copyright © 1969 by John Wiley & Sons, Inc. All rights reserved. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language without the written permission of the publisher. Library of Congress Catalog Card Number: 69-19241 SBN 471 73334 2 Printed in the United States of America

1 2 3 4 5 6 7 8 9 10

To My Mother and Father

PREFACE The purpose of this book is to present in an informal and easily readable style an introduction to those mathematical concepts and techniques involving analytic functions and distributions that have found wide application in theoretical physics and engineering in recent years. Originally written with an industrial educational program in mind, it has been my intention to present the material in a form suitable for students and research workers whose interest is primarily in the application of mathematical theories rather than in these theories themselves. I have avoided therefore abstract and involved mathematical arguments and have not attempted to present the foundations of the concepts and techniques in their most general form. Instead I have attempted to present an easily accessible introduction on the mathematical level of most research papers concerned with the applications of analytic functions and distributions to the solution of concrete physical problems. I have assumed that the reader has some knowledge of the elements of the theory of functions of a complex variable. However, in an effort to make the book reasonably self-contained, I included a chapter on the more important aspects of this theory, and, in a few instances, presented detailed discussions of some rather elementary mathematical concepts. The physical problems that serve to illustrate the mathematical techniques pertain to relevant current research in theoretical physics and are far from trivial. They have been taken from research papers on hydrodynamics, electrodynamics, neutron transport theory, and plasma physics. However, to make the book accessible to readers with a varied background in physics, I assumed no specialized knowledge of physics on the part of the reader and presented discussions of the physical fundamentals and derivations of the basic equations. Furthermore, it has been my policy to treat a few subjects in detail rather than many subjects superficially. As criteria for including a particular subject I used the frequency of occurrence of relevant applications or analytic functions and distributions in the research papers on the subject, vII

viii

PREFACE

the current or future relevance of these applications to other subjects, and the degree of my familiarity with the particular subjects. I included therefore rather extensive discussions of neutron transport theory and of longitudinal plasma oscillations. In addition, I have attempted to introduce the reader to those concepts and techniques of the theory of analytic functions and distributions that are important for an understanding of the applications of dispersion relations and causality arguments in classical and modern theoretical physics. I agree with the observation made by T. Teichmann in one of his book reviews in Physics Today that the majority of Western books on advanced mathematical topics, which have appeared in the last few years, are either elegant but abstract descriptions of the theory or encyclopedic descriptions of applications. This has made it rather difficult for those readers who wish to learn more about the subject with no view to conducting research in it or solving arrays of engineering or physics problems. The present book is aimed precisely at such readers in the field of analytic functions and distributions. It is not a standard textbook on the theory of functions of a complex variable and its applications. Instead it stresses the more advanced theories and techniques seldom found in the current textbook literature. It is definitely applied in nature and is directed to those physicists, engineers, and applied mathematicians who are interested in becoming acquainted with the subject rather than to specialists, although I hope, of course, that the book will also be useful to the latter. I have drawn upon many sources for my material and I feel a depth of gratitude to the authors of the books and articles from which I learned. I am also indebted to my friends and colleagues who made the John Jay Hopkins Laboratory a place where it is enjoyable to do research and creative work. Particularly the encouragements of Dr. Lothar W. Nordheim contributed much to the completion of this book. It is with great pleasure that I acknowledge the support received from Gulf General Atomic, Inc., and from its President, Dr. Frederic de Hoffman and its Vice President for Research and Development, Dr. Edward C. Creutz. Finally I wish to express my sincere gratitude to my wife Nardina for her constant encouragement and for her untiring help in editing, proofreading, and indexing the manuscript.

Rancho Santa Fe February 23, 1969

BERNARD W. ROOS

CONTENTS

CHAPTER I

ANALYTIC FUNCTIONS

1

1.1 Integral Representations of Analytic Functions 1.2 Series Representations of Analytic Functions 1.3 Singularities at Isolated Points A. Isolated Singular Points in the Finite Plane B. Isolated Singularities at Infinity 1.4 Entire and Meromorphic Functions; Theorem of Liouville 1.5 The Lemma of Jordan 1.6 Analytic Continuation A. Definition and Uniqueness of an Analytic Continuation B. Techniques of Analytic Continuation C. Illustrative Examples 1.7 Multivalued Functions A. The Multivalued Function B. Multivalued Functions in Electrostatics C. Multivalued Functions in the Propagation of Harmonic Plane Waves D. The Multfvalued Function log A E. Evaluation of an Integral in Neutron Transport Theory F. Evaluation of the Inversion Integral of a Unilateral Fourier Transform

F

Ix

1 13 17 18 24 27 29 30

32 35 39 43 44 47 49 54 56 60

x

CONTENTS

CHAPTER II

FOURIER TRANSFORMS, CAUSALITY, AND DISPERSION RELATIONS 64 2.0 2.1 2.2 2.3 2.4 2.5

2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14

2.15

CHAPTER III

Introduction Some Remarks on Integration and Cauchy Principal Value Integrals Infinite Integrals Depending on a Parameter Representation of an Analytic Function by an Infinite Integral Bilateral Fourier Transforms Some Elementary Properties of Bilateral Fourier Transforms Bilateral Fourier Transforms, Linear Physical Systems, and Green's Functions Small Oscillations of a Physical System about a Position of Stable Equilibrium Electric Polarization and Dispersion Relations Classical Motion of a Bound Electron in an Electric Field Unilateral Fourier Transforms Causal Fourier Transforms, Dispersion Relations, and Titchmarsh's Theorem Applications to Linear Physical Systems Analytic Continuation of a Function Defined by a Hilbert Transform Integral Asymptotic Representations of Unilateral Fourier Transforms A. Asymptotic Representations of Functions of a Real or Complex Variable B. Asymptotic Representations of Unilateral Fourier Transforms C. Asymptotic Behavior of an Inversion Integral for Large Values of t Applications to Integral Equations with a Displacement Kernel A. Homogeneous Integral Equations B. Inhomogeneous Integral Equations

THE WIENER-HOPF TECHNIQUE

3.0 Introduction 3.1 The Mittag-Leffler Expansion

64 69 73 79 83

87

94 98 109

111 114 118 134 136 142 143 148 151

153 155 159

162

162 167

CONTENTS

xi

3.2 The Decomposition of a Function that is Analy3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

CHAPTER IV

tic in a Strip Weierstrass's Factorization Formula Wiener-Hopf Factorization Formulas Wiener-Hopf Integral Equations of the Second Kind Wiener-Hopf Integral Equations of the First Kind Remarks on Some Partial Differential Equations of Mathematical Physics Boundary Value Problems Involving the Helmholtz Equation Small Amplitude Theory of Water Waves Semi-Infinite Dock on a Sea of Finite Depth

169 173 176 184 190

198 202

207 210

BOUNDARY VALUE PROBLEMS FOR SECTIONALLY ANALYTIC FUNCTIONS 218

4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

Introduction Holder Continuous Functions Sectionally Continuous Functions Sectionally Analytic Functions Cauchy Integral Representations of Sectionally Analytic Functions The Principal Value of a Cauchy Integral The Formulas of Plemelj Boundary Value Problems of Plemelj Representation of Sectionally Analytic Functions by Cauchy Integrals Over an Infinite Line The Poincare-Bertrand Transformation Formula Boundary Value Problems for Sectionally Analytic Functions in a Plane Cut Along a Contour A. The Homogeneous Hilbert pr~em for a Contour \ B. The Inhomogeneous Hilbert Problem for a Contour

218 220

222 225 226 228 231 235 236 242 243 244 248

4.11 Singular Integral Equations of the Cauchy Type 250 4.12 Boundary Value Problems for Sectionally Analy-

tic Functions in a Plane Cut Along a Finite Path 253 A. The Homogeneous Hilbert Problem for a Finite Path

253

CONTENTS

4.13

4.14

4.15 4.16 4.17

CHAPTER V

B. The Inhomogeneous Hilbert Problem for a Finite Path Boundary Value Problems for Sectionally Analytic Functions in a Plane Cut Along an Infinite Straight Line A. The Homogeneous Hilbert Problem for an Infinite Line B. The Inhomogeneous Hilbert Problem for an Infinite Line Wiener-Hopf Integral Equations and Hilbert Problems A. Homogeneous Wiener-Hopf Integral Equations of the Second Kind B. Inhomogeneous Wiener-Hopf Integral Equations of the Second Kind C. Wiener-Hopf Integral Equations of the First Kind Green's Functions and the Solution of Boundary Value Problems in Electrostatics Green's Function for a Point Charge on the Axis of an Infinite Cylinder Green's Function for a Point Charge on the Axis of a Semi-Infinite Cylinder

DISTRIBUTIONS

256

258

258 262 263

263 265 267 269 276 279

289

5.0 Introduction 5.1 Definition of a Schwartz Distribution 5.2 Algebraic and Analytic Operations on Distributions 5.3 The Derivative of a Distribution 5.4 Tempered Distributions and Their Fourier Transforms 5.5 The Cauchy Representation of Distributions A. Cauchy Representations of Distributions on the Space E B. Cauchy Representations of Distributions on the Space B C. Cauchy Representations and Plemelj Formulas 5.6 Generalized Analytic Functions 5.7 Distributions and Boundary Values of Analytic Functions

289 290 299 306 313 322

323 329 335 339 349

CONTENTS

CHAPTER VI

APPLICATIONS IN NEUTRON TRANSPORT THEORY 361 6.0 Introduction 6.1 Properties of Angular Eigenfunctions: Orthogonality 6.2 Properties of Angular Eigenfunctions: Completeness 6.3 Full Range Expansion Coefficients 6.4 Half Range Expansion Coefficients 6.5 Summary 6.6 Some One-Dimensional Boundary Value Problems of Time-Independent Neutron Transport Theory A. The Infinite Medium Green's Function B. The Half Space Green's Function C. The Half Space Albedo Problem D. The Milne Problem E. Half Space With Constant Isotropic Source F. Bare Slab Criticality Problem

CHAPTER VII APPLICATIONS IN PLASMA PHYSICS

xiii

361 365

370 377 382 386

389 390 394 398 401 405 406

411

7.0 Introduction 7.1 The Collisionless Boltzmann Equation 7.2 Longitudinal Oscillations in an Unbounded Plasma: Problem Statement 7.3 Landau's Transform Solution of the Initial Value Problem 7.4 The Time-Dependent Behavior of the Potential of the Self-Consistent Field 7.5 Criteria for Growing or Damped Plasma Oscillations A. Jackson's Criterion for Bell-shaped Equilibrium Velocity Distributions B. Penrose's Criterion for N~Maxwellian Equilibrium Velocity Distributi~n1...C. The Criterion of Nyquist D. McGune's Exact Inversion Procedure 7.6 Long and ShortWave Length Plasma Oscillations A. Long Wave Length Plasma Oscillations B. Short Wave Length Plasma Oscillations

411 413 418 421 422 436

437 439 444 448 449 449 453

xiv

CONTENTS

7.7

Some Special Equilibrium Velocity Distributions A. A Zero Temperature Plasma B. The Two-Stream Velocity Distribution C. The Lorentzian Equilibrium Velocity Distribution D. The Maxwellian Equilibrium Velocity Distribution 7.8 Van Kampen's Normal Mode Solution of the Initial Value Problem for a Maxwellian Equilibrium Velocity Distribution 7.9 Case's Normal Mode Solution of the Initial Value Problem for General Equilibrium Velocity Distributions

APPENDIX A

PATHS, CONTOURS, AND REGIONS IN THE COMPLEX PLANE 481

APPENDIX B

ORDER RELATIONS

BIBLIOGRAPHY INDEX

499

487

484

454 454 456

457 458

465

472