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1 Texts in Applied Mathematics 41 Editors JE. Marsden L. Sirovich M. Golubitsky S.S. Antman Advisors G.Iooss P. Holmes D. Barkley M. Dellnitz P. Newton Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo

2 Texts in Applied Mathematics 1. Sirovich: Introduction to Applied Mathematics. 2. Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos. 3. Hale/KO(;ak: Dynamics and Bifurcations. 4. Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics, 3rd ed. 5. HubbardIWest: Differential Equations: A Dynamical Systems Approach: Ordinary Differential Equations. 6. Sontag: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nded. 7. Perko: Differential Equations and Dynamical Systems, 3rd ed. 8. Seaborn: Hypergeometric Functions and Their Applications. 9. Pipkin: A Course on Integral Equations. 10. HoppensteadtiPeskin: Modeling and Simulation in Medicine and the Life Sciences, 2nded. 11. Braun: Differential Equations and Their Applications, 4th ed. 12. StoerlBulirsch: Introduction to Numerical Analysis, 2nd ed. 13. RenardylRogers: An Introduction to Partial Differential Equations. 14. Banks: Growth and Diffusion Phenomena: Mathematical Frameworks and Applications. 15. Brenner/Scott: The Mathematical Theory of Finite Element Methods. 16. Van de Velde: Concurrent Scientific Computing. 17. MarsdenlRatiu: Introduction to Mechanics and Symmetry, 2nd ed. 18. HubbardIWest: Differential Equations: A Dynamical Systems Approach: Higher- Dimensional Systems. 19. Kaplan/Glass: Understanding Nonlinear Dynamics. 20. Holmes: Introduction to Perturbation Methods. 21. CurtainlZwan: An Introduction to Infinite-Dimensional Linear Systems Theory. 22. Thomas: Numerical Partial Differential Equations: Finite Difference Methods. 23. Taylor: Partial Differential Equations: Basic Theory. 24. Merkin: Introduction to the Theory of Stability of Motion. 25. Naber: Topology, Geometry, and Gauge Fields: Foundations. 26. PoldermanIWillems: Introduction to Mathematical Systems Theory: A Behavioral Approach. 27. Reddy: Introductory Functional Analysis with Applications to Boundary-Value Problems and Finite Elements. 28. GustafsonIWilcox: Analytical and Computational Methods of Advanced Engineering Mathematics. 29. TveitolWinther: Introduction to Partial Differential Equations: A Computational Approach. 30. GasquetIWitomski: Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets. (continued after index)

3 Brian Davies Integral Transforms and Their Applications Third Edition With 59 Illustrations Springer

4 Brian Davies Mathematics Department School of Mathematical Sciences Australian National University Canberra, ACT 0200 Australia Series Editors JE. Marsden Control and Dynamical Systems, California Institute of Technology Pasadena, CA USA M. Golubitsky Department of Mathematics University of Houston Houston, TX USA L. Sirovich Division of Applied Mathematics Brown University Providence, RI USA S.S. Antman Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD USA Mathematics Subject Classification (2000): 44-01, 44A10, 44A15, 65R10 Library of Congress Cataloging-in-Publication Data Davies, B. (Brian), Integral transforms and tbeir applications/brian Davies.-3rd ed. p. cm. - (Texts in Applied Mathematics; v. 41) Includes bibliographical references and index. ISBN Integral transforms. I. Title. II. Texts in Applied Mathematics (Springer-Verlag New York, Inc.); v. 41. QAl.A647 vol. 41, 2001 [QA432] 510 s-dc21 [515'.723] Printed on acid-free paper. 2002, 1985, 1978 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 3rd edition 2002 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of tbe publisher (Springer-Verlag New York, Inc., 175 Fiftb Avenue, New York, NY 10010, USA), except for brief excerpts in connection witb reviews or scholarly analysis. Use in connection witb any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar metbodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in tbis publication, even if the former are not especially identified, is not to be taken as a sign tbat such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Francine McNeill; manufacturing supervised by Jacqui Ashri. Photocomposed copy prepared from tbe autbor's 1l\TEiX files using Springer's svsing.sty macro ISBN ISBN (ebook) DOl / Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH

5 Series Preface Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs. Pasadena, California Providence, Rhode Island Houston, Texas College Park, Maryland J.E. Marsden L. Sirovich M. Golubitsky S.S. Antman

6 Preface to the Third Edition It is more than 25 years since I finished the manuscript of the first edition of this volume, and it is indeed gratifying that the book has been in use over such a long period and that the publishers have requested a third edition. After such a long time, a simple coat of paint will not substitute for a more thorough renovation; in fact, I have pondered for some considerable time the question of what form a new edition should take. Consequently, this volume is about 20% longer than the previous one and organized somewhat differently, into chapters, rather than into four parts, each divided into sections. That said, I should state the principles by which I was guided in preparing the present edition. They are: (i) To abide by the precept set out in the Preface to the First Edition: "to produce a work whose scope is selective rather than encyclopedic [so as not to] make the book too long." (ii) To include applications that are both interesting and illustrative of the wide utility of integral transforms. (iii) To provide a volume that can be used either in teaching an advancedlevel course or for self-study. The first chapter is largely new, giving a rather brisk account of those aspects of complex variable theory that will be needed in the sequel. It replaces the appendixes of the previous editions. I have crystallized this material over the past few years, when teaching advanced-level students. Whereas, in the 1980s, one could require such students to have a good

7 viii Preface to the Third Edition grounding in complex variable techniques, today that is not always possible. For such students, the first chapter is intended to be a guide to what they need to master before they begin. Chapters 2 to 6 correspond to the previous first part and are devoted to the Laplace transform. I have reorganized this material somewhat; it originally formed six sections. Chapter 3 incorporates all the material on the inversion integral, originally Sections 2 and 6. I have completely rewritten the material on ordinary differential equations (Chapter 4), to emphasize the use of linear algebra and to include an introduction to the role of the Laplace transform in linear control theory. Chapter 5, on partial differential equations, has also been expanded somewhat over the original treatment, although in this case the purpose was to improve the exposition rather than to expand the subject matter. Chapters 7 to 11 correspond to the previous second part and are concerned with the Fourier transform. Compared with the material on the Laplace transform, I have not seen fit to change the basic structure here, only to try to improve on the exposition and to refurbish the applications somewhat. Experience in teaching students convinces me that it is necessary to introduce generalized functions at this point and that one cannot rely on this as prerequisite knowledge. The material on the Mellin transform (previously the first three sections of Part III) has been combined into Chapters 12 and 13, and some material has been removed. The techniques seem still to be relatively unknown, despite their power and elegance. The material on Hankel transforms and dual integral equations, being so tightly related, form a single new Chapter 14. For the rest, each chapter of the present volume corresponds to a section of the previous edition, since the subject matter of each is quite distinct. The last chapter, however, on the numerical inversion of the Laplace transform, has been completely reorganized and rewritten, to reflect the enormous progress in this area. The emphasis is on an exposition of the theoretical basis of various computational schemes, although I have tried also to reflect the changes that flow from the explosion of computational power over the past two decades. I would like to thank Dr. J.H. Knight for many helpful discussions about the chapter on the numerical inversion of the Laplace transform. I would also like to thank the editorial staff at Springer-Verlag New York, Inc., for their assistance in the preparation of this new edition and, in particular, the copy editor for helping to eliminate many small, but irritating, errors from the final version of the manuscript. Canberra, Australia 2001 Brian Davies

8 Preface to the Second Edition In preparing this second edition, I have restricted myself to making small corrections and changes to the first edition. Two chapters have had extensive changes made. First, the material of Sections 14.1 and 14.2 has been rewritten to make explicit reference to the book of Bleistein and Handelsman [12J, which appeared after the original Chapter 14 had been written. Second, Chapter 21, on numerical methods, has been rewritten to take account of comparative work that was done by the author and Brian Martin a]1d that was published as a review paper. The material for all these chapters was in fact prepared for a translation of the book. Considerable thought has been given to a much more comprehensive revision and expansion of the book. In particular, there have been spectacular advances in the solution of some nonlinear problems using isospectral methods, which may be regarded as a generalization of the Fourier transform. However, the subject is a large one, and even a modest introduction would have added substantially to the book. Moreover, the recent book by Dodd et al. [21 J is at a similar level to the present volume. Similarly, I have refrained from expanding the chapter on numerical methods into a complete new part of the book, since a specialized monograph on numerical methods is in preparation with a colleague. Canberra, Australia 1984 Brian Davies

9 Preface to the First Edition This book is intended to serve as introductory and reference material for the application of integral transforms to a range of common mathematical problems. It has its immediate origin in lecture notes prepared for seniorlevel courses at the Australian National University, although lowe a great deal to my colleague Barry W. Ninham, a matter to which I refer below. In preparing the notes for publication as a book, I have added a considerable amount of material additional to the lecture notes, with the intention of making the book more useful, particularly to the graduate student involved in the solution of mathematical problems in the physical, chemical, engineering, and related sciences. Any book is necessarily a statement of the author's viewpoint and involves a number of compromises. My prime consideration has been to produce a work whose scope is selective rather than encyclopedic; consequently, there are many facets of the subject that have been omitted-in not a few cases after a preliminary draft was written-because I believe that their inclusion would make the book too long. Some of the omitted material is outlined in various problems and should be useful in indicating possible approaches to certain problems. I hav:elaid gi"eat stressqil- the use ef GeInplex variable techniques, an area of mathematics often unfashionable, but frequently of great power. I have been particularly severe in excising formal proofs, even though there is a considerable amount of "pure mathematics" associated with the understanding and use of generalized functions, another area of enormous utility in mathematics. Thus, for the formal aspects of the theory of integral transforms I must refer the reader to one of the many excellent books addressed to this area; I have chosen an approach that is

10 Xli Preface to the First Edition more common in published research work in applications. I can only hope that the course I have steered will be of great interest and help to students and research workers who wish to use integral transforms. It was my privilege as a student to attend lectures on mathematical physics by Professor Barry W. Ninham, now at the Australian National University. For several years, it was his intention to publish a comprehensive volume on mathematical techniques in physics, and he prepared draft material on several important topics to this end. In 1972, we agreed to work on this project jointly and continued to do so until During that period, it became apparent that the size, and therefore cost, of such a large volume would be inappropriate to the current situation, and we decided to each publish a smaller book in our particular area of interest. I must record my gratitude to him for agreeing that one of his special intereststhe use of the Mellin transform in asymptotics-should be included in the present book. In addition, there are numerous other debts that I owe him for guidance and criticism. References to sources of material have been made in two ways, since this is now a fairly old subject area. First, there is a selected bibliography of books, and I have referred, in various places, to those books that have been of particular assistance to me in preparing lectures or in pursuing research. Second, where a section is based directly on an original paper, the reference is given as a footnote. Apart from this, I have not burdened the reader with tedious lists of papers, especially as there are some comprehensive indexing and citation systems now available. A great deal of the final preparation was done while I was a visitor at the Unilever Research Laboratories (UK) and at Liverpool University in 1975, and I must thank those establishments for their hospitality, and the Australian National University for the provision of study leave. Most of the typing and retyping of the manuscript have been done by Betty Hawkins of the Mathematics Department, while the figures were prepared by Mrs. L. Wittig of the Photographic Services Department, ANU. Timothy Lewis, of Applied Mathematics at Brown University, has proofread the manuscript and suggested a number of useful changes. To these people, I express my gratitude and also to Professor Lawrence Sirovich for his encouragement and helpful suggestions. This book is dedicated to my respected friend and colleague, Barry W. Ninham. Canberra, AustraJia 1977 Brian Davies

11 Contents Preface to the Third Edition Preface to the Second Edition Preface to the First Edition 1 Functions of a Complex Variable 1.1 Analytic Functions Contour Integration 1.3 Analytic Continuation 1.4 Residue Theory Loop Integrals 1.6 Liouville's Theorem 1.7 The Factorial Function. 1.8 Riemann's Zeta Function 2 The Laplace Transform 2.1 The Laplace Integral. 2.2 Important Properties. 2.3 Simple Applications 2.4 Asymptotic Properties: Watson's Lemma Problems The Inversion Integral 3.1 The Riemann-Lebesgue Lemma. vii ix xi

12 xiv Contents 3.2 Dirichlet Integrals 3.3 The Inversion Inversion of Rational Functions 3.5 Taylor Series Expansion Inversion of Meromorphic Functions 3.7 Inversions Involving a Branch Point 3.8 Watson's Lemma for Loop Integrals 3.9 Asymptotic Forms for Large t Heaviside Series Expansion Problems Ordinary Differential Equations 4.1 Elementary Examples Higher-Order Equations Transfer Functions and Block Diagrams 4.4 Equations with Polynomial Coefficients 4.5 Simultaneous Differential Equations 4.6 Linear Control Theory Realization of Transfer Functions Problems Partial Differential Equations I 5.1 Heat Diffusion: Semi-Infinite Region 5.2 Finite Thickness 5.3 Wave Propagation 5.4 Transmission Line Problems... 6 Integral Equations 6.1 Convolution Equations of Volterra Type 6.2 Convolution Equations over an Infinite Range. 6.3 The Percus-Yevick Equation Problems The Fourier Transform 7.1 Exponential, Sine, and Cosine Transforms 7.2 Important Properties Spectral Analysis Kramers-Kronig Relations. Problems Partial Differential Equations II 8.1 Potential Problems Water Waves: Basic Equations Waves Generated by a Surface Displacement

13 Contents xv 8.4 Waves Generated by a Periodic Disturbance Problems Generalized Functions 9.1 The Delta Function Test Functions and Generalized Functions 9.3 Elementary Properties Analytic Functionals Fourier Transforms of Generalized Functions Problems Green's Functions 10.1 One-Dimensional Green's Functions 10.2 Green's Functions as Generalized Functions 10.3 Poisson's Equation in Two Dimensions Helmholtz's Equation in Two Dimensions Problems Transforms in Several Variables 11.1 Basic Notation and Results Diffraction of Scalar Waves Retarded Potentials of Electromagnetism Problems The Mellin Transform 12.1 Definitions Simple Examples Elementary Properties 12.4 Potential Problems in Wedge-Shaped Regions 12.5 Transforms Involving Polar Coordinates 12.6 Hermite Functions Problems Application to Sums and Integrals 13.1 Mellin Summation Formula A Problem of Ramanujin Asymptotic Behavior of Power Series Integrals Involving a Parameter Ascending Expansions for Fourier Integrals Problems Hankel Transforms 14.1 The Hankel Transform Pair 14.2 Elementary Properties Some Examples Boundary-Value Problems

14 xvi Contents 14.5 Weber's Integral The Electrified Disc Dual Integral Equations of Titchmarsh Type 14.8 Erdelyi-Kober Operators Problems Integral Transforms Generated by Green's Functions 15.1 The Basic Formula Finite Intervals Some Singular Problems Kontorovich-Lebedev Transform 15.5 Boundary-Value Problems in a Wedge 15.6 Diffraction of a Pulse by a Two-Dimensional Half-Plane Problems The Wiener-Hopf Technique 16.1 The Sommerfeld Diffraction Problem Wiener-Hopf Procedure: Half-Plane Problems Integral and Integro-Differential Equations. Problems Methods Based on Cauchy Integrals 17.1 Wiener-Hopf Decomposition by Contour Integration 17.2 Cauchy Integrals The Riemann-Hilbert Problem Problems in Linear Transport Theory 17.5 The Albedo Problem A Diffraction Problem Problems Laplace's Method for Ordinary Differential Equations 18.1 Laplace's Method Hermite Polynomials Hermite Functions Bessel Functions: Integral Representations 18.5 Bessel Functions of the First Kind Functions of the Second and Third Kind PoisSQll and Relat@d RepI"@sentatiQIlS 18.8 Modified Bessel Functions Problems Numerical Inversion of Laplace Transforms 19.1 General Considerations Gaver-Stehfest Method 19.3 Mobius Transformation

15 Contents xvii 19.4 Use of Chebyshev Polynomials Use of Laguerre Polynomials 19.6 Representation by Fourier Series 19.7 Quotient-Difference Algorithm 19.8 Talbot's Method.... Bibliography Index