Integral Transforms and Their Applications

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 re­ search 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 numeri­ cal 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 Mathe­ matical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs. Pasadena, California J.E. Marsden Providence, Rhode Island L. Sirovich Houston, Texas M. Golubitsky College Park, Maryland S.S. Antman 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.

Table of Contents:
1 Functions of a Complex Variable.- 1.1 Analytic Functions.- 1.2 Contour Integration.- 1.3 Analytic Continuation.- 1.4 Residue Theory.- 1.5 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.- 3 The Inversion Integral.- 3.1 The Riemann-Lebesgue Lemma.- 3.2 Dirichlet Integrals.- 3.3 The Inversion.- 3.4 Inversion of Rational Functions.- 3.5 Taylor Series Expansion.- 3.6 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.- 3.10 Heaviside Series Expansion.- Problems.- 4 Ordinary Differential Equations.- 4.1 Elementary Examples.- 4.2 Higher-Order Equations.- 4.3 Transfer Functions and Block Diagrams.- 4.4 Equations with Polynomial Coefficients.- 4.5 Simultaneous Differential Equations.- 4.6 Linear Control Theory.- 4.7 Realization of Transfer Functions.- Problems.- 5 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.- 7 The Fourier Transform.- 7.1 Exponential, Sine, and Cosine Transforms.- 7.2 Important Properties.- 7.3 Spectral Analysis.- 7.4 Kramers-Krönig Relations.- Problems.- 8 Partial Differential Equations II.- 8.1 Potential Problems.- 8.2 Water Waves: Basic Equations.- 8.3 Waves Generated by a Surface Displacement.- 8.4 Waves Generated by a Periodic Disturbance.- Problems.- 9 GeneralizedFunctions.- 9.1 The Delta Function.- 9.2 Test Functions and Generalized Functions.- 9.3 Elementary Properties.- 9.4 Analytic Functionals.- 9.5 Fourier Transforms of Generalized Functions.- Problems.- 10 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.- 10.4 Helmholtz’s Equation in Two Dimensions.- Problems.- 11 Transforms in Several Variables.- 11.1 Basic Notation and Results.- 11.2 Diffraction of Scalar Waves.- 11.3 Retarded Potentials of Electromagnetism.- Problems.- 12 The Mellin Transform.- 12.1 Definitions.- 12.2 Simple Examples.- 12.3 Elementary Properties.- 12.4 Potential Problems in Wedge-Shaped Regions.- 12.5 Transforms Involving Polar Coordinates.- 12.6 Hermite Functions.- Problems.- 13 Application to Sums and Integrals.- 13.1 Mellin Summation Formula.- 13.2 A Problem of Ramanujin.- 13.3 Asymptotic Behavior of Power Series.- 13.4 Integrals Involving a Parameter.- 13.5 Ascending Expansions for Fourier Integrals.- Problems.- 14 Hankel Transforms.- 14.1 The Hankel Transform Pair.- 14.2 Elementary Properties.- 14.3 Some Examples.- 14.4 Boundary-Value Problems.- 14.5 Weber’s Integral.- 14.6 The Electrified Disc.- 14.7 Dual Integral Equations of Titchmarsh Type.- 14.8 Erdelyi-Köber Operators.- Problems.- 15 Integral Transforms Generated by Green’s Functions.- 15.1 The Basic Formula.- 15.2 Finite Intervals.- 15.3 Some Singular Problems.- 15.4 Kontorovich-Lebedev Transform.- 15.5 Boundary-Value Problems in a Wedge.- 15.6 Diffraction of a Pulse by a Two-Dimensional Half-Plane.- Problems.- 16 The Wiener-Hopf Technique.- 16.1 The Sommerfeld Diffraction Problem.- 16.2 Wiener-Hopf Procedure: Half-Plane Problems.- 16.3 Integral and Integro-DifferentialEquations.- Problems.- 17 Methods Based on Cauchy Integrals.- 17.1 Wiener-Hopf Decomposition by Contour Integration.- 17.2 Cauchy Integrals.- 17.3 The Riemann-Hilbert Problem.- 17.4 Problems in Linear Transport Theory.- 17.5 The Albedo Problem.- 17.6 A Diffraction Problem.- Problems.- 18 Laplace’s Method for Ordinary Differential Equations.- 18.1 Laplace’s Method.- 18.2 Hermite Polynomials.- 18.3 Hermite Functions.- 18.4 Bessel Functions: Integral Representations.- 18.5 Bessel Functions of the First Kind.- 18.6 Functions of the Second and Third Kind.- 18.7 Poisson and Related Representations.- 18.8 Modified Bessel Functions.- Problems.- 19 Numerical Inversion of Laplace Transforms.- 19.1 General Considerations.- 19.2 Gaver-Stehfest Method.- 19.3 Möbius Transformation.- 19.4 Use of Chebyshev Polynomials.- 19.5 Use of Laguerre Polynomials.- 19.6 Representation by Fourier Series.- 19.7 Quotient-Difference Algorithm.- 19.8 Talbot’s Method.