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.

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3rd edition

Reviews

From the reviews of the third edition: B. Davies Integral Transforms and their Applications "Extremely well-written and a joy to read . . . Whether the reader is seeking a useful text for a graduate course or a valuable reference on integral transforms, I would highly recommend Brian Davies' book." —THE AUSTRALIAN MATHEMATICAL SOCIETY "A testament to the quality and usefulness of this monograph is that a third edition has now appeared. … In addition to the material previously included, there have been significant extensions in some of the topics … . The writing style and the overall structure of the presentation have been modified … and, as a consequence, the current edition is extremely well-written and a joy to read. … I would highly recommend Brian Davies’ book." (W.P. Wood, The Australian Mathematical Society Gazette, Vol. 30 (1), 2003) "In its present third edition this well known work has undergone considerable reconstruction or augmentation … . Due to the general character of this volume, the many examples worked out or provided, this volume will be of use to readers intending to make actual use of integral transforms." (H. Muthsam, Monatshefte für Mathematik, Vol. 139 (2), 2003) "Nevertheless that the new edition has less pages, it is now … about 20% longer which can find a seemingly explanation by its more compact and now modern photocomposed copy preparation on the basis of a LaTeX file. … As conclusion … the new edition gained considerably … it can be highly recommended to a broad readership." (J. Synnatzschke, ZAA, Vol. 21 (3), 2002)