Integral Transforms and Their Applications

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.