Integral Transforms and Their Applications

Preface
Integral Transforms
Brief Historical Introduction
Basic Concepts and Definitions
Fourier Transforms
Introduction
The Fourier Integral Formulas
Definition of the Fourier Transform and Examples
Basic Properties of the Fourier Transforms
Applications of Fourier Transforms to Ordinary Differential Equations
Solutions of Integral Equations
Solutions of Partial Differential Equations
Fourier Cosine and Sine Transforms with Examples
Properties of Fourier Cosine and Sine Transforms
Applications of Fourier Cosine and Sine Transforms to Partial Differential Equations
Evaluation of Definite Integrals
Applications of Fourier Transforms in Mathematical Statistics
Multiple Fourier Transforms and Their Applications
Exercises
Laplace Transforms
Introduction
Definition of the Laplace Transform and Examples
Existence Conditions for the Laplace Transform
Basic Properties of the Laplace Transforms
The Convolution Theorem and Properties of Convolution
Differentiation and Integration of Laplace Transforms
The Inverse Laplace Transform and Examples
Tauberian Theorems and Watson's Lemma
Laplace Transforms of Fractional Integrals and Fractional Derivatives
Exercises
Applications of Laplace Transforms
Introduction
Solutions of Ordinary Differential Equations
Partial Differential Equations, and Initial and Boundary Value Problems
Solutions of Integral Equations
Solutions of Boundary Value Problems
Evaluation of Definite Integrals
Solutions of Difference and Differential-Difference Equations
Applications of the Joint Laplace and Fourier Transform
Summation of Infinite Series
Exercises
Hankel Transforms
Introduction
The Hankel Transform and Examples
Operational Properties of the Hankel Transform
Applications of Hankel Transforms to Partial Differential Equations
Exercises
Mellin Transforms
Introduction
Definition of the Mellin Transform and Examples
Basic Operational Properties
Applications of Mellin Transforms
Mellin Transforms of the Weyl Fractional Integral and the Weyl Fractional Derivative
Application of Mellin Transforms to Summation of Series
Generalized Mellin Transforms
Exercises
Hilbert and Stieltjes Transforms
Introduction
Definition of the Hilbert Transform and Examples
Basic Properties of Hilbert Transforms
Hilbert Transforms in the Complex Plane
Applications of Hilbert Transforms
Asymptotic Expansions of the One-Sided Hilbert Transforms
Definition of the Stieltjes Transform and Examples
Basic Operational Properties of Stieltjes Transforms
Inversion Theorems for Stieltjes Transforms
Applications of Stieltjes Transforms
The Generalized Stieltjes Transform
Basic Properties of the Generalized Stieltjes Transform
Exercises
Finite Fourier Cosine and Sine Transforms
Introduction
Definitions of the Finite Fourier Sine and Cosine Transforms and Examples
Basic Properties of Finite Fourier Sine and Cosine Transforms
Applications of Finite Fourier Sine and Cosine Transforms
Multiple Finite Fourier Transforms and Their Applications
Exercises
Finite Laplace Transforms
Introduction
Definition of the Finite Laplace Transform and Examples
Basic Operational Properties of the Finite Laplace Transform
Applications of Finite Laplace Transforms
Tauberian Theorems
Exercises
Z Transforms
Introduction
Dynamic Linear Systems and Impulse Response
Definition of the Z Transform and Examples
Basic Operational Properties
The Inverse Z Transform and Examples
Applications of Z Transforms to Finite Difference Equations
Summation of Infinite Series
Exercises
Finite Hankel Transforms
Introduction
Definition of the Finite Hankel Transform and Examples
Basic Operational Properties
Applications of Finite Hankel Transforms
Exercises
Legendre Transforms
Introduction
Definition of the Legendre Transform and Examples
Basic Operational Properties of Legendre Transforms
Applications of Legendre Transforms to Boundary Value Problems
Exercises
Jacobi and Gegenbauer Transforms
Introduction
Definition of the Jacobi Transform and Examples
Basic Operational Properties
Applications of Jacobi Transforms to the Generalized Heat Conduction Problem
The Gegenbauer Transform and Its Basic Operational Properties
Application of the Gegenbauer Transform
Laguerre Transforms
Introduction
Definition of the Laguerre Transform and Examples
Basic Operational Properties
Applications of Laguerre Transforms
Exercises
Hermite Transforms
Introduction
Definition of the Hermite Transform and Examples
Basic Operational Properties
Exercises
Appendix A: Some Special Functions and Their Properties
Gamma, Beta, and Error Functions
Bessel and Airy Functions
Legendre and Associated Legendre Functions
Jacobi and Gegenbauer Polynomials
Laguerre and Associated Laguerre Functions
Hermite and Weber-Hermite Functions
Appendix B: Tables of Integral Transforms
Fourier Transforms
Fourier Cosine Transforms
Fourier Sine Transforms
Laplace Transforms
Hankel Transforms
Mellin Transforms
Hilbert Transforms
Stieltjes Transforms
Finite Fourier Cosine Transforms
Finite Fourier Sine Transforms
Finite Laplace Transforms
Z Transforms
Finite Hankel Transforms
Answers and Hints to Selected Exercises
Bibliography
Index