Integral Transforms and Their Applications, Second Edition

Keeping the style, content, and focus that made the first edition a bestseller, Integral Transforms and their Applications, Second Edition stresses the development of analytical skills rather than the importance of more abstract formulation. The authors provide a working knowledge of the analytical methods required in pure and applied mathematics, physics, and engineering. The second edition includes many new applications, exercises, comments, and observations with some sections entirely rewritten. It contains more than 500 worked examples and exercises with answers as well as hints to selected exercises.

The most significant changes in the second edition include:

New chapters on fractional calculus and its applications to ordinary and partial differential equations, wavelets and wavelet transformations, and Radon transform

Revised chapter on Fourier transforms, including new sections on Fourier transforms of generalized functions, Poissons summation formula, Gibbs phenomenon, and Heisenbergs uncertainty principle

A wide variety of applications has been selected from areas of ordinary and partial differential equations, integral equations, fluid mechanics and elasticity, mathematical statistics, fractional ordinary and partial differential equations, and special functions

A broad spectrum of exercises at the end of each chapter further develops analytical skills in the theory and applications of transform methods and a deeper insight into the subject

A systematic mathematical treatment of the theory and method of integral transforms, the book provides a clear understanding of the subject and its varied applications in mathematics, applied mathematics, physical sciences, and engineering.

INTEGRAL TRANSFORMS
Brief Historical Introduction
Basic Concepts and Definitions

FOURIER TRANSFORMS AND THEIR APPLICATIONS
Introduction
The Fourier Integral Formulas
Definition of the Fourier Transform and Examples
Fourier Transforms of Generalized Functions
Basic Properties of Fourier Transforms
Poisson's Summation Formula
The Shannon Sampling Theorem
Gibbs' Phenomenon
Heisenberg's Uncertainty Principle
Applications of Fourier Transforms to Ordinary Differential Eqn
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 DE
Evaluation of Definite Integrals
Applications of Fourier Transforms in Mathematical Statistics
Multiple Fourier Transforms and Their Applications
Exercises

LAPLACE TRANSFORMS AND THEIR BASIC PROPERTIES
Introduction
Definition of the Laplace Transform and Examples
Existence Conditions for the Laplace Transform
Basic Properties of 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
Exercises

APPLICATIONS OF LAPLACE TRANSFORMS
Introduction
Solutions of Ordinary Differential Equations
Partial Differential Equations, 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
Transfer Function and Impulse Response Function
Exercises

FRACTIONAL CALCULUS AND ITS APPLICATIONS
Introduction
Historical Comments
Fractional Derivatives and Integrals
Applications of Fractional Calculus
Exercises

APPLICATIONS OF INTEGRAL TRANSFORMS TO FRACTIONAL DIFFERENTIAL EQUATIONS
Introduction
Laplace Transforms of Fractional Integrals
Fractional Ordinary Differential Equations
Fractional Integral Equations
Initial Value Problems for Fractional Differential Equations
Green's Functions of Fractional Differential Equations
Fractional Partial Differential Equations
Exercises

HANKEL TRANSFORMS AND THEIR APPLICATIONS
Introduction
The Hankel Transform and Examples
Operational Properties of the Hankel Transform
Applications of Hankel Transforms to Partial Differential Equations
Exercises

MELLIN TRANSFORMS AND THEIR APPLICATIONS
Introduction
Definition of the Mellin Transform and Examples
Basic Operational Properties
Applications of Mellin Transforms
Mellin Transforms of the Weyl Fractional Integral and 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 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 SINE AND COSINE 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

THE RADON TRANSFORM AND ITS APPLICATION
Introduction
Radon Transform
Properties of Radon Transform
Radon Transform of Derivatives
Derivatives of Radon Transform
Convolution Theorem for Radon Transform
Inverse of Radon Transform
Exercises

WAVELETS AND WAVELET TRANSFORMS
Brief Historical Remarks
Continuous Wavelet Transforms
The Discrete Wavelet Transform
Examples of Orthonormal Wavelets
Exercises

Appendix A Some Special Functions and Their Properties
A-1 Gamma, Beta, and Error Functions
A-2 Bessel and Airy Functions
A-3 Legendre and Associated Legendre Functions
A-4 Jacobi and Gegenbauer Polynomials
A-5 Laguerre and Associated Laguerre Functions
A-6 Hermite and Weber-Hermite Functions
A-7 Hurwitz and Riemann zeta Functions

Appendix B Tables of Integral Transforms
B-1 Fourier Transforms
B-2 Fourier Cosine Transforms
B-3 Fourier Sine Transforms
B-4 Laplace Transforms
B-5 Hankel Transforms
B-6 Mellin Transforms
B-7 Hilbert Transforms
B-8 Stieltjes Transforms
B-9 Finite Fourier Cosine Transforms
B-10 Finite Fourier Sine Transforms
B-11 Finite Laplace Transforms
B-12 Z Transforms
B-13 Finite Hankel Transforms

Answers and Hints to Selected Exercises
Bibliography
Index