Integral Transforms and Their Applications

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