INTEGRAL TRANSFORMS and THEIR APPLICATIONS Lokenath Debnath Professor and Chair of Mathematics and Professor of Mechanical and Aerospace Engineering University of Central Florida Orlando, Florida CRC Press Boca Raton New York London Tokyo
Preface 1. Integral Transforms 1 1.1 Brief Historical Introduction 1 1.2 Basic Concepts and Definitions 3 2. Fourier Transforms 5 2.1 Introduction 5 2.2 The Fourier Integral Formulas 5 2.3 Definition of the Fourier Transform and Examples 7 2.4 Basic Properties of Fourier Transforms 14 2.5 Applications of Fourier Transforms to Ordinary Differential Equations 25 2.6 Solutions of Integral Equations 29 2.7 Solutions of Partial Differential Equations 32 2.8 Fourier Cosine and Sine Transforms with Examples 50 2.9 Properties of Fourier Cosine and Sine Transforms 52 2.10 Applications of Fourier Cosine and Sine Transforms to Partial Differential Equations 54 2.11 Evaluation of Definite Integrals 58 2.12 Applications of Fourier Transforms in Mathematical Statistics 60 2.13 Multiple Fourier Transforms and Their Applications 64 2.14 Exercises 72 3. Laplace Transforms 8 3 3.1 Introduction 83 3.2 Definition of the Laplace Transform and Examples 83 3.3 Existence Conditions for the Laplace Transform 87 3.4 Basic Properties of Laplace Transforms 88
viii Contents 3.5 The Convolution Theorem and Properties of Convolution 92 3.6 Differentiation and Integration of Laplace Transforms 96 3.7 The Inverse Laplace Transform and Examples 98 3.8 Tauberian Theorems and Watson's Lemma 109 3.9 Laplace Transforms of Fractional Integrals and Fractional Derivatives 113 3.10 Exercises 117 4. Applications of Laplace Transforms 123 4.1 Introduction 123 4.2 Solutions of Ordinary Differential Equations 123 4.3 Partial Differential Equations, and Initial and Boundary Value Problems 147 4.4 Solutions of Integral Equations 159 4.5 Solutions of Boundary Value Problems 163 4.6 Evaluation of Definite Integrals 166 4.7 Solutions of Difference and Differential-Difference Equations 168 4.8 Applications of the Joint Laplace and Fourier Transform 173 4.9 Summation of Infinite Series 183 4.10 Exercises 185 5. Hankel Transforms 193 5.1 5.2 5.3 5.4 5.5 Introduction The Hankel Transform and Examples Operational Properties of the Hankel Transform Applications of Hankel Transforms to Partial Differential Equations Exercises 193 193 195 198 205 Mellin Transforms 211 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 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 211 211 214 218 222 227 229 232
i x 7. Hilbert and Stieltjes Transforms 237 7.1 Introduction 237 7.2 Definition of the Hilbert Transform and Examples 237 7.3 Basic Properties of Hilbert Transforms 239 7.4 Hilbert Transforms in the Complex Plane 242 7.5 Applications of Hilbert Transforms 243 7.6 Asymptotic Expansions of One-Sided Hilbert Transforms 248 7.7 Definition of the Stieltjes Transform and Examples 250 7.8 Basic Operational Properties of Stieltjes Transforms 252 7.9 Inversion Theorems for Stieltjes Transforms 254 7.10 Applications of Stieltjes Transforms 257 7.11 The Generalized Stieltjes Transform 259 7.12 Basic Properties of the Generalized Stieltjes Transform 260 7.13 Exercises 261 8. Finite Fourier Cosine and Sine Transforms 265 8.1 Introduction 265 8.2 Definitions of the Finite Fourier Sine and Cosine Transforms and Examples 265 8.3 Basic Properties of Finite Fourier Sine and Cosine Transforms 267 8.4 Applications of Finite Fourier Sine and Cosine Transforms 272 8.5 Multiple Finite Fourier Transforms and Their Applications 277 8.6 Exercises 280 Finite Laplace Transforms 283 9.1 Introduction 283 9.2 Definition of the Finite Laplace Transform and Examples 283 9.3 Basic Operational Properties of the Finite Laplace Transform 288 9.4 Applications of Finite Laplace Transforms 290 9.5 Tauberian Theorems 294 9.6 Exercises 294 10. Z Transforms 295 10.1 Introduction 295 10.2 Dynamic Linear Systems and Impulse Response 295 10.3 Definition of the Z Transform and Examples 298 10.4 Basic Operational Properties 301
10.5 The Inverse Z Transform and Examples 306 10.6 Applications of Z Transforms to Finite Difference Equations 308 10.7 Summation of Infinite Series 311 10.8 Exercises 313 11. Finite Hankel Transforms 317 11.1 Introduction 317 11.2 Definition of the Finite Hankel Transform and Examples 317 11.3 Basic Operational Properties 319 11.4 Applications of Finite Hankel Transforms 319 11.5 Exercises 323 12. Legendre Transforms 325 12.1 Introduction 325 12.2 Definition of the Legendre Transform and Examples 325 12.3 Basic Operational Properties of Legendre Transforms 328 12.4 Applications of Legendre Transforms to Boundary Value Problems 333 12.5 Exercises 335 13. Jacobi and Gegenbauer Transforms 337 13.1 Introduction 337 13.2 Definition of the Jacobi Transform and Examples 337 13.3 Basic Operational Properties 339 13.4 Applications of Jacobi Transforms to the Generalized Heat Conduction Problem 340 13.5 The Gegenbauer Transform and its Basic Operational Properties 341 13.6 Application of the Gegenbauer Transform 344 14. Laguerre Transforms 345 14.1 Introduction 345 14.2 Definition of the Laguerre Transform and Examples 345 14.3 Basic Operational Properties 348 14.4 Applications of Laguerre Transforms 352 14.5 Exercises 354
XI 15. Hermite Transforms 15.1 Introduction 15.2 Definition of the Hermite Transform and Examples 15.3 Basic Operational Properties 15.4 Exercises 355 355 355 358 365 Appendix A Some Special Functions and Their Properties 367 A-l 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 367 372 377 379 383 385 Appendix B Table B-1 Table B-2 Table B-3 Table B-4 Table B-5 Table B-6 Table B-7 Table B-8 Table B-9 Table B-10 Table B-11 Table B-12 Table B-13 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 387 387 391 393 395 400 403 406 409 413 415 417 420 422 Answers and Hints to Selected Exercises 423 Bibliography 441 Index 449