Integral Transforms and Their Applications

IntegralTransforms and Their Applications Second Edition Lokenath Debnath Dambaru Bhatta Contents 1 Integral Transforms 1 1.1 Brief Historical Introduction . . . . . . . . . . . . . . . . . . . 1 1.2 . . . . . . . . . . . . . . . . . 2 Fourier Transforms and Their Applications 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 2.3 Definition of the Fourier Transform and Examples . . . . . . 2.4 Fourier Transforms of Generalized Functions . . . . . . . . . . 2.5 Basic Properties of Fourier Transforms . . . . . . . . . . . . . 28 2.6 2.7 The Shannon Sampling Theorem . . . . . . . . . . . . . . . . 44 2.8 Gibbs’ Phenomenon 2.9 Heisenberg’s Uncertainty Principle . . . . . . . . . . . . . . . 57 2.10 Applications of Fourier Transforms to Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.11 Solutions of Integral Equations . . . . . . . . . . . . . . . . . 65 2.12 Solutions of Partial Differential Equations . . . . . . . . . . . 68 2.13 Fourier Cosine and Sine Transforms with Examples . . . . . . 91 2.14 Properties of Fourier Cosine and Sine Transforms . . . . . . . 93 2.15 Applications of Fourier Cosine and Sine Transforms to Partial Differential Equations 2.16 Evaluation of Definite Integrals . . . . . . . . . . . . . . . . . 100 2.17 Applications of Fourier Transforms in Mathematical Statistics 103 2.18 Multiple Fourier Transforms and Their Applications . . . . . 109 3 Laplace Transforms and Their Basic Properties 133 3.1 3.2 Definition of the Laplace Transform and Examples . . . . . . 134 3.3 Existence Conditions for the Laplace Transform . . . . . . . . 139 3.4 Basic Properties of Laplace Transforms . . . . . . . . . . . . . 140 3.5 The Convolution Theorem and Properties of Convolution . . 145 3.6 Differentiation and Integration of Laplace Transforms . . . . 151 3.7 The Inverse Laplace Transform and Examples . . . . . . . . . 154 3.8 Tauberian Theorems and Watson’s Lemma . . . . . . . . . . 168 Basic Concepts and Definitions The Fourier Integral Formulas . . . . . . . . . . . . . . . . . . 10 6 17 12 9 Poisson’s Summation Formula . . . . . . . . . . . . . . . . . . 37 . . . . . . . . . . . . . . . . . . . . . . . 54 . . . . . . . . . . . . . . . . . . . . . . 96 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 2.19 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4 Applications of Laplace Transforms 181 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 4.2 Solutions of Ordinary Differential Equations . . . . . . . . . . 182 4.3 Partial Differential Equations, Initial and Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 4.4 Solutions of Integral Equations . . . . . . . . . . . . . . . . . 222 4.5 Solutions of Boundary Value Problems . . . . . . . . . . . . . 225 4.6 Evaluation of Definite Integrals . . . . . . . . . . . . . . . . . 228 4.7 Solutions of Difference and Differential-Difference Equations . 230 4.8 Applications of the Joint Laplace and Fourier Transform . . . 237 4.9 Summation of Infinite Series . . . . . . . . . . . . . . . . . . . 248 4.10 Transfer Function and Impulse Response Function of a Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 4.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 5 Fractional Calculus and Its Applications 269 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 5.2 Historical Comments . . . . . . . . . . . . . . . . . . . . . . . 270 5.3 Fractional Derivatives and Integrals . . . . . . . . . . . . . . . 272 5.4 Applications of Fractional Calculus . . . . . . . . . . . . . . . 279 5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 6 Applications of Integral Transforms to Fractional Differential and Integral Equations 283 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 6.2 Laplace Transforms of Fractional Integrals and Fractional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 6.3 Fractional Ordinary Differential Equations . . . . . . . . . . . 287 6.4 Fractional Integral Equations . . . . . . . . . . . . . . . . . . 290 6.5 Initial Value Problems for Fractional Differential Equations . 295 6.6 Green’s Functions of Fractional Differential Equations . . . . 298 6.7 Fractional Partial Differential Equations . . . . . . . . . . . . 299 6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 7 Hankel Transforms and Their Applications 315 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 7.2 The Hankel Transform and Examples . . . . . . . . . . . . . . 316 7.3 Operational Properties of the Hankel Transform . . . . . . . . 319 7.4 Applications of Hankel Transforms to Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 © 2007 by Taylor & Francis Group, LLC 8 Mellin Transforms and Their Applications 339 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 8.2 Definition of the Mellin Transform and Examples . . . . . . . 340 8.3 Basic Operational Properties of Mellin Transforms . . . . . . 343 8.4 Applications of Mellin Transforms . . . . . . . . . . . . . . . 349 8.5 Mellin Transforms of the Weyl Fractional Integral and the Weyl Fractional Derivative . . . . . . . . . . . . . . . . . 353 8.6 Application of Mellin Transforms to Summation of Series . . 358 8.7 Generalized Mellin Transforms . . . . . . . . . . . . . . . . . 361 8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 9 Hilbert and Stieltjes Transforms 371 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 9.2 Definition of the Hilbert Transform and Examples . . . . . . 372 9.3 Basic Properties of Hilbert Transforms . . . . . . . . . . . . . 375 9.4 Hilbert Transforms in the Complex Plane . . . . . . . . . . . 378 9.5 Applications of Hilbert Transforms . . . . . . . . . . . . . . . 380 9.6 Asymptotic Expansions of One-Sided Hilbert Transforms . . . 388 9.7 Definition of the Stieltjes Transform and Examples . . . . . . 391 9.8 Basic Operational Properties of Stieltjes Transforms . . . . . 394 9.9 Inversion Theorems for Stieltjes Transforms . . . . . . . . . . 396 9.10 Applications of Stieltjes Transforms . . . . . . . . . . . . . . . 399 9.11 The Generalized Stieltjes Transform . . . . . . . . . . . . . . 401 9.12 Basic Properties of the Generalized Stieltjes Transform . . . . 403 9.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 10 Finite Fourier Sine and Cosine Transforms 407 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 10.2 Definitions of the Finite Fourier Sine and Cosine Transforms and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 10.3 Basic Properties of Finite Fourier Sine and Cosine Transforms 410 10.4 Applications of Finite Fourier Sine and Cosine Transforms . . 416 10.5 Multiple Finite Fourier Transforms and Their Applications . 422 10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 11 Finite Laplace Transforms 429 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 11.2 Definition of the Finite Laplace Transform and Examples . . 430 11.3 Basic Operational Properties of the Finite Laplace Transform 436 11.4 Applications of Finite Laplace Transforms . . . . . . . . . . . 439 11.5 Tauberian Theorems . . . . . . . . . . . . . . . . . . . . . . . 443 11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443© 2007 by Taylor & Francis Group, LLC 12 Z Transforms 445 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 12.2 Dynamic Linear Systems and Impulse Response . . . . . . . . 445 12.3 Definition of the Z Transform and Examples . . . . . . . . . . 449 12.4 Basic Operational Properties of Z Transforms . . . . . . . . . 453 12.5 The Inverse Z Transform and Examples . . . . . . . . . . . . 459 12.6 Applications of Z Transforms to Finite Difference Equations . 463 12.7 Summation of Infinite Series . . . . . . . . . . . . . . . . . . . 466 12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 13 Finite Hankel Transforms 473 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 13.2 Definition of the Finite Hankel Transform and Examples . . . 473 13.3 Basic Operational Properties . . . . . . . . . . . . . . . . . . 476 13.4 Applications of Finite Hankel Transforms . . . . . . . . . . . 476 13.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 14 Legendre Transforms 485 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 14.2 Definition of the Legendre Transform and Examples . . . . . 486 14.3 Basic Operational Properties of Legendre Transforms . . . . . 489 14.4 Applications of Legendre Transforms to Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 14.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 15 Jacobi and Gegenbauer Transforms 501 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 15.2 Definition of the Jacobi Transform and Examples . . . . . . . 501 15.3 Basic Operational Properties . . . . . . . . . . . . . . . . . . 504 15.4 Applications of Jacobi Transforms to the Generalized Heat Conduction Problem . . . . . . . . . . . . . . . . . . . . . . . 505 15.5 The Gegenbauer Transform and Its Basic Operational Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 15.6 Application of the Gegenbauer Transform . . . . . . . . . . . 510 16 Laguerre Transforms 511 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 16.2 Definition of the Laguerre Transform and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 16.3 Basic Operational Properties . . . . . . . . . . . . . . . . . . 516 16.4 Applications of Laguerre Transforms . . . . . . . . . . . . . . 520 16.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523© 2007 by Taylor & Francis Group, LLC 17 Hermite Transforms 525 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 17.2 Definition of the Hermite Transform and Examples . . . . . . 526 17.3 Basic Operational Properties . . . . . . . . . . . . . . . . . . 529 17.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 18 The Radon Transform and Its Applications 539 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 18.2 The Radon Transform . . . . . . . . . . . . . . . . . . . . . . 541 18.3 Properties of the Radon Transform . . . . . . . . . . . . . . . 545 18.4 The Radon Transform of Derivatives . . . . . . . . . . . . . . 550 18.5 Derivatives of the Radon Transform . . . . . . . . . . . . . . 551 18.6 Convolution Theorem for the Radon Transform . . . . . . . . 553 18.7 Inverse of the Radon Transform and the Parseval Relation . . 554 18.8 Applications of the Radon Transform . . . . . . . . . . . . . . 560 18.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 19 Wavelets and Wavelet Transforms 563 19.1 Brief Historical Remarks . . . . . . . . . . . . . . . . . . . . . 563 19.2 Continuous Wavelet Transforms . . . . . . . . . . . . . . . . . 565 19.3 The Discrete Wavelet Transform . . . . . . . . . . . . . . . . 573 19.4 Examples of Orthonormal Wavelets . . . . . . . . . . . . . . . 575 19.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 Appendix A Some Special Functions and Their Properties 587 A-1 Gamma, Beta, and Error Functions . . . . . . . . . . . . . . . 587 A-2 Bessel and Airy Functions . . . . . . . . . . . . . . . . . . . . 592 A-3 Legendre and Associated Legendre Functions . . . . . . . . . 598 A-4 Jacobi and Gegenbauer Polynomials . . . . . . . . . . . . . . 601 A-5 Laguerre and Associated Laguerre Functions . . . . . . . . . . 605 A-6 Hermite Polynomials and Weber-Hermite Functions . . . . . . 607 A-7 Mittag Leffler Function . . . . . . . . . . . . . . . . . . . . . . 609 Appendix B Tables of Integral Transforms 611 B-1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . 611 B-2 Fourier Cosine Transforms . . . . . . . . . . . . . . . . . . . . 615 B-3 Fourier Sine Transforms . . . . . . . . . . . . . . . . . . . . . 617 B-4 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . 619 B-5 Hankel Transforms . . . . . . . . . . . . . . . . . . . . . . . . 624 B-6 Mellin Transforms . . . . . . . . . . . . . . . . . . . . . . . . 627 B-7 Hilbert Transforms . . . . . . . . . . . . . . . . . . . . . . . . 630 B-8 Stieltjes Transforms . . . . . . . . . . . . . . . . . . . . . . . 633 B-9 Finite Fourier Cosine Transforms . . . . . . . . . . . . . . . . 636 B-10 Finite Fourier Sine Transforms . . . . . . . . . . . . . . . . . 638 B-11 Finite Laplace Transforms . . . . . . . . . . . . . . . . . . . . 640B-12 Z Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 B-13 Finite Hankel Transforms . . . . . . . . . . . . . . . . . . . . 644 Answers and Hints to Selected Exercises 645 2.19 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 4.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 9.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 13.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 16.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 17.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 18.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 19.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 Bibliography 673 Other Core of CS Books Integral equation - Wikipedia, the free encyclopedia Integral Equations and Operator Theory Handbook of Integral Equations Inequalities for Differential and Integral Equations Download