# Integral Transforms and Their Applications, Second Edition

CRC Press, Oct 11, 2006 - Mathematics - 722 pages
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.
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### Contents

 Integral Transforms 1 12 Basic Concepts and Definitions 6 Fourier Transforms and Their Applications 9 22 The Fourier Integral Formulas 10 23 Definition of the Fourier Transform and Examples 12 24 Fourier Transforms of Generalized Functions 17 25 Basic Properties of Fourier Transforms 28 26 Poissons Summation Formula 37
 104 Applications of Finite Fourier Sine and Cosine Transforms 416 105 Multiple Finite Fourier Transforms and Their Applications 422 106 Exercises 425 Finite Laplace Transforms 429 112 Definition of the Finite Laplace Transform and Examples 430 113 Basic Operational Properties of the Finite Laplace Transform 436 114 Applications of Finite Laplace Transforms 439 115 Tauberian Theorems 443

 27 The Shannon Sampling Theorem 44 28 Gibbs Phenomenon 54 29 Heisenbergs Uncertainty Principle 57 210 Applications of Fourier Transforms to Ordinary Differential Equations 60 211 Solutions of Integral Equations 65 212 Solutions of Partial Differential Equations 68 213 Fourier Cosine and Sine Transforms with Examples 91 214 Properties of Fourier Cosine and Sine Transforms 93 215 Applications of Fourier Cosine and Sine Transforms to Partial Differential Equations 96 216 Evaluation of Definite Integrals 100 217 Applications of Fourier Transforms in Mathematical Statistics 103 218 Multiple Fourier Transforms and Their Applications 109 219 Exercises 119 Laplace Transforms and Their Basic Properties 133 32 Definition of the Laplace Transform and Examples 134 33 Existence Conditions for the Laplace Transform 139 34 Basic Properties of Laplace Transforms 140 35 The Convolution Theorem and Properties of Convolution 145 36 Differentiation and Integration of Laplace Transforms 151 37 The Inverse Laplace Transform and Examples 154 38 Tauberian Theorems and Watsons Lemma 168 39 Exercises 173 Applications of Laplace Transforms 181 42 Solutions of Ordinary Differential Equations 182 43 Partial Differential Equations Initial and Boundary Value Problems 207 44 Solutions of Integral Equations 222 45 Solutions of Boundary Value Problems 225 46 Evaluation of Definite Integrals 228 47 Solutions of Difference and DifferentialDifference E quations 230 48 Applications of the Joint Laplace and Fourier Transform 237 49 Summation of Infinite Series 248 410 Transfer Function and Impulse Response Function of a Linear System 251 411 Exercises 256 Fractional Calculus and Its Applications 269 52 Historical Comments 270 53 Fractional Derivatives and Integrals 272 54 Applications of Fractional Calculus 279 55 Exercise 282 Applications of Integral Transforms to Fractional Differential and Integral Equations 283 62 Laplace Transforms of Fractional Integrals and Fractional Derivatives 284 63 Fractional Ordinary Differential Equations 287 64 Fractional Integral Equations 290 65 Initial Value Problems for Fractional Differential Equations 295 66 Greens Functions of Fractional Differential Equations 298 67 Fractional Partial Differential Equations 299 68 Exercises 312 Hankel Transforms and Their Applications 315 72 The Hankel Transform and Examples 316 73 Operational Properties of the Hankel Transform 319 74 Applications of Hankel Transforms to Partial Differential Equations 322 75 Exercises 331 Mellin Transforms and Their Applications 339 82 Definition of the Mellin Transform and Examples 340 83 Basic Operational Properties of Mellin Transforms 343 84 Applications of Mellin Transforms 349 85 Mellin Transforms of the Weyl Fractional Integral and the Weyl Fractional Derivative 353 86 Application of Mellin Transforms to Summation of Series 358 87 Generalized Mellin Transforms 361 88 Exercises 365 Hilbert and Stieltjes Transforms 371 92 Definition of the Hilbert Transform and Examples 372 93 Basic Properties of Hilbert Transforms 375 94 Hilbert Transforms in the Complex Plane 378 95 Applications of Hilbert Transforms 380 96 Asymptotic Expansions of OneSided Hilbert Transforms 388 97 Definition of the Stieltjes Transform and Examples 391 98 Basic Operational Properties of Stieltjes Transforms 394 99 Inversion Theorems for Stieltjes Transforms 396 910 Applications of Stieltjes Transforms 399 911 The Generalized Stieltjes Transform 401 912 Basic Properties of the Generalized Stieltjes Transform 403 913 Exercises 404 Finite Fourier Sine and Cosine Transforms 407 102 Definitions of the Finite Fourier Sine and Cosine Transforms and Examples 408 103 Basic Properties of Finite Fourier Sine and Cosine Transforms 410
 Z Transforms 445 123 Definition of the Z Transform and Examples 449 124 Basic Operational Properties of Z Transforms 453 125 The Inverse Z Transform and Examples 459 126 Applications of Z Transforms to Finite Difference Equations 463 127 Summation of Infinite Series 466 128 Exercises 469 Finite Hankel Transforms 473 133 Basic Operational Properties 476 135 Exercises 481 Legendre Transforms 485 142 Definition of the Legendre Transform and Examples 486 143 Basic Operational Properties of Legendre Transforms 489 144 Applications of Legendre Transforms to Boundary Value Problems 497 145 Exercises 498 Jacobi and Gegenbauer Transforms 501 153 Basic Operational Properties 504 154 Applications of Jacobi Transforms to the Generalized Heat Conduction Problem 505 155 The Gegenbauer Transform and Its Basic Operational Properties 507 156 Application of the Gegenbauer Transform 510 Laguerre Transforms 511 163 Basic Operational Properties 516 164 Applications of Laguerre Transforms 520 165 Exercises 523 Hermite Transforms 525 172 Definition of the Hermite Transform and Examples 526 173 Basic Operational Properties 529 174 Exercises 538 The Radon Transform and Its Applications 539 182 The Radon Transform 541 183 Properties of the Radon Transform 545 184 The Radon Transform of Derivatives 550 185 Derivatives of the Radon Transform 551 186 Convolution Theorem for the Radon Transform 553 187 Inverse of the Radon Transform and the Parseval Relation 554 188 Applications of the Radon Transform 560 189 Exercises 561 Wavelets and Wavelet Transforms 563 192 Continuous Wavelet Transforms 565 193 The Discrete Wavelet Transform 573 194 Examples of Orthonormal Wavelets 575 195 Exercises 584 Some Special Functions and Their Properties 587 A2 Bessel and Airy Functions 592 A3 Legendre and Associated Legendre Functions 598 A4 Jacobi and Gegenbauer Polynomials 601 A5 Laguerre and Associated Laguerre Functions 605 A6 Hermite Polynomials and WeberHermite Functions 607 A7 Mittag Leffler Function 609 Tables of Integral Transforms 611 TABLE B2 Fourier Cosine Transforms 615 TABLE B3 Fourier Sine Transforms 617 TABLE B4 Laplace Transforms 619 TABLE B5 Hankel Transforms 624 TABLE B6 Mellin Transforms 627 TABLE B7 Hilbert Transforms 630 TABLE B8 Stieltjes Transforms 633 TABLE B9 Finite Fourier Cosine Transforms 636 TABLE B10 Finite Fourier Sine Transforms 638 TABLE B11 Finite Laplace Transforms 640 TABLE B12 Z Transforms 642 TABLE B13 Finite Hankel Transforms 644 Answers and Hints to Selected Exercises 645 39 Exercises 651 411 Exercises 655 68 Exercises 662 88 Exercises 663 913 Exercises 664 106 Exercises 665 116 Exercises 667 135 Exercises 670 189 Exercises 671 Bibliography 673 Index 689 Copyright