## Integral Transforms and Their Applications, Second EditionKeeping 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: 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 |

689 | |

### Common terms and phrases

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