Integral Transforms and Their Applications

Cover......Page 1
Editors......Page 2
List of Publications......Page 3
Title......Page 4
LCC  2001032818 ......Page 5
Series Preface ......Page 6
Preface to the Third Edition ......Page 8
Preface to the Second Edition ......Page 10
Preface to the First Edition ......Page 11
Contents ......Page 13
1.1 Analytic Functions ......Page 18
1.2 Contour Integration ......Page 23
1.3 Analytic Continuation ......Page 26
1.4 Residue Theory ......Page 30
1.5 Loop Integrals ......Page 32
1.6 Liouville\'s Theorem ......Page 35
1. 7 The Factorial Function ......Page 36
1.8 Riemann\'s Zeta Function ......Page 40
2.1 The Laplace Integral ......Page 44
2.2 Important Properties ......Page 45
2.3 Simple Applications ......Page 49
2.4 Asymptotic Properties: Watson\'s Lemma ......Page 50
Problems ......Page 53
3.1 The Riemann-Lebesgue Lemma ......Page 56
3.3 The Inversion ......Page 58
3.4 Inversion of Rational Functions ......Page 61
3.5 Taylor Series Expansion ......Page 63
3.6 Inversion of Meromorphic Functions ......Page 64
3.7 Inversions Involving a Branch Point ......Page 66
3.8 Watson\'s Lemma for Loop Integrals ......Page 67
3.9 Asymptotic Forms for Large t ......Page 69
3.10 Heaviside Series Expansion ......Page 70
Problems ......Page 71
4.1 Elementary Examples ......Page 74
4.2 Higher-Order Equations ......Page 76
4.3 Transfer Functions and Block Diagrams ......Page 78
4.4 Equations with Polynomial Coefficients ......Page 82
4.5 Simultaneous Differential Equations ......Page 84
4.6 Linear Control Theory ......Page 89
4. 7 Realization of Thansfer Functions ......Page 96
Problems ......Page 99
5  Partial Differential Equations I ......Page 102
5.1 Heat Diffusion: Semi-Infinite Region ......Page 103
5.2 Finite Thickness ......Page 106
5.3 Wave Propagation ......Page 107
5.4 Thansmission Line ......Page 109
Problems ......Page 111
6.1 Convolution Equations of Volterra Type ......Page 114
6.2 Convolution Equations over an Infinite Range ......Page 118
6.3 The Percus- Yevick Equation ......Page 121
Problems ......Page 124
7.1 Exponential, Sine, and Cosine Thansforms ......Page 128
7.2 Important Properties ......Page 133
7.3 Spectral Analysis ......Page 136
7.4 Kramers-Kronig Relations ......Page 138
Problems ......Page 140
8.1 Potential Problems ......Page 146
8.2 Water Waves: Basic Equations ......Page 149
8.3 Waves Generated by a Surface Displacement ......Page 152
8.4 Waves Generated by a Periodic Disturbance ......Page 154
Problems ......Page 157
9.1 The Delta Function ......Page 160
9.2 Test Functions and Generalized Functions ......Page 161
9.3 Elementary Properties ......Page 165
9.4 Analytic Functionals ......Page 170
9.5 Fourier Transforms of Generalized Functions ......Page 172
Problems ......Page 174
10.1 One-Dimensional Green\'s Functions ......Page 180
10.2 Green\'s Functions as Generalized Functions ......Page 184
10.3 Poisson\'s Equation in Two Dimensions ......Page 186
10.4 Helmholtz\'s Equation in Two Dimensions ......Page 190
Pro blems ......Page 193
11.1 Basic Notation and Results ......Page 198
11.2 Diffraction of Scalar Waves ......Page 202
11.3 Retarded Potentials of Electromagnetism ......Page 204
Problems ......Page 206
12.1 Definitions ......Page 212
12.2 Simple Examples ......Page 213
12.3 Elementary Properties ......Page 217
12.4 Potential Problems in Wedge-Shaped Regions ......Page 219
12.5 Thansforms Involving Polar Coordinates ......Page 220
12.6 Hermite Functions ......Page 222
Problems ......Page 224
13.1 Mellin Summation Formula ......Page 228
13.2 A Problem of Ramanujin ......Page 230
13.3 Asymptotic Behavior of Power Series ......Page 232
13.4 Integrals Involving a Parameter ......Page 235
13.5 Ascending Expansions for Fourier Integrals ......Page 238
Problems ......Page 240
14.1 The Hankel Transform Pair ......Page 244
14.2 Elementary Properties ......Page 247
14.3 Some Examples ......Page 248
14.4 Boundary-Value Problems ......Page 249
14.5 Weber\'s Integral ......Page 251
14.6 The Electrified Disc ......Page 253
14.7 Dual Integral Equations of Titchmarsh Type ......Page 254
14.8 Erdelyi-Kober Operators ......Page 256
Problems ......Page 259
15.1 The Basic Formula ......Page 266
15.2 Finite Intervals ......Page 268
15.3 Some Singular Problems ......Page 270
15.4 Kontorovich-Lebedev Transform ......Page 273
15.5 Boundary- Value Problems in a Wedge ......Page 275
15.6 Diffraction of a Pulse by a Two- Dimensional Half- Plane ......Page 276
Problems ......Page 279
16.1 The Sommerfeld Diffraction Problem ......Page 284
16.2 Wiener-Hopf Procedure: Half- Plane Problems ......Page 290
16.3 Integral and Integra-Differential Equations ......Page 291
Problems ......Page 295
17.1 Wiener-Hopf Decomposition by Contour Integration ......Page 300
17.2 Cauchy Integrals ......Page 302
17.3 The Riemann-Hilbert Problem ......Page 306
17.4 Problems in Linear Thansport Theory ......Page 308
17.5 The Albedo Problem ......Page 312
17.6 A Diffraction Problem ......Page 314
Pro blems ......Page 319
18.1 Laplace\'s Method ......Page 320
18.2 Hermite Polynomials ......Page 322
18.3 Hermite Functions ......Page 324
18.4 Bessel Functions: Integral Representations ......Page 327
18.6 Functions of the Second and Third Kind ......Page 331
18.7 Poisson and Related Representations ......Page 336
18.8 Modified Bessel Functions ......Page 337
Pro blems ......Page 338
19.1 General Considerations ......Page 344
19.2 Gaver-Stehfest Method ......Page 346
19.3 Mobius Thansformation ......Page 348
19.4 Use of Chebyshev Polynomials ......Page 352
19. 5 Use of Laguerre Polynomials ......Page 355
19.6 Representation by Fourier Series ......Page 360
19.7 Quotient-Difference Algorithm ......Page 366
19.8 Talbot\'s Method ......Page 369
Bibliography ......Page 374
Index ......Page 380
Texts in Applied Mathematics ......Page 385