## Integral Transforms and Their ApplicationsMathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in re search and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numeri cal and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathe matical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs. Pasadena, California J.E. Marsden Providence, Rhode Island L. Sirovich Houston, Texas M. Golubitsky College Park, Maryland S.S. Antman Preface to the Third Edition It is more than 25 years since I finished the manuscript of the first edition of this volume, and it is indeed gratifying that the book has been in use over such a long period and that the publishers have requested a third edition. |

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### Contents

IV | 1 |

V | 6 |

VI | 9 |

VII | 13 |

VIII | 15 |

IX | 18 |

X | 19 |

XI | 23 |

LXVIII | 185 |

LXIX | 187 |

LXX | 189 |

LXXI | 195 |

LXXII | 196 |

LXXIII | 200 |

LXXIV | 202 |

LXXV | 203 |

XII | 27 |

XIII | 28 |

XIV | 32 |

XV | 33 |

XVI | 36 |

XVII | 39 |

XVIII | 41 |

XIX | 42 |

XX | 44 |

XXI | 46 |

XXII | 47 |

XXIII | 49 |

XXIV | 50 |

XXV | 52 |

XXVI | 53 |

XXVII | 54 |

XXVIII | 57 |

XXIX | 59 |

XXX | 61 |

XXXI | 65 |

XXXII | 67 |

XXXIII | 72 |

XXXIV | 79 |

XXXV | 82 |

XXXVI | 85 |

XXXVII | 86 |

XXXVIII | 89 |

XXXIX | 90 |

XL | 92 |

XLI | 94 |

XLII | 97 |

XLIII | 101 |

XLIV | 104 |

XLV | 107 |

XLVI | 111 |

XLVII | 116 |

XLVIII | 119 |

XLIX | 121 |

L | 123 |

LI | 129 |

LII | 132 |

LIII | 135 |

LIV | 137 |

LV | 140 |

LVI | 143 |

LVII | 144 |

LVIII | 148 |

LIX | 153 |

LX | 155 |

LXI | 157 |

LXII | 163 |

LXIII | 167 |

LXIV | 169 |

LXV | 173 |

LXVI | 176 |

LXVII | 181 |

LXXVI | 205 |

LXXVII | 207 |

LXXVIII | 211 |

LXXIX | 213 |

LXXX | 215 |

LXXXI | 218 |

LXXXII | 221 |

LXXXIII | 223 |

LXXXIV | 227 |

LXXXV | 230 |

LXXXVI | 231 |

LXXXVII | 232 |

LXXXVIII | 234 |

LXXXIX | 236 |

XC | 237 |

XCI | 239 |

XCII | 242 |

XCIII | 249 |

XCIV | 251 |

XCV | 253 |

XCVI | 256 |

XCVII | 258 |

XCVIII | 259 |

XCIX | 262 |

C | 265 |

CI | 273 |

CII | 274 |

CIII | 278 |

CIV | 283 |

CV | 285 |

CVI | 289 |

CVII | 291 |

CVIII | 295 |

CIX | 297 |

CX | 302 |

CXI | 303 |

CXII | 305 |

CXIII | 307 |

CXIV | 310 |

CXV | 312 |

CXVI | 314 |

CXVII | 319 |

CXVIII | 320 |

CXIX | 321 |

CXX | 327 |

CXXI | 329 |

CXXII | 331 |

CXXIII | 335 |

CXXIV | 338 |

CXXV | 343 |

CXXVI | 349 |

CXXVII | 352 |

357 | |

363 | |

### Common terms and phrases

analytic continuation analytic function applied arg(z asymptotic expansion asymptotic form behavior Bessel functions boundary conditions boundary-value problem bounded branch cut branch point Cauchy integrals Chapter choose close the contour coefficients consider convergence convolution deform the contour denote derive diffraction entire function error evaluate example exponential factor finite follows formula Fourier transform func function f(x given gives Green's function half-plane Hankel transform Hence Im(u infinite integral equation integral representation integral transforms integrand interval inversion contour inversion integral J-oo Laplace transform linear loop integral Math matrix Mellin transform method notation numerical obtain origin pair parameters plane polynomial potential properties radius Re(a Re(p Re(z real axis region relation residues restriction result satisfies Section shown in Figure simple pole singularity solution solved substitution Suppose techniques test function theorem theory tion transfer function truncation Watson's lemma wave write zero