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Davies B. — Integral Transforms and Their Applications
Davies B. — Integral Transforms and Their Applications

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Название: Integral Transforms and Their Applications

Автор: Davies B.

Аннотация:

This book is an introduction to and a reference for the applications of integral problems to a wide range of common mathematical problems. Emphasis is placed on the development of techniques and on the connection between properties of transforms and the kinds of problems for which they provide tools. Over 400 problems accompany the text, illustrating areas of applications. This third edition has been substantially updated, extended, and reorganized. Graduate students and researchers working in mathematics and physics will find this book useful.

The author is Reader in Mathematics at the Australian National University, with interests in applied mathematics, mathematical physics, scientific computation, nonlinear dynamics, and chaos. He has also held posts in physics and theoretical physics, and in private and government research organizations. He is the author of a recent book chaotic systems, as well as of numerous research articles in many areas.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Third Edition

Год издания: 2002

Количество страниц: 392

Добавлена в каталог: 25.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abel's integral equation      107
Abscissa of convergence      28 331
Adjoint boundary conditions      165
Adjoint problem      165
Advanced potential      188
Airy functions      325
Albedo problem      295
Analytic      4
Analytic continuation      9-12
Analytic function      4
Analytic functionals      153-156
Anomalous system      71
Asymptotic behavior of      215-217
Asymptotic expansion      34 50-53 197-200 215-222
Asymptotically equal      34
Barnes      205
Bernoulli's equation      133
Bessel functions      310-318
Bessel functions, Fourier transform of      115
Bessel functions, integral representations      310-314 318-320
Bessel functions, integrals involving      324
Bessel functions, of the first kind      66 115 312
Bessel functions, of the second and third kind      314-318
Bessel's equation      65 310
Bessel's integral      115 184 313
beta function      23
Block Diagrams      61
Branch cut      5
Branch point, definition      4
Branch point, integrals around      15
Branch point, inversions involving      49-52 100 116 174 256 298 307
Bromwich contour      43 353
Carleman      265
Case and Zweifel      295
Cauchy - Riemann relations      3
Cauchy integral formula      9
Cauchy integrals      283-288
Cauchy's theorems      8
Causality      121
Chebyshev polynomials      335-338
Clenshaw's algorithm      343
Complementary error function      88 208
Continued fractions      350
Continuity of linear functionals      147
contour      2
Contour integration      6
Controllability      75
Convergence of generalized functions      150
Convergence of test functions      147
Convolution equations      97-104
Convolutions      32 58 87 117 122 182 201 218
Cosine transform      111
Coulomb gauge      187
Cramer's rule      71
Cylinder functions      227 315
D'Alembert's method      194
Delta function      143 147
Diagonal Pad$\acute{e}$ approximation      350
Diffraction problems      185-187 265- 272 297-301
Diffusion problems      85-90
Dirac's delta function      143
Direct correlation function      105
Dirichlet conditions      41
Dirichlet integrals      41
Discontinuity theorem      288
Distributions      154
Double Laplace transforms      192- 194
Dual integral equations      236-239
Eigenfunction expansion      253
Electric circuit problems      68-70
Electron gas      220
Electrostatic problems      129-131 191 209 233 236
Entire function      18
Epsilon algorithm      347
Erdelyi - K$\ddot{o}$ber operators      239-242
Euler's constant      12 199 221
Exponential integral      12
Factorial function      19-22
Factorial function, asymptotic expansion      216
Factorial function, functional relations      20
Factorial function, Hankel's integral representation      22
Fast Fourier Transform      341
feedback loop      63
Fourier integrals, ascending expansions for      221
Fourier series      229 252 258 343
Fourier transform, application to partial differential equations      129-140
Fourier transform, definition      111
Fourier transform, in two or more variables      181 189
Fourier transform, inverse of      112
Fourier transform, of generalized functions      155
Fourier transform, of test functions      145
Fourier transform, properties of      116-118
Fourier transform, relation to Green's functions      254
Fourier transform, relation to Hankel transform      229
Fourier transform, relation to Laplace transform      111
Fourier transform, sine and cosine transforms      112
Fractional integration      239
Fraunhofer diffraction      186
Fresnel diffraction      186
Functional      144
Functional, analytic      153-156
Functional, continuous      147
Functional, linear      146
Functional, regular      147
Functional, singular      147
Generalized functions      143-157 167 276
Generalized functions, convergence of      150
Generalized functions, definition      146
Generalized functions, differentiation of      149
Generalized functions, Fourier transforms of      155
Generalized functions, on finite interval      148
Generalized functions, properties of      147-151
Generalized functions, regular      147
Generalized functions, sequences of      150
Generalized functions, singular      147
Green's functions      163-166
Green's functions, as generalized functions      167
Green's functions, for adjoint      165
Green's functions, for Helmholtz's equation      173
Green's functions, for Poisson's equation      169
Green's functions, integral transforms generated by      249
Green's functions, one-dimensional      163
Green's functions, symmetry of      171
Green's theorem      7
H$\ddot{o}$lder condition      285
Hankel functions      173 228 272 317-320
Hankel transform      227
Hankel transform, application to boundary-value problems      232
Hankel transform, connection with Fourier transform      229
Hankel transform, definition      227
Hankel transform, inverse of      227
Hankel transform, properties of      230 231
Hankel transform, relation to Green's functions      255
Hankel's loop integral      17
Harmonic function      9 133 246 258
Heat conduction      86-90
Heat diffusion kernel      87
Heaviside distortionless line      93
Heaviside expansion theorem      48
Heaviside series expansion      53
Heaviside step function      28
Helmholtz's equation      173-176 266
Helmholtz's equation, elementary solution      173
Helmholtz's equation, Green's function for      176
Hermite equation      305
Hermite functions      307-310
Hermite functions, asymptotic forms      205-207
Hermite polynomials      305-307
Hopf      265
Hydrodynamic equations      132
Images      172
Impedance      93
Influence function      122
Integral equations      97-107 274 292
Integral equations, classification      97
Integral equations, dual      236-239
Integrals, Fourier      221
Integrals, involving a parameter      218
Integro-differential equations      274
Inverse Fourier transform      112
Inverse Fourier transform, sine and cosine transform      113
Inverse Laplace transform      39
Inverse Laplace transform, asymptotic forms of      50-54
Inverse Laplace transform, involving a branch point      49
Inverse Laplace transform, numerical evaluation of      327-355
Inverse Laplace transform, of meromorphic functions      47
Inverse Laplace transform, of rational functions      44
Inverse Laplace transform, Taylor series of      46
Jacobi polynomials      335
Kirchhoff      185
Kontorovich - Lebedev transform      256-262
Kontorovich - Lebedev transform, relation to Mellin transform      258
Kramers - Kr$\ddot{o}$nig relations      121
Lagrangian interpolation      333
Laguerre polynomials      210 334 338
Laplace transform, application to ordinary differential equations      59
Laplace transform, application to partial differential equations      85-93
Laplace transform, application to simultaneous differential equations      67
Laplace transform, asymptotic properties      33 52
Laplace transform, definition      27
Laplace transform, differential equations with polynomial coefficients      65
Laplace transform, double      192-194
Laplace transform, inverse of      39
Laplace transform, inversion theorem      42
Laplace transform, properties of      28-32
Laplace transform, relation to Fourier transform      111
Laplace transform, Watson's lemma      35 50
Laplace's equation      129 190 233
Laplace's method      66 303-321
Laurent expansion      13
Lienard - Wiechert potential      189
Linear control theory      72-82
Linear control theory, controllability      75
Linear control theory, equivalent systems      77
Linear control theory, minimal realization      79
Linear control theory, observability      78
Linear control theory, realization      79
Linear functionals      146
Linear transport theory      291-297
Liouville's theorem      18
Lommel's integral      228
Loop integrals      15 49-53
M$\ddot{o}$bius transformation      331
MacDonald's function      321
MacRobert      227
Matrix exponential      73
Maxwell's equations      187
Mellin transform      195
Mellin transform, application to differential equations      205
Mellin transform, application to potential problems      202
Mellin transform, definition      195
Mellin transform, in asymptotics      197-200 215- 222
Mellin transform, in summation      211-216
Mellin transform, inverse of      196
Mellin transform, properties of      200-203
Mellin transform, relation to Fourier transform      195
Mellin transform, relation to Green's functions      255
Meromorphic functions      14 47
Meromorphic functions, inverse Laplace transform of      47
Method of images      172
Milne's equation      274 284
Minimal realization of transfer function      79
Mittag - LefHer theorem      281
Modified Bessel functions      320
Newton's law of cooling      89
Newton's second law      132
Normal system      71
Numerical inversion of Laplace transforms      327-355
Numerical inversion of Laplace transforms, collocation methods      333
Numerical inversion of Laplace transforms, Fourier series methods      343
Numerical inversion of Laplace transforms, Gaver - Stehfest method      329
Numerical inversion of Laplace transforms, Korrectur method      346
Numerical inversion of Laplace transforms, Lyness and Giunta's method      340
Numerical inversion of Laplace transforms, method of de Hoog, Knight, and Stokes      349
Numerical inversion of Laplace transforms, Talbot's method      352
Numerical inversion of Laplace transforms, Weeks's methods      339
Observability      78
Ordinary differential equations, Green's functions for      163-169
Ordinary differential equations, Laplace transform methods for      57-77 79-82
Ordinary differential equations, Laplace's method for      303-321
Ordinary differential equations, stability of solutions      60
Pad$\acute{e}$ approximation      349
Pair distribution function      104
Parseval relations      118 121 231
Partial differential equations, Fourier transform methods for      129-140
Partial differential equations, Laplace transform methods for      85-93
Partial fractions      45
Percus - Yevick equation      104
Plemelj formulae      286-289
Poisson integral representation      225 319
Poisson summation formula      128 153
Poisson's equation      169
Pole      13
Polynomial interpolation      333
potential problems      129-132 187- 202 233-237
Power series      10
Principal value integral      122 152 286
Quotient-difference algorithm      351
Radiation condition      139 184 188
Rational functions, inverse Laplace transform of      44
Realization of transfer functions      79
Recurrence relations      336 342 351
Regular generalized functions      147
Regularization      328
Residue theory      13-15
Resolvent kernel      98
Retarded potential      187
Riemann - Hilbert problem      289-291
Riemann zeta function      23-26
Riemann zeta function, asymptotic forms      26
Riemann zeta function, functional relation      25
Riemann zeta function, in summation      212
Self-adjoint      166 176 249
Shrinking a contour      15
Simple pole      13
Sine transform      111
Singular generalized functions      147
Singular point      4
Singularity      4
Sommerfeld diffraction problem      265-272
Sonine's integrals      243
Spectral analysis      119-121
Stability of solutions      60
Stirling's series      216
Stretched string      90-93
Sturm - Liouville problem      251
Symmetry of Green's functions      171
Taylor series of inverse Laplace transform      46
Test functions      144-146
Titchmarsh      237 251
Transfer functions      61
transmission line      92 93
Trapezoidal rule      340 344 353
Two-point boundary-value problem      164
Ultradistributions      154
Variation of parameters      164
Watson's lemma      33-36
Watson's lemma, for loop integrals      50-53
Wave equation      85 90 173 185 259 265 297
Wave propagation      90
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