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| Davies B. — Integral Transforms and their Applications |
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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 expansion 34 50—53 197—200 215—222
Asymptotically equal 34
Barnes 205
Bernoulli's equation 133
Bessel functions 310—318
Bessel functions of the first kind 66 115 312
Bessel functions of the second and third kind 314—318
Bessel functions, Fourier transform of 115
Bessel functions, integral representations 310—314 318—320
Bessel functions, integrals involving 324
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 integral formula 9
Cauchy integrals 283—288
Cauchy — Riemann relations 3
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 Pade approximation 350
Diffraction problems 185—187 265—265 297—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—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 — Koeber 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 in two or more variables 181—181 189
Fourier transform of generalized functions 155
Fourier transform of test functions 145
Fourier transform, application to partial differential equations 129—140
Fourier transform, definition 111
Fourier transform, inverse of 112
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—157 167 188 276
Generalized functions on finite interval 148
Generalized functions, convergence of 150
Generalized functions, definition 146
Generalized functions, differentiation of 149
Generalized functions, Fourier transforms of 155
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
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
Hoelder condition 285
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 of meromorphic functions 47
Inverse Laplace transform of rational functions 44
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, Taylor series of 46
Jacobi polynomials 335
Kirchhoff 185
Kontorovich — Lebedev transform 256—262
Kontorovich — Lebedev transform, relation to Mellin transform 258
Kramers — Kroenig 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
MacDonald's function 321
MacRobert 227
Matrix exponential 73
Maxwell's equations 187
Mellin transform 195
Mellin transform in asymptotics 197—200 215—215
Mellin transform in summation 211—216
Mellin transform, application to differential equations 205
Mellin transform, application to potential problems 202
Mellin transform, definition 195
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-Lefller theorem 281
Modified Bessel functions 320
Moebius transformation 331
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
Pade 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—187 202 233—237
Power series 10
Power series, asymptotic behavior of 215—217
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 zeta function 23—26
Riemann zeta function in summation 212
Riemann zeta function, asymptotic forms 26
Riemann zeta function, functional relation 25
Riemann — Hilbert problem 289—291
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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