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Wong R. — Asymptotic approximations of integrals
Wong R. — Asymptotic approximations of integrals

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Название: Asymptotic approximations of integrals

Автор: Wong R.

Аннотация:

Asymptotic methods are frequently used in many branches of both pure and applied mathematics, and this classic text remains the most up-to-date book dealing with one important aspect of this area, namely, asymptotic approximations of integrals. In Asymptotic Approximations of Integrals , all results are proved rigorously, and many of the approximation formulas are accompanied by error bounds. A thorough discussion on multidimensional integrals is given, and references are provided. Asymptotic Approximations of Integrals contains the "distributional method," which is not available elsewhere. Most of the examples in this text come from concrete applications.

Since its publication twelve years ago, significant developments have occurred in the general theory of asymptotic expansions, including smoothing of the Stokes phenomenon, uniform exponentially improved asymptotic expansions, and hyperasymptotics. These new concepts belong to the area now known as "exponential asymptotics." Expositions of these new theories are available in papers published in various journals, but not yet in book form.



Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 543

Добавлена в каталог: 05.12.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abel limit      198
Abel summability      198
Abramowitz, A.      82 222
Abscissas      222
Airy function      46 51 90 133 370
Airy function of negative argument      220
Airy function of negative argument, asymptotic expansion of      220
Airy function of negative argument, relation to Bessel functions      220
Airy function, asymptotic expansion of      47 93
Airy function, differential equation for      91
Airy function, generalized      386
Airy function, integral of      46 91
Airy function, Maclaurin expansion of      91
Airy integral      see "Airy function"
Amplitude function      429
Anger function $A_{\nu}(x)$      418
Anger function $A_{\nu}(x)$, uniform asymptotic expansion of      418
Anger's function $J_{\nu}(x)$      82
Armstrong, J.A.      144
Asymptotic equalities      2
Asymptotic expansions      5
Asymptotic expansions of Poincare type      5 11
Asymptotic expansions of power series form      11
Asymptotic expansions, compound      11
Asymptotic expansions, failure of      18
Asymptotic expansions, generalized      10
Asymptotic expansions, uniform      353 356
Asymptotic formulas      see "Asymptotic equalities"
Asymptotic sequences      10
Asymptotics      1
Asymptotics, paradox in      146
Atiyah, M.F.      492
Bakhoom, N.G.      144
Barnes' Lemma      48 229 355
Barnes, E.W.      48 54
Bender, C.M.      82
Bender, E.A.      4 121
Berg, L.      54
Berghuis, J.      475
Bernoulli numbers      2 32
Bernoulli polynomials      32
Bernoulli polynomials, generalized      47
Bessel function      24 45
Bessel function of large argument      24 26 112—116 134
Bessel function of large order      98 100 102—103
Bessel function of the second kind      236
Bessel function, asymptotic formulas      205 206
Bessel function, integral representation of      24 97 112 134 417
Bessel function, modified      50 225
Bessel function, modified, asymptotic expansion of      227
Bessel function, relation to Hankel functions      97 211 227
Bessel function, Schlaefii's integral representation for $J_{\nu}(\nu a)$      97
Bessel function, series representation      231
Bessel transforms      221
Binet — Cauchy product formula      510
Bleistein, N.      54 144 148 149 171 191 192 364 368 421 422 423 476 514 515
Boersma, J.      352
Bremermann, H.J.      265 290 291 313 314
Bremmer, H.      475
Bruening, J.      177 191 192
Carlson, B.C.      176 345 346
Catastrophy theory      388
Cauchy limit      500
Cauchy principal value      313
Cauchy, A.L.      76
Caustics      392
Central limit theorem      141
Chako, N.      423 475 515
Chen, T.C.      224
Chester, C.      367 368 421
Complementary error function      42 218
Complementary error function, asymptotic expansion of      42
Completeness of $\mathscr{D}^{'}$      248
Confluent hypergeometric function      46
Confluent hypergeometric function, asymptotic expansion of      46
Connor, J.N.L.      388
Convergence in $\mathscr{D}$      242 265
Convergence in $\mathscr{D}^{'}$      247 265
Convergence in $\mathscr{S}$      261 267
Convergence in $\mathscr{S}^{'}$      262
Convolutions      254
Convolutions of distributions      241 255
Convolutions, differentiation of      255
Cooke, J.C.      475 476
Copson, E.T.      3 9 54 61 64 90 93 103 111 144 382 388 422
Cosine integral      513
Courant, R.      280
Critical points      281 424 478
Critical points of the first kind      426
Critical points of the second kind      426 442 478
Critical points of the third kind      426 448
Cubic transformation of Chester, Friedman & Ursell      356 367
Curvature of an arc      510
Darboux approximant      117
Darboux's method      116 146 146
Darboux, G.M.      117
Davis, P.J.      222
de Bruijn, N.G.      54 144 146 476 515
de Jager, E.M.      242
de Kok, F.      475
Debye's method      84 87 see
Debye, P.      84 87 98
Delta function      see "Dirac delta function"
Derivatives (of distributions)      249 266
Derivatives of convolutions      see "Convolutions"
Derivatives of Fourier transforms      274 276
Dieudonne, J.      134 146
Diffeomorphisms      479
Differentiation under an integral sign      327
Dingle, R.B.      54
Dirac delta function      244 266
Dirac delta function, derivative of      252 283
Dirac delta function, Fourier transform of      287
Dirac delta function, translated      244 285
Dirichlet — Mehler integral      420
Dispersive wave theory      83
Distributional approach      293 351
Distributional/Weak limit      247
Distributional/Weak limit in $\mathscr{D}^{'}$      247
Distributional/Weak limit in $\mathscr{S}^{'}$      262
Distributions      243 265 see
Distributions in $\mathbb{R}^{n}$      265
Distributions, $r^{\lambda}$      268
Distributions, $x^{\lambda}_{+} log^{k} x_{+}$      270 272
Distributions, $x^{\lambda}_{+}$      258 263 272
Distributions, concentrated on a set      280 282
Distributions, convergence of      see "Convergence"
Distributions, convolution of      see "Convolution"
Distributions, derivative of      see "Derivatives"
Distributions, equal in an open set      245
Distributions, multiplication by a $C^{\infty}$-function      248
Distributions, regular      243
Distributions, singular      243
Distributions, support of      245
Distributions, surface      241 280
Distributions, translation of      249
Distributions, vanishing of      247
Distributions, zero      244
Divided differences      223
Doetsch, G.      148 230
Dominant (exponentially increasing)      94
Double integrals      176 423
Duffin, R.J.      515
Durbin, P.      351
Edgeworth series      141 146
Edwards, C.H., Jr.      288 487 488 490 509 510
Elementary symmetric functions in theory of equations      385
Elliptic function (of Weierstrass)      391
Elliptic integrals of the first kind      345
Erdelyi, A.      9 10 18 54 64 77 82 137 143 144 145 154 171 172 192 219 277 333 406 421 422 475
Error function      see "Complementary error function"
Error test      38
Euler — Maclaurin summation formula      32
Euler's limit formula      51
Evgrafov, M.A.      54 146 152
Exponential integral      14 29
Exponential integral, asymptotic expansion of      15 30
Exponentially small      6 355
Farrell, R.A.      192
Fedoryuk, M.V.      54 475 483 514 515
Feller, W.      55 146
Fields, J.L.      54 146
Fleming, W.H.      274
Focke, J.      423 476
Fourier integral      15 195 198 210 221
Fourier integral, asymptotic expansion of      16 199
Fourier integral, error bound for      201
Fourier transforms      147 221 274 276 323 see
Fourier transforms of $r^{\lambda}$      278 279
Fourier transforms of $x^{\lambda}_{+}$      277
Fourier transforms of a tempered distribution      276
Fourier transforms, inverse      229
Fourier transforms, n-dimensional      274 276 478 500
Fourier transforms, n-dimensional, asymptotic expansion of      503
Fourier transforms, n-dimensional, inverse      287
Fourier transforms, near the origin      321 323 362
Fourier transforms, Parseval identity for      287
Fractional integral      326
Fractional integral, asymptotic expansion of      328 330
Fractional integral, error bound for      332
Frank, R.M.      224
Franklin, J.      54
Frenzen, C.L.      54 129 215 240 421
Fresnel integral      218 332 352 411
Fresnel Integral, Mellin transform of      219
Friedman, B.      54 367 368 421
Fu, J.C.      146
Fulks, W.      143 515
Functional, continuous linear      242
Functional, continuous linear on $\mathscr{D}$      243 265
Functional, continuous linear on $\mathscr{S}$      262 267
Gabutti, B.      210 240
Gamma function      60 see
Gamma function, asymptotic expansion of      62 110 135
Gamma function, asymptotic expansion of ratio      47
Gamma function, asymptotic formula for large imaginary argument      154
Gamma function, residues of      154
Gamma function, singularities of      321
Gauss sum      203
Gauss — Laguerre quadrature formulae      222
Gaussian quadrature      221
Gautschi, W.      228
Gelfand, I.M.      241 248 273 290 291
Generalized functions      240 see
Geometric optics      388
Gilmore, R.      388
Glauber, R.J.      210
Gradshteyn, I.S.      169 434 437 439
Green, G.      3
Greene, D.H.      2
Greenhill, A.G.      391
Grosjean, C.C.      352
Grosswald, E.      146
Group velocity      83
Grzesik, J.      515
Gustafson, J.L.      176 346
Handelsman, R.A.      54 148 149 152 157 158 171 191 192 423 476 514 515
Hankel function      94
Hankel function, asymptotic expansion of      134
Hankel function, integral representation of      94
Hankel function, Mellin transform of      171
Hankel transform      147 203 210 502
Hankel transform, asymptotic expansion of      204—207
Hankel transform, error bound for      207—209
Hardy, G.H.      54 197
Harmonic numbers      1 38
Harris, B.      142 146
Hayman's admissible functions      123
Hayman's formula      4 122 145
Hayman, W.K.      4 122 123 126 146
Heaviside function      252 282
Heaviside function, distributional derivative of      262
Henrici, P.      54 56 137 462 515
Hessian matrix      478
Hessian of a function      435
Hilbert transform      187 312
Hilbert transform, asymptotic expansion of      314—316
Hildebrand, F.B.      221 223 226 227
Hills      87
Hironaka, H.      492
Hironaka, theorem of      492
Hoermander, L.      482 483
Hsu, L.C.      54 476 514 515
Incomplete Gamma function, complementary      28 43 108 232 365
Incomplete Gamma function, complementary, asymptotic expansion of      43
Incomplete Gamma function, complementary, uniform asymptotic expansion of      365 366
Jeanquartier, P.      291
Jeffreys, H.      54
John, F.      280
Jones, D.S.      143 242 351 421 423 475 476 484 514 516
Kaminski, D.      385 389 391 392 394 422
Kaper, H.G.      352
Karatygin, V.A.      452
Klein — Gordon equation      84 144
Kline, M.      423 475 476
Knuth, D.E.      2 141
Kontorovich — Lebedev transform      324
Kontorovich, M.I.      452
Lagrange resolvents for zeros of quarries      391
Lagrange's inversion formula      61 64 93
Laguerre polynomials      222 372 396 421
Laguerre polynomials, uniform asymptotic expansion of      381 402
Lam, P.F.      282
Landau, E.      4
Laplace transforms      30 69 147 321
Laplace transforms, inverse      30 229
Laplace transforms, near the origin      167—171 322
Laplace transforms, Tauberian theorem for      230
Laplace's approximation (or Laplace's formula)      3 56
Laplace's approximation (or Laplace's formula), n-dimensional      494 508 515
Laplace's approximation (or Laplace's formula), two-dimensional      459
Laplace's method      55
Laplace-type integrals      424 459 467 494
Laplace-type integrals, asymptotic expansions of      461 467 495 498
LaSalle, J.P.      282
Laurent series      270
Laurent series for $r^{\lambda}$      273
Laurent series for $x^{\lambda}_{+}$      272
Lauwerier, H.A.      54 351
Le Marquis de Laplace      3 56
Lebedev, N.N.      154
Lebesgue constants      1 2 40—42
Lebesgue constants, associated with polynomial interpolation      64
Lefschetz, S.      282
Legendre function of the first kind      406 422
Legendre function of the first kind, uniform asymptotic expansion of      408
Legendre function of the second kind      47
Legendre polynomials      141 146 420
Legendre polynomials, asymptotic expansion of      141
Legendre polynomials, uniform asymptotic expansion of      420
Leighton, W.      282
Lepora, P.      240
Levey, H.C.      352
Levinson, N.      383
Lew, J.S.      148 152 157 158 191 192
Lewis, R.M.      422
Lighthill, M.J.      240 351
Lin, J.F.      144 146
Liouville — Green approximation      4
Liouville, J.      3
1 2
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