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Wong R. — Asymptotic approximations of integrals
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Название: Asymptotic approximations of integralsАвтор: 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: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2001Количество страниц: 543Добавлена в каталог: 05.12.2013Операции: Положить на полку |
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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 418 Anger function 418 Anger's function 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 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 248 Confluent hypergeometric function 46 Confluent hypergeometric function, asymptotic expansion of 46 Connor, J.N.L. 388 Convergence in 242 265 Convergence in 247 265 Convergence in 261 267 Convergence in 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 247 Distributional/Weak limit in 262 Distributions 243 265 see Distributions in 265 Distributions, 268 Distributions, 270 272 Distributions, 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 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 278 279 Fourier transforms of 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 243 265 Functional, continuous linear on 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 273 Laurent series for 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
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