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Wong K. Asymptotic Approximations of Integrals
Wong K.  Asymptotic Approximations of Integrals









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: Asymptotic Approximations of Integrals

: Wong K.

:

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.


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ed2k: ed2k stats

: 2001

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: 13.12.2009

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Abel limit      198
Abel summability      198
Abscissas      222
Airy function      46 51 90 133 370
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, of negative argument      220
Airy function, relation to Bessel functions      220
Airy integral      (see Airy function)
Amplitude function      429
Anger function $A_{v}(x)$      418
Anger function $A_{v}(x)$, uniform asymptotic expansion of      418
Angers function $J_{v}(x)$      82
Asymptotic of Poincare type      5 11
Asymptotic of power series form      11
Asymptotic, compound      11
Asymptotic, equalities      2
Asymptotic, expansions      5
Asymptotic, failure of      18
Asymptotic, formulas      (see Asymptotic equalities)
Asymptotic, generalized      10
Asymptotic, sequences      10
Asymptotic, uniform      353 356
Asymptotics      1
Asymptotics, paradox in      146
Barnes lemma      48 229 355
Bernoulli numbers      2 32
Bernoulli polynomials      32
Bernoulli polynomials, generalized      47
Bessel function      24 45
Bessel function of large argument      2426 112116 134
Bessel function of large order      98 100 102103
Bessel function of the second kind      236
Bessel function, asymptotic expansion of      227
Bessel function, asymptotic formulas      205 206
Bessel function, integral representation of      24 97 112 134 417
Bessel function, modified      50 225
Bessel function, relation to Hankel functions      97 211 227
Bessel function, Schlaeflis integral representation for $J_{v}(\nu\alpha)$      97
Bessel function, series representation      231
Bessel transforms      221
Binet Cauchy product formula      510
Catastrophy theory      388
Cauchy limit      500
Cauchy principal value      313
Caustics      392
Central limit theorem      141
Complementary error function      42 218
Complementary error function, asymptotic expansion of      42
Completeness of $\mathcal{D}'$      248
Confluent hypergeometric function      46
Confluent hypergeometric function, asymptotic expansion of      46
Convergence in $\mathcal{D}'$      247 265
Convergence in $\mathcal{Y}$      261 267
Convolutions      254
Convolutions differentiation of      255
Convolutions of distributions      241 255
Cosine integral      513
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
Darbouxs method      116 145 146
Debyes method      84 87
Delta function      (see Dirac delta function)
Derivatives (of distributions)      249 266
Derivatives (of distributions) of convolutions      (see Convolutions)
Derivatives (of distributions) of Fourier transforms      274 276
Diffeomorphisms      479
Differentiation under an integral sign      327
Dirac delta function      244 266
Dirac delta function, derivative of      252 283
Dirichlet Mehler integral      420
Dispersive wave theory      83
Distributional approach      293 351
Distributional/Weak limit      247
Distributional/Weak limit in $\mathcal{D}'$      247
Distributional/Weak limit in $\mathcal{Y}'$      262
Distributions      243 265
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, in $\mathbb{R}^{n}$      265
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
Dominant (exponentially increasing)      94
Double integrals      176 423
Edgeworth series      141 146
Elementary symmetric functions in theory of equations      385
Elliptic function (of Weierstrass)      391
Elliptic integrals of the first kind      345
Error function      (see Complementary error function)
Error test      38
Euler Maclaurin summation formula      32
Eulers limit formula      51
Exponential integral      14 29
Exponential integral, asymptotic expansion of      15 30
Exponentially small      6 355
Fourier integral      15 195 198 210 221
Fourier integral, asymptotic expansion of      16 199
Fourier integral, error bound for      201
Fourier transform of      287
Fourier transform of translated      244 285
Fractional integral      326
Fractional integral, asymptotic expansion of      328 330
Fractional integral, error bound for      332
Fresnel integral      218 332 362 411
Fresnel Integral, Mellin transform of      219
Functional, continuous linear      242
Functional, continuous linear on $\mathcal{D}$      243 265
Functional, continuous linear on $\mathcal{Y}$      262 267
Furier transforms      147 221 274 276 323
Furier transforms of $r^{\lambda}$      278 279
Furier transforms of $x^{\lambda}_{+}$      277
Furier transforms of a tempered distribution      276
Furier transforms, n-dimensional      274 276 478 500
Furier transforms, n-dimensional, asymptotic expansion of      603
Furier transforms, n-dimensional, inverse      287
Furier transforms, near the origin      321 323 362
Furier transforms, Parseval identity for      287
Gamma function      60 (see also Incomplete Gamma function)
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 Laguerre quadrature formulae      222
Generalized functions      240 (see also Distributions)
Geometric optics      388
Group velocity      83
Guass sum      203
Guassian quadrature      221
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      204207
Hankel transform, error bound for      207209
Harmonic numbers      1 38
Haymans admissible functions      123
Haymans admissible functions, distributional derivative of      262
Haymans formula      4 122 145
Heaviside function      252 282
Hessian matrix      478
Hessian of a function      436
Hilbert transform      187 312
Hilbert transform, asymptotic expansion of      314316
Hills      87
Hironaka, theorem of      492
Incomplete Gamma function, asymptotic expansion of      43
Incomplete Gamma function, complementary      28 43 108 232 365
Incomplete Gamma function, uniform asymptotic expansion of      365 366
Klein Gordon equation      84 144
Kontorovich Lebedev transform      324
Lagrange resolvents for zeros of quartics      391
Lagranges inversion formula      61 64 93
Laguerre polynomials      222 372 396 421
Laguerre polynomials, uniform asymptotic expansion of      381 402
Laplace transforms      30 69 147 321
Laplace transforms, inverse      30 229
Laplace transforms, near the origin      167171 322
Laplace-type integrals      424 459 467 494
Laplace-type integrals, asymptotic expansions of      461 467 495 498
Laplaces approximation (or Laplaces formula)      3 56
Laplaces approximation (or Laplaces formula), n-dimensional      494 508 516
Laplaces approximation (or Laplaces formula), two-dimensional      459
Laplaces method      55
Laurent series      270
Laurent series for $r^{\lambda}$      273
Laurent series for $x^{\lambda}_{+}$      272
Lebesgue constants      1 2 4042
Lebesgue constants, associated with polynomial interpolation      54
Legendre ftinction of the first kind      406 422
Legendre ftinction of the second kind      47
Legendre ftinction, uniform asymptotic expansion of      408
Legendre polynomials      141 146 420
Legendre polynomials uniform asymptotic expansion of      420
Legendre polynomials, asymptotic expansion of      141
Liouville Green approximation      4
Locally absolutely continuous function      260
Logarithmic integral      43
Logarithmic integral, asymptotic expansion of      43
Logarithmic singularity      66 144 146 186
Maclaurin coefficients, asymptotic behavior of      116127 145
Maclaurin expansion of      91
Mellin transform technique      147 351
Mellin transforms      147
Mellin transforms (relation with) bilateral Laplace transform      149
Mellin transforms, analytic continuation      153
Mellin transforms, convolution integral      147 151 333 352
Mellin transforms, generalized      337
Mellin transforms, inversion formula      147 151
Mellin transforms, Parseval formula (or Parseval identity)      148 189
Mellin transforms, properties of      149
Mellin transforms, table of      192194
Mellin Barnes type integrals      228
Method of steepest descents      84 144 145
Morses Lemma      479
Multiple integrals      177 477
Multiple Integrals, resolution of      280
Nesting property      281
Neutralizers      246
Open covering      246
Oscillatory kernels      216
Parabolic cylinder function      236 364
Parseval formula      (see Mellin transform & Fourier transform)
Partition of unity      246
Pearcey integral      388 395
Pearcey integral, uniform asymptotic expansion of      416 417
Perrons method      103 145 406
Phase function      76
phase velocity      83
Phragm      6
Phragmen Lindeloef theorem      166
Plemelj formula      313
Pochhammer notation      47
Positive definite functions      281
Principle of Stationary Phase      76 144 240
Quadrature formulae for Bessel transforms      226
Quadrature formulae for Fourier transforms      222
Quadrature formulae, Gauaa Laguerre      222
Quartic transformation      392
Rapidly decreasing functions      261 266
Recessive (exponentially decreasing)      94
Regularization      267
Regularization at infinity      260 261
Regularization in finite interval      259
Regularization of a function      258
Regularization of an integral      258 336
Regularization, method of      333 352
Regularization, nonuniqueness      272
Resolution of singularities      (see Singularities)
Riemann-Lebesgue lemma, generalized      200
Riemann-Liouville fractional integral      (see Fractional integral)
Robinsons conjecture      120
Saddle point      86 432
Saddle point of order m      87
Saddle point, many coalescing      383
Saddle point, near a pole      356
Saddle point, near an endpoint      360
Saddle point, relevant      89
Saddle point, two coalescing      366
Saddle-point method      103
Schlaefiis integral      97
Schroedinger spectral kernel      230
Scorers function      419
Second fundamental form of differential, geometry      511
Signature of a matrix      480
Sine integral      513
Singularities, resolution of      492
Splitting lemma      491 512
Stationary phase      (see Principle of stationary phase)
Stationary phase approximation      77 240
Stationary phase approximation for double integrals      431
Stationary phase approximation, n-dimensional analogue      478
Stationary points      77 424 478
Stationary points, boundary      440 484
Stationary points, curve of      452
Stationary points, degenerate      435 491
Stationary points, non-degenerate      487
Steepest ascent      87
Steepest descent      86 87
Stiltjes transform      147 295 304 351
Stiltjes transform, asymptotic expansion of      186
Stiltjes transform, error bounds for      302304 309
Stiltjes transform, generalized      185
Stirling formula      1 51 62
Stirling numbers      143
Stirling numbers, asymptotic expansion of      143
Stirling series      51
Stokes line      94
Stokes phenomenon      94
Summability method      195 216
Support of a distribution      245
Support of a function      242
Surface distribution      241 280
Swallowtail integral      388 389
Swallowtail integral, uniform asymptotic expansion of      395
Tangential contact, points of      487
Tauberian theorem      (see Laplace transforms)
Tauberian theorem for      230
Taylor series for $r^{\lambda}$      273
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