Asymptotics and Mellin-Barnes Integrals provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typically of interest in classical analysis and mathematical physics. After developing the properties of these integrals, their use in determining the asymptotic behavior of special functions is detailed. Although such integrals have a long history, the book's account includes recent research results in analytic number theory and hyperasymptotics. The book also fills a gap in the literature on asymptotic analysis and special functions by providing a thorough account of the use of Mellin-Barnes integrals that is otherwise not available in standard references on asymptotics.
Author(s): R. B. Paris, D. Kaminski
Series: Encyclopedia of Mathematics and its Applications
Publisher: Cambridge University Press
Year: 2001
Language: English
Pages: 440
Half-title......Page 3
Series-title......Page 5
Title......Page 7
Copyright......Page 8
Contents......Page 9
Preface......Page 15
1.1.1 Order Relations......Page 19
1.1.2 Asymptotic Expansions......Page 22
Robert Hjalmar Mellin......Page 43
Ernest William Barnes......Page 44
2.1.1 The Asymptotic Expansion of Ghe(z)......Page 47
2.1.2 The Stirling Coefficients......Page 50
2.1.3 Bounds for Ghe(z)......Page 51
2.2 Expansion of Quotients of Gamma Functions......Page 53
2.2.1 Inverse Factorial Expansions......Page 54
2.2.2 A Recursion Formula when Alpha = Beta = 1......Page 58
2.2.3 Examples......Page 62
2.2.4 An Algebraic Method for the Determination of the A......Page 64
2.2.5 Special Cases......Page 67
2.3 The Asymptotic Expansion of Integral Functions......Page 73
2.3.1 An Example......Page 76
2.4 Convergence of Mellin-Barnes Integrals......Page 81
2.5.1 An Example......Page 87
2.5.2 Lemmas......Page 89
3.1.1 Definition......Page 97
3.1.2 Translational and Differential Properties......Page 99
3.1.3 The Parseval Formula......Page 100
3.2 Analytic Properties......Page 103
3.3.1 Integrals Connected with e......Page 107
3.3.2 Some Standard Integrals......Page 109
3.3.3 Discontinuous Integrals......Page 111
3.3.4 Gamma-Function Integrals......Page 114
3.3.5 Ramanujan-Type Integrals......Page 117
3.3.6 Barnes’ Lemmas......Page 121
3.4 Mellin-Barnes Integral Representations......Page 124
3.4.1 The Confluent Hypergeometric Functions......Page 125
3.4.2 The Gauss Hypergeometric Function......Page 128
3.4.3 Some Special Functions......Page 130
4.1.1 The Mellin Transform Method......Page 135
4.1.2 The Poisson-Jacobi Formula......Page 138
4.2.1 An Infinite Series......Page 140
4.2.2 A Smoothed Dirichlet Series......Page 143
4.2.3 A Finite Sum......Page 146
4.3.1 A Harmonic Sum......Page 151
4.3.2 Euler’s Product......Page 154
4.3.3 Ramanujan’s Function......Page 155
4.3.4 Some Other Number-Theoretic Sums......Page 159
4.4.1 Potential Problems in Wedge-Shaped Regions......Page 164
4.4.2 Ordinary Differential Equations......Page 167
4.4.3 Inverse Mellin Transform Solutions......Page 170
4.5 Solution of Integral Equations......Page 174
4.5.1 Kernels of the Form k(xt)......Page 175
4.5.2 Kernels of the Form k(x/t)......Page 178
4.6 Solution of Difference Equations......Page 182
4.6.1 Solution by Mellin Transforms......Page 184
4.6.2 The Hypergeometric Difference Equation......Page 185
4.6.3 Solution of the Inhomogeneous First-Order Equation......Page 190
4.7 Convergent Inverse Factorial Series......Page 192
5.1 Algebraic Asymptotic Expansions......Page 197
5.1.1 The Exponential Integral E1(z)......Page 198
5.1.2 The Parabolic Cylinder Function D(z)......Page 200
5.1.3 A Bessel Function Integral......Page 202
5.1.4 The Mittag-Leffler Function Epsilona(z)......Page 204
5.2.1 Error Bounds......Page 208
5.2.2 Numerical Evaluation......Page 213
5.3 Saddle-Point Approximation of Integrals......Page 215
5.3.1 An Integral Due to Heading and Whipple......Page 216
5.3.2 The Bessel Function J(nx)......Page 218
5.3.3 A Gauss Hypergeometric Function......Page 220
5.4 Exponential Asymptotic Expansions......Page 223
5.4.1 The Exponential Integral E1(z)......Page 224
5.4.2 The Bessel Function J(z)......Page 226
5.4.3 The Parabolic Cylinder Function D(z)......Page 228
5.4.4 An Infinite Sum......Page 231
5.5 Faxén’s Integral......Page 234
5.6 Integrals with a ‘Contour Barrier’......Page 238
5.6.1 An Illustrative Example......Page 239
5.6.2 An Integral Involving a Bessel Function......Page 243
5.6.3 Example in 4.2.3 Revisited......Page 247
6.1.1 A Qualitative Description......Page 252
6.1.2 The Modified Bessel Function K(z)......Page 254
6.2 Mellin-Barnes Theory......Page 258
6.2.1 Exponentially-Improved Expansion......Page 259
6.2.2 Estimates for…......Page 262
6.2.3 The Stokes Multiplier......Page 264
6.2.4 The Stokes Multiplier for a High-Order Differential Equation......Page 267
6.2.5 Numerical Examples......Page 274
6.2.6 Asymptotics of the Terminant T(z)......Page 277
6.3 Hyperasymptotics......Page 283
6.3.1 Mellin-Barnes Theory of Hyperasymptotics......Page 284
6.3.2 Optimal Truncation Schemes......Page 289
6.3.3 A Numerical Example......Page 295
6.4 Exponentially-Improved Asymptotics for Ghe(z)......Page 297
6.4.1 Origin of the Exponentially Small Terms......Page 298
6.4.2 The Expansion of Omega(z)......Page 300
6.4.3 A Numerical Example......Page 304
7.1 Some Double Integrals......Page 307
7.2 Residues and Double Integrals......Page 313
7.3 Laplace-Type Double Integrals......Page 317
7.3.1 Mellin-Barnes Integral Representation......Page 321
7.3.2 The Newton Diagram......Page 322
7.4 Asymptotics of I(Lambda)......Page 323
7.4.1 Two Internal Points......Page 324
7.4.2 Three and More Internal Points......Page 329
7.4.3 Other Double Integrals......Page 340
7.5 Geometric Content......Page 341
7.5.1 Remoteness......Page 342
7.5.2 Asymptotic Scales......Page 344
7.6.1 Representation of Treble Integrals......Page 346
7.6.2 Asymptotics with One Internal Point......Page 347
7.6.3 Asymptotics with Two Internal Points......Page 350
7.6.4 Other Considerations for Treble Integrals......Page 357
7.7 Numerical Examples......Page 358
8 Application to Some Special Functions......Page 370
8.1.1 Introduction......Page 371
8.1.2 The Saddle Point Approach......Page 374
8.1.3 The Asymptotics of S(a)......Page 379
8.1.4 Mellin-Barnes Integral Approach......Page 383
8.1.5 The Asymptotic Expansion of…......Page 386
8.1.6 The Coefficients A......Page 389
8.1.7 The Stokes Phenomenon for p Asymptotically 2......Page 392
8.2.1 Introduction......Page 398
8.2.2 An Expansion for Zeta(s)......Page 400
8.2.3 An Exponentially-Smoothed Gram-Type Expansion......Page 402
8.2.4 A Riemann-Siegel-Type Expansion......Page 403
8.3.1 Introduction......Page 407
8.3.2 A Mellin-Barnes Integral Representation......Page 409
8.3.3 Asymptotics of…......Page 411
8.3.4 Asymptotics of…......Page 416
8.3.5 Asymptotics of P(X, Y) for Real X, Y......Page 418
8.3.6 Generalisations of P(X, Y)......Page 419
A Short Table of Mellin Transforms......Page 422
References......Page 427
Index......Page 437
