This page intentionally left blank Asymptotics and Mellin-Barnes Integrals is a comprehensive account of the proper- ties of Mellin-Barnes integrals and their application to problems involving special functions, primarily the determination of asymptotic expansions. An account of the basic analytical properties of Mellin-Barnes integrals and Mellin transforms and their use in applications ranging from number theory to differential and difference equations is followed by a systematic analysis of the asymptotics of Mellin-Barnes representations of many important special functions, including hypergeometric, Bessel and parabolic cylinder functions. An account of the recent developments in the understanding of the Stokes phenomenon and of hyperasymptotics in the set- ting of Mellin-Barnes integrals ensues. The book concludes with the application of ideas set forth in the earlier parts of the book to higher-dimensional Laplace-type integrals and sophisticated treatments of Euler-Jacobi series, the Riemann zeta function and the Pearcey integral. Detailed numerical illustrations accompany many of the results developed in the text. Richard Paris is a Reader in Mathematics at the University of Abertay Dundee. He is the author of numerous papers that have appeared in Proceedings A of the Royal Society of London, Methods and Applications of Analysis and the Journal of Computational and Applied Mathematics and is the co-author of Asymptotics of High Order Differential Equations. David Kaminski is an Associate Professor of Mathematics at the University of Lethbridge. He is the author of numerous papers that have appeared in such publi- cations as SIAM Journal on Mathematical Analysis, the Journal of Computational and Applied Mathematics, Utilitas Mathematica and Methods and Applications of Analysis. encyclopedia of mathematics and its applications EDITED BY G.-C. ROTA Editorial Board R. Doran, M. Ismail, T.-Y. Lam, E. Lutwak Volume 85 Asymptotics and Mellin-Barnes Integrals 6 H. Minc Permanents 19 G. G. Lorentz, K. Jetter, and S. D. Riemenschneider Birkhoff Interpolation 22 J. R. Bastida Field Extensions and Galois Theory 23 J. R. Cannon The One-Dimensional Heat Equation 24 S. Wagon The Banach–Tarski Paradox 25 A. Salomaa Computation and Automata 27 N. H. Bingham, C. M. Goldie, and J. L. Teugels Regular Variation 28 P. P. Petrushev and V. A. Popov Rational Approximation of Real Functions 29 N. White (ed.) Combinatorial Geometries 30 M. Pohst and H. Zassenhaus Algorithmic Algebraic Number Theory 31 J. Aczel and J. Dhombres Functional Equations in Several Variables 32 M. Kuczma, B. Choczewski, and R. Ger Iterative Functional Equations 33 R. V. Ambartzumian Factorization Calculus and Geometric Probability 34 G. Gripenberg, S.-O. Londen, and O. Staffans Volterra Integral and Functional Equations 35 G. Gasper and M. Rahman Basic Hypergeometric Series 36 E. Torgersen Comparison of Statistical Experiments 37 A. Neumaier Interval Methods for Systems of Equations 38 N. Korneichuk Exact Constants in Approximation Theory 39 R. Brualdi and H. Ryser Combinatorial Matrix Theory 40 N. White (ed.) Matroid Applications 41 S. Sakai Operator Algebras in Dynamical Systems 42 W. Hodges Basic Model Theory 43 H. Stahl and V. Totik General Orthogonal Polynomials 44 R. Schneider Convex Bodies 45 G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions 46 A. Björner et al. Oriented Matroids 47 G. Edgar and L. Sucheston Stopping Times and Directed Processes 48 C. Sims Computation with Finitely Presented Groups 49 T. Palmer Banach Algebras and the General Theory of *-Algebras 50 F. Borceux Handbook of Categorical Algebra 1 51 F. Borceux Handbook of Categorical Algebra 2 52 F. Borceux Handbook of Categorical Algebra 3 53 V. F. Kolchin Random Graphs 54 A. Katok and B. Hasselblatt Intorduction to the Modern Theory of Dynamical Systems 55 V. N. Sachkov Combinatorial Methods in Discrete Mathematics 56 V. N. Sachkov Probabilistic Methods in Discrete Mathematics 57 P. M. Cohn Skew Fields 58 R. Gardner Geometric Topography 59 G. A. Baker Jr. and P. Graves-Morris Pade Approximations 60 J. Krajicek Bounded Arithmetic, Propositional Logic, and Complexity Theory 61 H. Groemer Geometric Applications of Fourier Series and Spherical Harmonics 62 H. O. Fattorini Infinite Dimensional Optimization and Control Theory 63 A. C. Thompson Minkowski Geometry 64 R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices with Applications 65 K. Engel Sperner Theory 66 D. Cvetkovic, P. Rowlinson, and S. Simic Eigenspaces of Graphs 67 F. Bergeron, G. Labelle, and P. Leroux Combinatorial Species and Tree-Like Structures 68 R. Goodman and N. Wallach Representations and Invariants of the Classical Groups 70 A. Pietsch and J. Wenzel Orthonormal Systems and Banach Space Geometry 71 G. E. Andrews, R. Askey, and R. Roy Special Functions ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Asymptotics and Mellin-Barnes Integrals R. B. PARIS D. KAMINSKI Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521790017 © Cambridge University Press 2001 This book is in copyright. Suect to statutory ception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2001 isbn-13- 978-0-511-06706-8 eBook (NetLibrary) isbn-10- 0-511-06706-2 eBook (NetLibrary) isbn-13- 978-0-521-79001-7 hardback isbn-10- 0-521-79001-8 hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for ternal or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Prefacepagexiii 1 Introduction 1 1.1 Introduction to Asymptotics 1 1.1.1 Order Relations 1 1.1.2 Asymptotic Expansions 4 1.1.3 Other Expansions 15 1.2 Biographies of Mellin and Barnes 25 2 Fundamental Results 29 2.1 The Gamma Function (z) 29 2.1.1 The Asymptotic Expansion of (z) 29 2.1.2 The Stirling Coefficients 32 2.1.3 Bounds for (z) 33 2.2 Expansion of Quotients of Gamma Functions 35 2.2.1 Inverse Factorial Expansions 36 2.2.2 A Recursion Formula when αr=βr=140 2.2.3 Examples 44 2.2.4 An Algebraic Method for the Determination of the Aj 46 2.2.5 Special Cases 49 2.3 The Asymptotic Expansion of Integral Functions 55 2.3.1 An Example 58 2.4 Convergence of Mellin-Barnes Integrals 63 2.5 Order Estimates for Remainder Integrals 69 2.5.1 An Example 69 2.5.2 Lemmas 71 vii viii Contents 3 Properties of Mellin Transforms 79 3.1 Basic Properties 79 3.1.1 Definition 79 3.1.2 Translational and Differential Properties 81 3.1.3 The Parseval Formula 82 3.2 Analytic Properties 85 3.3 Inverse Mellin Transforms 89 3.3.1 Integrals Connected with e−z 89 3.3.2 Some Standard Integrals 91 3.3.3 Discontinuous Integrals 93 3.3.4 Gamma-Function Integrals 96 3.3.5 Ramanujan-Type Integrals 99 3.3.6 Barnes’ Lemmas 103 3.4 Mellin-Barnes Integral Representations 106 3.4.1 The Confluent Hypergeometric Functions 107 3.4.2 The Gauss Hypergeometric Function 110 3.4.3 Some Special Functions 112 4 Applications of Mellin Transforms 117 4.1 Transformation of Series 117 4.1.1 The Mellin Transform Method 117 4.1.2 The Poisson-Jacobi Formula 120 4.2 Examples 122 4.2.1 An Infinite Series 122 4.2.2 A Smoothed Dirichlet Series 125 4.2.3 A Finite Sum 128 4.3 Number-Theoretic Examples 133 4.3.1 A Harmonic Sum 133 4.3.2 Euler’s Product 136 4.3.3 Ramanujan’s Function 137 4.3.4 Some Other Number-Theoretic Sums 141 4.4 Solution of Differential Equations 146 4.4.1 Potential Problems in Wedge-Shaped Regions 146 4.4.2 Ordinary Differential Equations 149 4.4.3 Inverse Mellin Transform Solutions 152 4.5 Solution of Integral Equations 156 4.5.1 Kernels of the Form k(xt) 157 4.5.2 Kernels of the Form k(x/t) 160 4.6 Solution of Difference Equations 164 4.6.1 Solution by Mellin Transforms 166 4.6.2 The Hypergeometric Difference Equation 167 4.6.3 Solution of the Inhomogeneous First-Order Equation 172 Contents ix 4.7 Convergent Inverse Factorial Series 174 5 Asymptotic Expansions 179 5.1 Algebraic Asymptotic Expansions 179 5.1.1 The Exponential Integral E1(z) 180 5.1.2 The Parabolic Cylinder Function Dν (z) 182 5.1.3 A Bessel Function Integral 184 5.1.4 The Mittag-Leffler Function Ea(z) 186 5.2 Remainder Integrals 190 5.2.1 Error Bounds 190 5.2.2 Numerical Evaluation 195 5.3 Saddle-Point Approximation of Integrals 197 5.3.1 An Integral Due to Heading and Whipple 198 5.3.2 The Bessel Function Jn(nx) 200 5.3.3 A Gauss Hypergeometric Function 202 5.4 Exponential Asymptotic Expansions 205 5.4.1 The Exponential Integral E1(z) 206 5.4.2 The Bessel Function Jν (z) 208 5.4.3 The Parabolic Cylinder Function Dν (z) 210 5.4.4 An Infinite Sum 213 5.5 Faxén’s Integral 216 5.6 Integrals with a ‘Contour Barrier’ 220 5.6.1 An Illustrative Example 221 5.6.2 An Integral Involving a Bessel Function 225 5.6.3 Example in §4.2.3 Revisited 229 6 The Stokes Phenomenon and Hyperasymptotics 234 6.1 The Stokes Phenomenon 234 6.1.1 A Qualitative Description 234 6.1.2 The Modified Bessel Function Kν (z) 236 6.2 Mellin-Barnes Theory 240 6.2.1 Exponentially-Improved Expansion 241 6.2.2 Estimates for |RN,M(z)| 244 6.2.3 The Stokes Multiplier 246 6.2.4 The Stokes Multiplier for a High-Order Differential Equation 249 6.2.5 Numerical