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• catalog abstract ""Asymptotics and Mellin-Barnes Integrals is a comprehensive account of the properties 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 setting 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."--Jacket.".
• catalog contributor b12249245.
• catalog contributor b12249246.
• catalog created "c2001.".
• catalog date "2001".
• catalog date "c2001.".
• catalog dateCopyrighted "c2001.".
• catalog description ""Asymptotics and Mellin-Barnes Integrals is a comprehensive account of the properties 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 setting of Mellin-Barnes integrals ensues.".
• catalog description "Includes bibliographical references (p. [409]-418) and index.".
• catalog description "Order Relations -- Asymptotic Expansions -- Other Expansions -- Biographies of Mellin and Barnes -- Fundamental Results -- The Gamma Function [Gamma] (z) -- The Asymptotic Expansion of [Gamma] (z) -- The Stirling Coefficients -- Bounds for [Gamma] (z) -- Expansion of Quotients of Gamma Functions -- Inverse Factorial Expansions -- A Recursion Formula when [alpha subscript r] = [beta subscript r] -- An Algebraic Method for the Determination of the A[subscript j] -- Special Cases -- The Asymptotic Expansion of Integral Functions -- Convergence of Mellin-Barnes Integrals -- Order Estimates for Remainder Integrals -- Lemmas -- Properties of Mellin Transforms -- Basic Properties -- Translational and Differential Properties -- The Parseval Formula -- Analytic Properties -- Inverse Mellin Transforms -- Integrals Connected with e[superscript -z] -- Some Standard Integrals -- Discontinuous Integrals -- Gamma-Function Integrals -- Ramanujan-Type Integrals -- Barnes' Lemmas -- Mellin-Barnes Integral Representations -- The Confluent Hypergeometric Functions -- The Gauss Hypergeometric Function -- Some Special Functions -- Applications of Mellin Transforms -- Transformation of Series -- The Mellin Transform Method -- The Poisson-Jacobi Formula -- An Infinite Series -- A Smoothed Dirichlet Series -- A Finite Sum -- Number-Theoretic Examples -- A Harmonic Sum -- Euler's Product -- Ramanujan's Function -- Some Other Number-Theoretic Sums -- Solution of Differential Equations -- Potential Problems in Wedge-Shaped Regions.".
• catalog description "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."--Jacket.".
• catalog extent "xvi, 422 p. :".
• catalog identifier "0521790018".
• catalog isPartOf "Encyclopedia of mathematics and its applications ; v. 85".
• catalog issued "2001".
• catalog issued "c2001.".
• catalog language "eng".
• catalog publisher "Cambridge, UK ; New York : Cambridge University Press,".
• catalog subject "515/.723 21".
• catalog subject "Asymptotic expansions.".
• catalog subject "Mellin transform.".
• catalog subject "QA432 .P34 2001".
• catalog tableOfContents "Order Relations -- Asymptotic Expansions -- Other Expansions -- Biographies of Mellin and Barnes -- Fundamental Results -- The Gamma Function [Gamma] (z) -- The Asymptotic Expansion of [Gamma] (z) -- The Stirling Coefficients -- Bounds for [Gamma] (z) -- Expansion of Quotients of Gamma Functions -- Inverse Factorial Expansions -- A Recursion Formula when [alpha subscript r] = [beta subscript r] -- An Algebraic Method for the Determination of the A[subscript j] -- Special Cases -- The Asymptotic Expansion of Integral Functions -- Convergence of Mellin-Barnes Integrals -- Order Estimates for Remainder Integrals -- Lemmas -- Properties of Mellin Transforms -- Basic Properties -- Translational and Differential Properties -- The Parseval Formula -- Analytic Properties -- Inverse Mellin Transforms -- Integrals Connected with e[superscript -z] -- Some Standard Integrals -- Discontinuous Integrals -- Gamma-Function Integrals -- Ramanujan-Type Integrals -- Barnes' Lemmas -- Mellin-Barnes Integral Representations -- The Confluent Hypergeometric Functions -- The Gauss Hypergeometric Function -- Some Special Functions -- Applications of Mellin Transforms -- Transformation of Series -- The Mellin Transform Method -- The Poisson-Jacobi Formula -- An Infinite Series -- A Smoothed Dirichlet Series -- A Finite Sum -- Number-Theoretic Examples -- A Harmonic Sum -- Euler's Product -- Ramanujan's Function -- Some Other Number-Theoretic Sums -- Solution of Differential Equations -- Potential Problems in Wedge-Shaped Regions.".
• catalog title "Asymptotics and Mellin-Barnes integrals / R.B. Paris, D. Kaminski.".
• catalog type "text".