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Asymptotic approximations of integrals. (English) Zbl 0679.41001
Computer Science and Scientific Computing. Boston, MA etc.: Academic Press, Inc. xiii, 543 p. $ 69.95 (1989).
The book focuses on the classical techniques-Laplace method, Perron method, the method of steepest decents, Darboux method (Chapters I and II) as well as the recent techniques, namely, the Mellin transform, the Hankel transform, the Stieltjes transform and the Hilbert transform (Chapters III, IV and VI) in the asymptotic evaluation of integrals. Chapter V is devoted to theory of distributions. Chapter VII deals with integrals which depend on auxiliary parameters in addition to the asymptotic variable. Multidimensional integrals are studied in Chapters VIII and IX. The following exercises, among many, indicate the scope of the book: (i) show that \(e^ x \cos (e^ x)\) is a tempered distribution. (ii) Construct an asymptotic expansion for the integral \(\int^{\infty}_{0}t^ n(\log t)^ m(e^{-t}/t+x)dt\) as \(x\to 0^+\), where n and m are nonnegative integers. (iii) Show that, if A is a real, symmetric and positive definite matrix, \(\int_{{\mathbb{R}}^ n}\exp (x^ TAx)dx=\pi^{n/2}/(\det A)^{1/2}.\) The get-up is nice. The book is highly recommended for students, and researchers, whose interests impinge on asymptotic approximation of integrals, as it is expertly written. Supplementary Notes, Bibliography, Symbol Index and Subject Index are given to help the reader. The purpose of the book is to provide an up-to-date account of methods used in asymptotic approximation of integrals.

41-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to approximations and expansions
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
44A15 Special integral transforms (Legendre, Hilbert, etc.)