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Asymptotics: integrals and series. (Asimptotika: integraly i ryady). (Russian) Zbl 0641.41001
Spravochnaya Matematicheskaya Biblioteka. Moskva: “Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury. 544 p. R. 1.90 (1987).
Basic methods for evaluation of the asymptotic behaviour of integrals and series are exposed and applied to obtain recent results by these methods. Content: (I) Elementary asymptotic estimates, asymptotic series and potential series in particular. (II) Laplace method in one and several dimensions. (III) Method of stationary phase in one and several dimensions. Contributions to nondegenerate inner and boundary points. Degnerate stationary points. (IV) Saddle-point method in one dimension. Existence theorems. The asymptotic behaviour of Laplace, Fourier and Mellin transforms. Saddle-point at infinity. (V) Saddle-point method in several dimensions. Saddle-points of polynomial and algebraic functions. Existence theorems. Asymptotic behaviour of fundamental solutions of equations correct in the sense of Petrovskij. Stability in C of the Cauchy problem for difference equations and equations with partial derivatives. (VI) Confluence of two or more saddle-points and confluence of a saddle-point and a pole. - In this reference book the basic asymptotic formulae are derived and many concrete applications are considered in detail.
Reviewer: S.Aljančić

MSC:
41-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to approximations and expansions
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
30E15 Asymptotic representations in the complex plane
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type