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Zbl 0764.30031
Berry, M.V.; Howls, C.J.
Hyperasymptotics for integrals with saddles. (English)
[J] Proc. R. Soc. Lond., Ser. A 434, No.1892, 657-675 (1991). ISSN 0080-4630

Integrals of the type $\int\sb{{\cal C}\sb n} g(z) \exp(-k f(z)) dz$ are considered, where $k$ is a large parameter. It is assumed that $f$ has several first order saddle points (simple zeros of $f'(z)$). ${{\cal C}\sb n}$ is an infinite contour through the saddle point $z\sb n$. With the aid of the principle of resurgence, inspired by the works of Dingle and Écalle, the divergence of the asymptotic expansions is interpreted in terms of the influence of other saddle points. Hyperasymptotic methods are used in order to obtain an improved asymptotic expansion. The term `hyperasymptotics' is introduced in an earlier paper by {\it M. V. Berry} and {\it C. J. Howls} [ibid. 430, No. 1880, 653-668 (1990; Zbl 0745.34052)]\ used for functions defined by differential equations.
[N.M.Temme (Amsterdam)]

MSC 2000:: *30E15 Asymptotic representations in the complex domain
41A60 Asymptotic problems in approximation

Keywords: saddle points; principle of resurgence; hyperasymptotics

Citations: Zbl 0745.34052

Cited in: Zbl 1039.33500 Zbl 0973.33002 Zbl 0953.65014 Zbl 0929.34045 Zbl 0829.33001 Zbl 0839.65021 Zbl 0773.30040

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