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Integrals with a large parameter: a double complex integral with four nearly coincident saddle-points

Published online by Cambridge University Press:  24 October 2008

F. Ursell
F. Ursell
Affiliation:
Department of Mathematics, University of Manchester, Manchester M 13 9PL
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Abstract

The method of steepest descents for finding the asymptotic expansion of contour integrals of the form ∫ g(z) exp (Nf(z)) dz where N is a real parameter tending to + ∞ is familiar. As is well known, the principal contributions to the asymptotic expansion come from certain critical points; the most important are saddle-points where df/dz = 0. The original contour is deformed into an equivalent contour consisting of paths of steepest descent through certain saddle-points, the relevant saddle-points. The determination of these is a global problem which can be solved explicitly only in simple cases. The function f (z) may also depend on parameters. The position of the saddle-points depends on the parameters and at a certain set of values of the parameters it may happen that two or more saddle-points coincide. The ordinary expansion is then non-uniform, but appropriate uniform expansions have been shown to exist in earlier work.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 87 , Issue 2 , March 1980 , pp. 249 - 273
Copyright
Copyright © Cambridge Philosophical Society 1980

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