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

  • F. Ursell (a1)
Abstract
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.

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(1) M. V. Berry Waves and Thom's theorem. Advances in Physics 25 (1976). 126.

(2) M. V. Berry , J. F. Nye and F. J. Wright The elliptic umbilic diffraction catastrophe. Phil. Trans. Roy. Soc. London Ser. A291 (1979), 453484.

(5) K. G. Budden Radio caustics and cusps in the ionosphere. Proc. Roy. Soc. London, Ser. A350 (1976), 143164.

(6) J. N. L. Connor Evaluation of multidimensional canonical integrals in semiclassical collision theory. Molecular Phys. 26 (1973), 13711377.

(9) J. F. Nye Optical caustics in the near field from liquid drops. Proc. Roy. Soc. London, Ser. A361 (1978), 2141.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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