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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
Affiliation:
Department of Mathematics, University of Manchester, Manchester M 13 9PL

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
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Berry, M. V.Waves and Thom's theorem. Advances in Physics 25 (1976). 126.Google Scholar
(2)Berry, M. V., Nye, J. F. and Wright, F. J.The elliptic umbilic diffraction catastrophe. Phil. Trans. Roy. Soc. London Ser. A 291 (1979), 453484.Google Scholar
(3)Bleistein, N.Uniform asymptotic expansions of integrals with many nearly stationary points and algebraic singularities. J. Math. Mech. 17 (1967), 533560.Google Scholar
(4)Bleistein, N. and Handelsman, R. A.Asymptotic expansions of integrals (New York, Holt, Rinehart & Winston, 1975).Google Scholar
(5)Budden, K. G.Radio caustics and cusps in the ionosphere. Proc. Roy. Soc. London, Ser. A 350 (1976), 143164.Google Scholar
(6)Connor, J. N. L.Evaluation of multidimensional canonical integrals in semiclassical collision theory. Molecular Phys. 26 (1973), 13711377.Google Scholar
(7)Gunning, R. G. and Rossi, H.Analytic functions of several complex variables (Englewood Cliffs, N. J., Prentice-Hall, 1965).Google Scholar
(8)Jeffreys, H.Asymptotic approximations (Oxford University Press, 1962).Google Scholar
(9)Nye, J. F.Optical caustics in the near field from liquid drops. Proc. Roy. Soc. London, Ser. A 361 (1978), 2141.Google Scholar
(10)Ursell, F.Integrals with a large parameter: paths of descent and conformal mapping. Proc. Cambridge Philos. Soc. 67 (1970), 371381.Google Scholar
(11)Ursell, F.Integrals with a large parameter. Several nearly coincident saddle-points. Proc. Cambridge Philos. Soc. 72 (1972), 4965.Google Scholar