Skip to main content
    • Aa
    • Aa

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

  • F. Ursell (a1)

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.

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

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 9 *
Loading metrics...

Abstract views

Total abstract views: 107 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th October 2017. This data will be updated every 24 hours.