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Unfolding the High Orders of Asymptotic Expansions with Coalescing Saddles: Singularity Theory, Crossover and Duality

M. V. Berry and C. J. Howls
Proceedings: Mathematical and Physical Sciences
Vol. 443, No. 1917 (Oct. 8, 1993), pp. 107-126
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/52382
Page Count: 20
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Unfolding the High Orders of Asymptotic Expansions with Coalescing Saddles: Singularity Theory, Crossover and Duality
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Abstract

We study the leading behaviour of the late coefficients (high orders r) of asymptotic expansions in a large parameter k, for contour integrals involving a cluster of coalescing saddles, and thereby establish the form of the divergence of the expansions. The two principal cases are: `saddle-to-cluster', where the integral is through a simple saddle and its expansion diverges because of a distant cluster; and `cluster-to-saddle', where the integral is through a cluster and its expansion diverges because of a distant simple saddle. In both, the large-r coefficients are dominated by the `factorial divided by power' familiar in asymptotics, but this changes its form as the saddles in the cluster are made to coalesce and separate by varying parameters A = {A1,A2....} in the integrand. The `crossover' between different forms is described by a series of canonical integrals, built from the cuspoid catastrophe polynomials of singularity theory that describe the geometry of the coalescence. The arguments of these integrals involve not only the A but also fractional powers of r, which by a curious duality replace the powers of the original large parameter k which occur in uniform approximations involving these integrals. A by-product of the cluster-to-saddle analysis is a new exact formula for the coefficients of uniform asymptotic expansions.

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