Unfolding the high orders of asymptotic expansions with coalescing saddles: singularity theory, crossover and duality

Abstract

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.