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

*(English)* Zbl 0803.58008
Summary: 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 $A = \{A\sb 1,A\sb 2,\dots\}$ 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.

##### MSC:

58C35 | Integration on manifolds; measures on manifolds |

58K35 | Catastrophe theory |