Globally exact asymptotics for integrals with arbitrary order saddles

TL;DR

This paper develops exact, rigorous remainder terms for asymptotic expansions around arbitrary order saddles, leading to sharper bounds and a hyperasymptotic theory for improved accuracy in complex integral analysis.

Contribution

It introduces the first practical, globally valid remainder terms for arbitrary order saddle asymptotics and develops a comprehensive hyperasymptotic framework for enhanced precision.

Findings

Sharper asymptotic bounds for truncated expansions

Development of a hyperasymptotic theory for arbitrary saddles

Efficient evaluation methods for generalized hyperterminants

Abstract

We derive the first exact, rigorous but practical, globally valid remainder terms for asymptotic expansions about saddles and contour endpoints of arbitrary order degeneracy derived from the method of steepest descents. The exact remainder terms lead naturally to sharper novel asymptotic bounds for truncated expansions that are a significant improvement over the previous best existing bounds for quadratic saddles derived two decades ago. We also develop a comprehensive hyperasymptotic theory, whereby the remainder terms are iteratively re-expanded about adjacent saddle points to achieve better-than-exponential accuracy. By necessity of the degeneracy, the form of the hyperasymptotic expansions are more complicated than in the case of quadratic endpoints and saddles, and require generalisations of the hyperterminants derived in those cases. However we provide efficient methods to evaluate them, and we remove all possible ambiguities in their definition. We illustrate this approach for three different examples, providing all the necessary information for the practical implementation of the method.