An extension of Laplace's method

TL;DR

This paper develops a rigorous asymptotic expansion method for a class of contour integrals, generalizing classical techniques like Laplace and Watson, with applications to special functions in transition regions.

Contribution

It extends Laplace's method to more general integrals involving complex parameters and provides a rigorous foundation for Dingle's earlier formal asymptotic results.

Findings

Unified asymptotic expansion framework for complex contour integrals

Rigorous justification of Dingle's formal methods

Applications demonstrated in special functions transition regions

Abstract

Asymptotic expansions are obtained for contour integrals of the form \[ \int_a^b \exp \left( - zp(t) + z^{\nu /\mu } r(t) \right)q(t)dt, \] in which $z$ is a large real or complex parameter, $p(t)$, $q(t)$ and $r(t)$ are analytic functions of $t$, and the positive constants $\mu$ and $\nu$ are related to the local behaviour of the functions $p(t)$ and $r(t)$ near the endpoint $a$. Our main theorem includes as special cases several important asymptotic methods for integrals such as those of Laplace, Watson, Erd\'elyi and Olver. Asymptotic expansions similar to ours were derived earlier by Dingle using formal, non-rigorous methods. The results of the paper also serve to place Dingle's investigations on a rigorous mathematical foundation. The new results have potential applications in the asymptotic theory of special functions in transition regions, and we illustrate this by two examples.