An explicit formula for the coefficients in Laplace's method

TL;DR

This paper derives an explicit formula for the coefficients in Laplace's method's asymptotic expansion, simplifying their computation using potential polynomials and classical theorems.

Contribution

It introduces a new explicit representation for the coefficients in Laplace's method, utilizing potential polynomials and classical mathematical tools.

Findings

Provides explicit formulas for asymptotic coefficients

Demonstrates the formulas with gamma function expansions

Simplifies computation of asymptotic series coefficients

Abstract

Laplace's method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion, arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron's formula gives them in terms of derivatives of an explicit function; Campbell, Fr\"oman and Walles simplified Perron's method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Fr\"oman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients, which contains ordinary potential polynomials. The proof is based on Perron's formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.