Higher-order asymptotic expansions for Laplace's integral and their error estimates
Ikki Fukuda, Yoshiki Kagaya
TL;DR
This work advances Laplace's method by deriving higher-order asymptotic expansions and rigorous error estimates for $I(t)=\int_{a}^{b} e^{t h(x)} g(x)\,dx$ as $t\to\infty$, including cases where the maximum point $c$ of $h$ coincides with zeros of $g$. The main result provides explicit leading terms for interior and endpoint maxima under smoothness and vanishing conditions on $g$, with concrete $O$-remainder bounds. The authors prove these expansions via Taylor expansions and Gaussian scaling around the maximum, and they compare the resulting asymptotics to numerical integration, notably composite Simpson's rule, showing regimes where the asymptotics yield smaller errors for large $t$. The results generalize prior work, offer practical error control, and enhance applicability to areas such as large deviations and particle filtering where Laplace-type integrals arise.
Abstract
We deal with the asymptotic analysis for Laplace's integral. For this problem, the so-called Laplace's method by P.S. Laplace (1812) is well-known and it has been developed in various forms over many years of studies. In this paper, we derive some formulas of the higher-order asymptotic expansions for that integral, with error estimates, which generalize previous results. Moreover, we discuss a comparison of these asymptotic formulas with approximations using numerical integral.