Higher-order asymptotic expansion with error estimate for the multidimensional Laplace-type integral under perturbations
Classical Analysis and ODEs
2025-12-11 v3 Probability
Abstract
We consider the asymptotic behavior of the multidimensional Laplace-type integral with a perturbed phase function. Under suitable assumptions, we derive a higher-order asymptotic expansion with an error estimate, generalizing some previous results including Laplace's method. The key points of the proof are a precise asymptotic analysis based on a lot of detailed Taylor expansions, and a careful consideration of the effects of the perturbations on the Hessian matrix of the phase function.
Cite
@article{arxiv.2504.01310,
title = {Higher-order asymptotic expansion with error estimate for the multidimensional Laplace-type integral under perturbations},
author = {Ikki Fukuda and Yoshiki Kagaya and Yuki Ueda},
journal= {arXiv preprint arXiv:2504.01310},
year = {2025}
}
Comments
14 pages. It has been significantly revised from the first manuscript, and the title has also been changed