Higher-order asymptotic expansion with error estimate for the multidimensional Laplace-type integral under perturbations

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.

14 pages. It has been significantly revised from the first manuscript, and the title has also been changed