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

2025/04/02 by Fukuda, Ikki, Kagaya, Yoshiki, Ueda, Yuki
#41A60 #41A80 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2504.01310

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.

Related

• Asymptotic Expansion of Laplace-Fourier-Type Integrals
• The Laplace asymptotic expansion in high dimensions
• The asymptotic behavior of degenerate oscillatory integrals in two dimensions
• Weighted stationary phase of higher orders
• Asymptotic expansions of oscillatory integrals with complex phase