标题： 关于多重Laplace型积分在扰动下的高阶渐近展开及其误差估计 显示英文标题

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

摘要： 我们研究了带受扰相位函数的多维Laplace型积分的渐近性态。在适当的假设下，我们推导出了一个带有误差估计的高阶渐近展开式，推广了一些先前的结果包括Laplace方法。证明的关键点在于基于大量详细的Taylor展开式的精确渐近分析，以及仔细考虑受扰对相位函数的Hessian矩阵的影响。 显示英文摘要

摘要： 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.