Higher-order asymptotic expansions for Laplace's integral and their error estimates

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.