On asymptotic expansions of oscillatory integrals with phase functions expressed by a product of positive real power function and real analytic function in one variable

TL;DR

This paper develops asymptotic expansion techniques for oscillatory integrals with phase functions that are products of positive real powers and real analytic functions, providing explicit coefficient computations in specific cases.

Contribution

It introduces a method to derive asymptotic expansions for a class of oscillatory integrals with complex phase functions involving products of powers and analytic functions, including explicit coefficient calculation.

Findings

Derived asymptotic expansions for the specified oscillatory integrals.

Provided a concrete example with explicit coefficient computation.

Extended existing methods to more complex phase functions.

Abstract

In this paper, by using asymptotic expansions of oscillatory integrals with positive real power phase functions in one variable, we obtain asymptotic expansions of oscillatory integrals with phase functions expressed by a product of positive real power function and real analytic function in one variable. Moreover we show an example which we can compute all coefficients of terms in asymptotic expansions concretely.