Oscillatory integrals with phase functions of positive real powers and asymptotic expansions

TL;DR

This paper extends the stationary phase method to degenerate oscillatory integrals with phase functions of positive real powers, providing new asymptotic expansion techniques without resolution of singularities.

Contribution

It generalizes Fresnel integrals and develops asymptotic expansions for degenerate phases with positive real powers, including multivariable cases with specific singularity types.

Findings

Derived asymptotic expansions for oscillatory integrals with positive real power phases.

Extended stationary phase method to degenerate phases without resolution of singularities.

Applied results to multivariable phases with specific singularity types.

Abstract

As to methods for expanding an oscillatory integral into an asymptotic series with respect to the parameter, the method of stationary phase for the non-degenerate phases and the method of using resolution of singularities for degenerate phases are known. The aim of this paper is to extend the former for degenerate phases with positive real powers without using resolution of singularities. For this aim, we first generalize Fresnel integrals as oscillatory integrals with phase functions of positive real powers. Next, by using this result, we have asymptotic expansions of oscillatory integrals for degenerate phases with positive real powers including moderate oscillations and for a wider amplitude class in one variable. Moreover, we obtain asymptotic expansions of oscillatory integrals for degenerate phases consisting of sums of monomials in each variable including the types $A_{k}$, $E_6$, $E_8$ in multivariable.