Weighted stationary phase of higher orders

TL;DR

This paper develops a higher-order stationary phase method for oscillatory integrals with a single stationary point, providing precise asymptotic expansions that improve classical results and have applications in analysis and number theory.

Contribution

It introduces an $n$th-order first derivative test for oscillatory integrals, extending classical results and offering sharper asymptotic expansions for arbitrary $n\geq1$.

Findings

Established an $n$th-order asymptotic expansion for weighted stationary phase integrals.

Sharpened the classical $n=1$ result by Huxley.

Potential applications in analysis and analytic number theory.

Abstract

An $n$th-order first derivative test for oscillatoric integrals is established. When the phase has a single stationary point, an $n$th-order asymptotic expansion of a weighted stationary phase integral is proved for arbitrary $n\geq1$. This asymptotic expansion sharpened the classical result for $n=1$ by Huxley. Possible applications include analysis and analytic number theory.