Stationary phase lemmas for general functions

TL;DR

This paper provides detailed proofs of stationary phase lemmas, including explicit high-order term derivations and a general case using Morse lemma, enhancing understanding of asymptotic analysis techniques.

Contribution

It introduces detailed proofs of stationary phase lemmas with explicit high-order terms and extends to a general case via Morse lemma, offering deeper insights.

Findings

Explicit derivation of high-order terms in stationary phase lemma

Extension of stationary phase lemma to general functions using Morse lemma

Enhanced understanding of asymptotic analysis methods

Abstract

We give some details about the stationary phase lemma. We first prove a special case where the high order terms are derived explicitly. Based on that, we prove a more general case by using Morse lemma.