Uniform Asymptotic Methods for Integrals

TL;DR

This paper reviews fundamental asymptotic methods for integrals, including their modifications for uniform expansions and applications involving special functions like Airy and Bessel functions.

Contribution

It provides a comprehensive overview of classical and modern uniform asymptotic techniques, comparing them with De Bruijn's developments in the field.

Findings

Comparison of classical methods with De Bruijn's developments

Examples of uniform asymptotic expansions using special functions

Modified methods for expansions with additional parameters

Abstract

We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson's lemma, Laplace's method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn's book {\em Asymptotic Methods in Analysis}. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are used as approximations in uniform asymptotic expansions.