Uniform asymptotic expansions for generalised trigonometric integrals and their zeros

TL;DR

This paper develops uniform asymptotic expansions for generalized trigonometric integrals and their zeros, using Liouville-Green methods for incomplete gamma functions, applicable for large parameters and complex arguments.

Contribution

It introduces new Liouville-Green asymptotic expansions for incomplete gamma functions and applies them to derive uniform asymptotics for zeros of generalized trigonometric integrals.

Findings

Asymptotic expansions valid for large parameter a

Uniform approximations for zeros of integrals

Extensions to complex argument values

Abstract

Asymptotic expansions for generalised trigonometric integrals are obtained in terms of elementary functions, which are valid for large values of the parameter $a$ and unbounded complex values of the argument. These follow from new Liouville-Green asymptotic expansions for incomplete gamma functions. Asymptotic expansions for the real zeros of the generalised trigonometric integrals are then constructed for large $a$ which are uniformly valid without restriction on their size (small or large).