Analytic formulas for the evaluation of the Pearcey integral

TL;DR

This paper develops new convergent and asymptotic formulas for the Pearcey integral, enabling accurate evaluation across large regions of the complex plane with simpler expressions than existing methods.

Contribution

It introduces novel, simpler expansions for the Pearcey integral that extend the evaluation range and improve computational efficiency.

Findings

New convergent and asymptotic expansions derived.

Expanded the region where the Pearcey integral can be accurately evaluated.

Numerical experiments confirm the accuracy of the new formulas.

Abstract

We can find in the literature several convergent and/or asymptotic expansions of the Pearcey integral $P(x,y)$ in different regions of the complex variables $x$ and $y$, but they do not cover the whole complex $x$ and $y$ planes. The purpose of this paper is to complete this analysis giving new convergent and/or asymptotic expansions that, together with the known ones, let the evaluation of the Pearcey integral in a large region of the complex $x$ and $y$ planes. The accuracy of the approximations derived in this paper is illustrated with some numerical experiments. Moreover, the expansions derived here are simpler compared with other known expansions, as they are derived from a simple manipulation of the integral definition of $P(x,y)$.