The Pearcey integral in the highly oscillatory region

TL;DR

This paper develops a simplified asymptotic expansion for the Pearcey integral in highly oscillatory regions using a modified saddle point method, providing explicit formulas and numerical validation.

Contribution

It introduces a modified saddle point method to derive complete asymptotic expansions of the Pearcey integral for large bs y, simplifying analysis in oscillatory regions.

Findings

Asymptotic expansion in three e2b0 y regions

Expansion in inverse powers of y^{2/3} with elementary coefficients

Numerical experiments confirm the approximation accuracy

Abstract

We consider the Pearcey integral $P(x,y)$ for large values of $\vert y\vert$ and bounded values of $\vert x\vert$. The integrand of the Pearcey integral oscillates wildly in this region and the asymptotic saddle point analysis is complicated. Then we consider here the modified saddle point method introduced in [Lopez, P\'erez and Pagola, 2009]. With this method, the analysis is simpler and it is possible to derive a complete asymptotic expansion of $P(x,y)$ for large $\vert y\vert$. The asymptotic analysis requires the study of three different regions for $\arg y$ separately. In the three regions, the expansion is given in terms of inverse powers of $y^{2/3}$ and the coefficients are elementary functions of $x$. The accuracy of the approximation is illustrated with some numerical experiments.