Zeroes of the Swallowtail Integral

TL;DR

This paper investigates the zeroes of the swallowtail integral, a key function in optics and asymptotic analysis, providing a detailed understanding of their structure beyond previous numerical observations.

Contribution

The paper offers a comprehensive analysis of the zeroes of the swallowtail integral, extending prior numerical findings with detailed structural insights.

Findings

Zeroes occur along specific lines in the parameter space.

Zeroes of S(0,y,z) are aligned along the line y=0.

The structure of zeroes is more intricate than previously observed.

Abstract

The swallowtail integral $S(x,y,z) = \int_{-\infty}^{\infty} \exp[i(u^5 + xu^3 + yu^2 + zu)] \, du$ is one of the so-called canonical diffraction integrals used in optics, and plays a role in the uniform asymptotics of integrals exhibiting a confluence of up to four saddle points. In a 1984 paper by Connor, Curtis and Farrelly, the authors make a number of remarkable observations regarding the zeroes of $S(x,y,z)$, including that its zeroes occur on lines in $xyz$-space, and that the zeroes of $S(0,y,z)$ lie along the line $y = 0$. These assertions are based on numerical evidence and the asymptotics of $S(0,0,z)$. We examine these assertions more completely and provide additional detail on the structure of the zeroes of $S(x,y,z)$.