Zeroes of the Swallowtail Integral

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 $S(x,y,z)$.