Zeroes of the Swallowtail Integral

Abstract

The swallowtail integral $S(x,y,z)={\int }_{-\infty }^{\infty }\exp [i(u^5+x{u}^3+y{u}^2+zu)]du$ 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 $S(x,y,z)$.

10 pages, 2 figures

References in corpus (1)

• Numerical Evaluation of Cuspoid and Bessoid Oscillating Integrals for Applications in Chemical Physics