Pearcey integral

In mathematics, the Pearcey integral is defined as^[1]

${\displaystyle \operatorname {Pe} (x,y)=\int _{-\infty }^{\infty }\exp(i(t^{4}+xt^{2}+yt))\,dt.}$

A plot of the absolute value of the Pearcey integral as a function of its two parameters.

A photograph of a cusp caustic produced by illuminating a flat surface with a laser beam through a droplet of water.

The Pearcey integral is a class of canonical diffraction integrals, often used in wave propagation and optical diffraction problems^[2] The first numerical evaluation of this integral was performed by Trevor Pearcey using the quadrature formula.^[3][4]

Reflective caustic generated from a circle and parallel rays. On one side, each point is contained in three light rays; on the other side, each point is contained in one light ray.

In optics, the Pearcey integral can be used to model diffraction effects at a cusp caustic, which corresponds to the boundary between two regions of geometric optics: on one side, each point is contained in three light rays; on the other side, each point is contained in one light ray.