The Pearcey function and the cusp catastrophe

Abstract

The subject of this work is a theoretical analysis of the Pearcey function. In optics, thin lens theory supposes that all rays focus at a unique point where the field converges. For a real lens, the focal point is replaced by a cusp, which is the end point of a caustic curve dividing the bright field region from the dark. My particular interest is the pattern of nodal points within the cusp. By investigating the stationary points for the cusp catastrophe, asymptotic approximations are found for the Pearcey function. This in turn leads to the development of finding the positions of nodal points inside, and outside a caustic. Also values for $|P|$ on a small circle surrounding a node are examined and show reasonable accuracy of order $10^{-8}$.