Evaluation of the Field near a Cusp of a Caustic

Abstract

The field in the neighbourhood of a cusp of a caustic can be expressed in terms of a function of two variables, P(X,Y), known as Pearcey's integral or the Pearcey function. In this paper we develop efficient algorithms for computing this function and its derivatives P/ X and P/ Y. They are based on a Taylor series expansion in a region close to the cusp at X = Y = 0, and on asymptotic approximations in regions far from the cusp. The asymptotic results are given in terms of contributions of isolated stationary points in regions far from the caustic, and in terms of the Airy integral function and its derivative in a region close to the caustic. Three terms in the asymptotic expansions are used, and they give an accuracy for P(X,Y) better than 0·006 in amplitude and 0·6° in phase at distances from the cusp greater than (X² +Y² )^1/2 = 4.