A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: application to Pearcey's integral and its derivatives

Abstract

A numerical method for the evaluation of the cuspoid canonical integrals and their derivatives is described. The method exploits Cauchy's theorem and Jordan's lemma to write the infinite integration path along different contours in the complex plane. The method is straightforward to implement on a computer and in many cases results of high accuracy can be obtained using standard quadrature techniques. Application is made to Pearcey's integral P(x,y) and its two partial derivatives and the method is shown to have some significant advantages over other techniques that have been applied to this problem. Tables of P(x,y), delta P(x,y)/ delta x, delta P(x,y)/ delta y and the real zeros of P(x,y) are presented for the grid -8.0<or=x<or=8.0 and 0<or=y<or=8.0.