The Pearcey integral in the highly oscillatory region II

Abstract

We consider the Pearcey integral $P(x,y)$ for large values of $\left|x\right|$ and bounded values of $\left|y\right|$. The standard saddle point analysis is difficult to apply because the Pearcey integral is highly oscillating in this region. To overcome this problem we use the modified saddle point method introduced in López et al. (2009). A complete asymptotic analysis is possible with this method, and we derive a complete asymptotic expansion of $P(x,y)$ for large $\left|x\right|$, accompanied by the exact location of the Stokes lines. There are two Stokes lines that divide the complex $x-$plane in two different sectors in which $P(x,y)$ behaves differently when $\left|x\right|$ is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of $x$ and $y$. Both of them are of Poincaré type; one of them is given in terms of inverse powers of $x$; the other one in terms of inverse powers of $x^{1/2}$, and it is multiplied by an exponential factor that behaves differently in the two mentioned sectors. Some numerical experiments illustrate the accuracy of the approximation.