Stationary phase with Cauchy singularity. A critical point of signature $(+,-)$

Abstract

Asymptotic expressions for an integral appearing in the solution of a d-bar problem are presented. The integral is a solid Cauchy transform of a function with a rapidly oscillating phase with a small parameter $h$, $0<h\ll 1$. Whereas standard steepest descent approaches can be applied to the case where the stationary points of the phase ${\omega }_k$, $k=1,\dots ,N$ are far from the singularity $\zeta$ of the integrand, a polarization approach is proposed for the case that $|\zeta -{\omega }_k|<\mathcal{O}(\sqrt{h})$ for some $k$. In this case the problem is studied in ${\mathbb{C}}^2$ ($\tilde{\omega }:=\overline{\omega }$ is treated as an independent variable) on steepest descent contours. An application of Stokes' theorem allows for a decomposition of the integral into three terms for which asymptotic expressions in terms of special functions are given.

Minor changes. To appear in Journal of Pseudo-Differential Operators and Applications