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 $\omega_{k}$, $k=1,\ldots, N$ 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 $\mathbb{C}^{2}$ ($\widetilde{\omega}:=\overline{\omega}$ 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.