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 $\mathbb{ C}^{ 2}$ ($\tilde{ \omega}:=\bar{ \omega}$ 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 asymptotics expressions in terms of special functions are given.
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