Gaussian Cauchy transform and its Stokes jump ($I(z)=i\pi e^{-z^2}\mathrm{erfc}(iz)$)

The integral ${\int }_{\mathbb{R}}{e}^{-t^2}(z-t{)}^{-1}dt$ in the lower half-plane extends to the displayed entire function. In the upper half-plane the continued function differs from the directly evaluated real-contour integral by $2\pi i{e}^{-z^2}$. Its algebraic expansion is $\sqrt{\pi }{z}^{-1}{\sum }_{m\ge 0}(2m-1)!!/(2z^2{)}^m$. The real axes are switching lines for the exponentially small term under the Stokes-multiplier convention; the diagonal rays $\arg z=\pi /4+j\pi /2$ are equal-magnitude boundaries for the exponential and algebraic mechanisms. Stating both the residue jump and the sector domains prevents confusion between switching and dominance conventions.