= Gaussian Cauchy transform and its Stokes jump
{title2=$I(z)=i\pi e^{-z^2}\operatorname{erfc}(iz)$}
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 $\sqrt\pi z^{-1}\sum_{m\geq0}(2m-1)!!/(2z^2)^m$. 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.
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