= Gaussian pole integral
{c}
{title2=$G(a)=\int_{-\infty}^{\infty}e^{-s^2/4}/(s-a)\,ds$}
For a pole off the real axis, substitution $s=2t$ into the <Faddeeva function> integral gives $G(a)=i\pi w(a/2)$ when $\operatorname{Im}a>0$, and $G(a)=-i\pi w(-a/2)$ when $\operatorname{Im}a<0$. The difference of boundary values is the <residue theorem> jump $2\pi i e^{-a^2/4}$. In particular, for a lower-half-plane pole, $G(a)+2\pi i e^{-a^2/4}=i\pi w(a/2)$, which packages a crossed residue and a nearby saddle into one smooth expression.
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