= Gaussian pole transition for a large gamma parameter
{c}
{title2=$I(\lambda,\gamma)\sim i\pi[1+\operatorname{erf}(\mu/\sqrt2)]$}
For the upper-pole integral $I=\int_{\mathcal C_+}t^{\gamma-1}e^{\lambda(1-t)}(1-t)^{-1}dt$, let $\gamma\to\infty$ and $\lambda=\gamma+i\mu\sqrt\gamma+O(1)$. Scaling $t=1+s/\sqrt\gamma$ gives $-\int_{\mathcal C_+}e^{-s^2/2-i\mu s}ds/s$ at leading order. At $\mu=0$ its indentation is $i\pi$. Differentiation with respect to $\mu$ gives $i\sqrt{2\pi}e^{-\mu^2/2}$, so integration gives the displayed <error function>. This is an <optimal truncation> transition; the underlying analytic integral has no literal discontinuity. The general mechanism is discussed in https://dlmf.nist.gov/2.11#iv[the DLMF account of Stokes smoothing].
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