= Gaussian saddle coalescing with a simple pole
{title2=$t=1+u/\sqrt n$}
For $z=2n+i\mu\sqrt n+O(1)$, the factor $t^{2n}e^{z(1-t)}$ has saddle scale $t=1+u/\sqrt n$ and becomes $e^{-u^2-i\mu u}$. If the amplitude has a simple pole there, its leading local integral contains $e^{-u^2-i\mu u}/u$. Differentiating its <Cauchy principal value> with respect to $\mu$ reduces it to a <Gaussian integral>, producing an <error function>. The nonsingular amplitude $(1+t)^{-1}$ gives $\frac12\sqrt{\pi/n}e^{-\mu^2/4}$.
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