= Imaginary-argument gamma asymptotic
{title2=$\Gamma(1+in)\sim\sqrt{2\pi n}e^{-\pi n/2}e^{i(n\log n-n+\pi/4)}$}
Rotating the integral defining the <Gamma function> toward the positive imaginary axis introduces the factor $ie^{-\pi n/2}$. After scaling by $n$, the remaining phase is $\log s-s$, whose unique stationary point is $s=1$ with second derivative $-1$. The <stationary phase method> gives $e^{-in-i\pi/4}\sqrt{2\pi/n}$ for this scaled integral, producing the displayed asymptotic. The boundary contour integral must be Abel-regularized; its off-saddle tails are smaller by integration by parts.
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