 Imaginary-argument gamma asymptotic

Rotating the integral defining the Gamma function toward the positive imaginary axis introduces the factor $i{e}^{-\pi n/2}$. 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.