Error-function smoothing of a Stokes multiplier ($\mathrm{erfc}(\mu /2)$)

The pole contour in a Gaussian saddle coalescing with a simple pole changes the multiplier smoothly. For the lower contour, ${\int }_{{\mathcal{C}}_{-}}{e}^{-u^2-i\mu u}du/u=i\pi \mathrm{erfc}(\mu /2)$. This follows by integrating $-i\sqrt{\pi }{e}^{-{\mu }^2/4}$ and adding the indentation contribution $i\pi$. The transition width is inverse square root of the large parameter; on a Stokes line the multiplier takes half its switched-on value.