Stationary phase endpoint cancellation in a half-strip contour

For fixed $\nu$, the vertical sides of the half-strip integral of $e^{-ix\sin z+i\nu z}$ contribute $-\sin (\pi \nu )/(\pi x)+O(x^{-2})$. Integration by parts on the nonstationary horizontal pieces contributes the opposite endpoint term. After cancellation, the two stationary points give $\sqrt{2/(\pi x)}\cos (x-\nu \pi /2-\pi /4)+O(x^{-3/2})$.