Stationary phase endpoint cancellation in a half-strip contour
= 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 $\sqrt{2/(\pi x)}\cos(x-\nu\pi/2-\pi/4)+O(x^{-3/2})$.