Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 30D / c

Deform the real ray upward to the saddle line, joining it near the endpoint and at a large right cutoff. The integrand is entire. The far bridge vanishes by Gaussian decay; the endpoint bridge is $O(x^{-1}{e}^{-3x/2})$ because $\Re \varphi (-1)=-3/2$.