Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 30D / a / Solution

Write the contour near the simple saddle aswith its given orientation. Since ${\varphi }^{\prime }(z_0)=0$,Along a steepest-descent tangent the quadratic coefficient is real and negative, soMoreover $dz=e^{i\alpha }ds$. Only a neighbourhood of width $O(x^{-1/2})$ contributes at leading order, and there $f(z)=f(z_0)+o(1)$. The local integral is consequentlyEvaluating the Gaussian proves the simple-saddle contribution in steepest descent: