Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 336 / 1 / c

At $\theta =1$, the two saddles have receded to infinity and the ordinary quadratic saddle approximation is nonuniform. One should first use an asymptotic expansion of the phase and amplitude for large $z$:soThe phase varies on the scale $z=O(\lambda )$ rather than in an $O({\lambda }^{-1/2})$ neighbourhood of a finite saddle. Rescaling $z=\lambda Z$ in the positive and negative tails, retaining the corresponding large-$z$ amplitude, and matching these tail integrals to the finite part of the deformed contour produces the leading approximation. This is an endpoint at infinity problem; a uniform calculation may equivalently begin from the $\theta <1$ saddle representation and take the coalescing-at-infinity limit.