Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 1 / b / Solution

The exponential phase isIts saddle points satisfyFor positive real $\lambda$, the original positive-real contour passes through $s=\lambda$; the singularity at $s=0$ separates it from the other saddle. At the contributing saddle,Set $s=\lambda +{\lambda }^{1/2}t$. ThenThe method of steepest descent therefore gives