Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 336 / 1 / c

For $F(t)=\mathrm{sech}\alpha \sinh t-t$, the relevant saddle point is $t=\alpha$, where $F(\alpha )=\tanh \alpha -\alpha$ and $F^{\prime \prime }(\alpha )=\tanh \alpha$. The method of steepest descent gives

At $\alpha =0$, the ordinary saddles coalesce at zero and $\sinh t-t=t^3/6+O(t^5)$. The contributing scale is $t=O({\nu }^{-1/3})$. The standard cubic saddle-point approximation givesEquivalently, the coefficient is $2^{1/3}\mathrm{Ai}(0)$.