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

Let $L=\log (1/ϵ)$. Taking logarithms gives $x^2+\log x=L$. Equivalently, $2x^2=W(2/{ϵ}^2)$ in terms of the Lambert W function. Iteration for large $L$ gives $x^2=L-\frac{1}{2}\log L+O((\log L)/L)$ and hence

Write $e^x=1+xg(x)$, where $g=(e^x-1)/x$ is smooth at zero. Thenand a uniformly integrable expansion givesBy integration by parts,Therefore

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 $2^{1/3}\mathrm{Ai}(0)$.