For $F(t)=\operatorname{sech}\alpha\sinh t-t$, the relevant <saddle point> is $t=\alpha$, where $F(\alpha)=\tanh\alpha-\alpha$ and $F''(\alpha)=\tanh\alpha$. The <method of steepest descent> gives
$$
\boxed{J_\nu(\nu\operatorname{sech}\alpha)
\sim\frac{\exp[\nu(\tanh\alpha-\alpha)]}
{\sqrt{2\pi\nu\tanh\alpha}}}
\qquad(\alpha>0).
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-336/1/c/i/solution