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> gives
$$
\boxed{J_\nu(\nu)
\sim\frac{2^{1/3}}{3^{2/3}\Gamma(2/3)}\nu^{-1/3}.}
$$
Equivalently, the coefficient is $2^{1/3}\operatorname{Ai}(0)$.

Solution

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

When two quadratic saddles coalesce and the first nonzero phase derivative is cubic, the contributing neighborhood has width $O(\lambda^{-1/3})$ rather than $O(\lambda^{-1/2})$. The leading contour integral reduces to a Gamma-function or Airy-function constant.


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