Solution
= Solution
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 $2^{1/3}\operatorname{Ai}(0)$.