Solution
= Solution
Substitute $u=nt$ and use the $\pi$-periodicity of $\cos(n\cos u)$ to get $J_n=\int_0^\pi\cos(n\cos u)du=\pi J_0(n)$. Stationary phase at $0,\pi$ gives
$$
J_n\sim\sqrt{\frac{2\pi}{n}}\cos(n-\pi/4).
$$
\b[Thus $A=\sqrt{2\pi/n}$, $\alpha=n-\pi/4$, and the next term is $O(n^{-3/2})$.]