Debye asymptotic for oscillatory Bessel functions ($J_n(n\sec \alpha )\approx [2/(\pi n\tan \alpha ){]}^{1/2}\cos [n(\tan \alpha -\alpha )-\pi /4]$)

For fixed $0<\alpha <\pi /2$, the Bessel function of the first kind has an interior nondegenerate stationary point in its cosine integral. The stationary phase method gives the displayed oscillatory leading term and an additive $O(n^{-3/2})$ remainder. The approximation loses uniformity near $\alpha =0$, where the stationary point reaches the endpoint and the curvature vanishes. For negative integer-order arguments, parity supplies a factor $(-1{)}^n$.