= Debye asymptotic for oscillatory Bessel functions
{c}
{title2=$J_n(n\sec\alpha)\approx[2/(\pi n\tan\alpha)]^{1/2}\cos[n(\tan\alpha-\alpha)-\pi/4]$}
= Debye Bessel asymptotic
{c}
{synonym}
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 $(-1)^n$.
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