Stationary-phase asymptotic for the generalized Bessel cosine integral
= Stationary-phase asymptotic for the generalized Bessel cosine integral
{title2=$Q_\alpha(x)$}
For
$$
Q_\alpha(x)=\frac1\pi\int_0^\pi
\cos(x\sin\theta-\alpha\theta)d\theta,
$$
fixed $\alpha$ and $x\to+\infty$, the stationary point near $\pi/2$ gives
$$
Q_\alpha(x)\sim\sqrt{\frac2{\pi x}}
\cos\left(x-\frac{\pi\alpha}{2}-\frac\pi4\right).
$$
When $\alpha=x$, the stationary point coalesces cubically with the endpoint at zero and the leading order changes to $x^{-1/3}$.