 Stationary-phase asymptotic for the generalized Bessel cosine integral

ID: stationary-phase-asymptotic-for-the-generalized-bessel-cosine-integral

For

Q_{α} (x) = \frac{1}{π} \int_{0}^{π} cos (x sin θ - α θ) d θ,

fixed

α

and

x \to + \infty

π /2

gives

Q_{α} (x) \sim \frac{2}{π x} cos (x - \frac{π α}{2} - \frac{π}{4}) .

When

α = x

, the stationary point coalesces cubically with the endpoint at zero and the leading order changes to

x^{- 1/3}

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