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I'm trying to make headway on the following ghastly integral:

[tex] \int_0^{\infty} x^{\frac{3}{2}}e^{-xd} J_o(rx) \frac{\sin (\gamma \sqrt{x}\sqrt{x^2+\alpha^2}t)}{\sqrt{x^2+\alpha^2}}\ dx [/tex]

where [itex] d,r, \alpha, \gamma ,t \in \mathbb{R}^+[/itex] and [itex]J_o[/itex] is the zeroth order Bessel function of the first kind. Normally I wouldn't think I'd have a shot at finding this in closed form, but because there is a (removable) singularity, maybe there's a chance to exploit some complex analysis.

My attempts so far have all focused on making a branch cut along the positive real axis and integrating around a modified keyhole, which goes along the real axis, then follows an arc to the imaginary axis, going around the singularities, before arcing back to the real axis. This has not led to anything productive. It reminds of the contour used in a Bromwich integral.

Any suggestions? They'd be greatly appreciated.

Cheers,

Nick