In an inverse Fourier transformation, I have to deal with the following integal over the wavenumber $k$: $$ I = \int_0^\infty \exp \left( ik\cos\theta \cos\theta_0 \right) \, J_0 \left( k \sin\theta \sin\theta_0 \right) \, \mathrm{d}k \, , $$ wherein $\theta$ and $\theta_0$ are both real numbers between 0 and $\pi$ (polar angles in spherical coordinates). Maple gives: $$ I = \frac{1}{\sqrt{\sin^2\theta_0 - \cos^2\theta}} \, . $$ Clearly, the latter is really defined only for $\cos\theta < \sin\theta_0$. A subsequent integration has to be performed over $\theta$ from 0 to $\pi$ and the resulting value MUST be really valued (physical quantity).
My question is whether the result given above is valid only under some conditions that Maple overlooked.
Any help will be highly rated and appreciated.
Thanks
Fede