I need help with the following integral:
$$ \int_{0}^{\infty} \frac{J_0(ax)xe^{-bx^2}}{1-cx^2}dx $$
Where $ J_0(x) $ is a Bessel function of the first kind (of zero order).
I've looked up a few books containing tables of integrals and can't find this exact one.
I tried using contour integration over a semi-circle with $ f(z) = \frac{J_0(az)ze^{-bz^2}}{1-cz^2} $ but because it's an odd function of z then the principle part of the integration disappears and I don't know what to do.
Any help will be appreciated!