# Integral of Bessel function with Gaussian over a quadratic

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!

• The integral does not converge for $c\in \mathbb{R}$. – Ron Gordon Nov 19 '14 at 18:34
• You see from @RonGordon's answer that for $c>0$ it is easy to "evaluate" the integral. It might be good if you could provide some context (where does this problem come from) and tell us what values you want the parameters $a,b,c$ to obtain. – Fabian Nov 19 '14 at 18:42
• It comes from parameterizing ocean eddies in an abstract sense so it's origins are not helpful. The constants are not important either to be honest, I care more about a functional form of the solution. – TomBolton Nov 19 '14 at 18:55
• @TomBolton: the constants seem to be important as for $c>0$ the integral simply diverges. It also diverges for all $\text{Re} \,b <0$. So unless you tell us where in the complex plane your parameters are it will be hard to help you. – Fabian Nov 19 '14 at 18:58
• @Fabian: Ah I see what you mean now. All the constants a,b,c are real and positive. – TomBolton Nov 19 '14 at 19:01