I was wondering, if there is a general solution for integrals involving Bessel functions of the form:
$\int_0^\infty \frac{p(x)}{q(x)} BesselJ(0,x*r) dx $
where p(x) and q(x) are polynomals of order n and m, and BesselJ is the Bessel function of the first kind.
I have some solutions for special types of polynomes, but is there a general approach by partial fraction decomposition?
Thank you.
Michael
Such integral can be computed by first computing the inverse Laplace transform of $\frac{p(x)}{q(x)}$ through partial fraction decomposition, then exploiting the fact that: $$\mathcal{L}\left( J_0(rx)\right) = \frac{1}{\sqrt{r^2+s^2}}.$$ In such a way, the original integral can be expressed through values of the Bessel $Y$ and Struve $H$ functions.