solution for integral $\int_0^{\infty} \frac{k}{k^3-a}J_0\left(k \, r\right) dk $ involving Bessel function (Hankel transform)

Question

during my research I am facing for the first time integrals involving Bessel functions. In particular i need to evaluate the following integral:

$\int_0^{\infty} \frac{k}{k^3-a}J_0\left(k \, r\right) dk $

with $a$ and $r$ being two real positive numbers. $J_0$ is Bessel function of the first kind and order zero.

I know this can be seen as an Hankel transform of the function $\frac{1}{k^3-a}$ however I was not able to find reference for this transformation. Maybe it is a known one.

Unfortunately Mathematica is not helping in finding the solution to the problem. Any help or hint is appreciated.

Maybe a way of solving this could be using a complex decomposition of the fraction?

Answer 1

We may go through the Laplace transform and partial fraction decomposition: $$\mathcal{L}\left(J_0(kr)\right) = \frac{1}{\sqrt{r^2+s^2}},\qquad \mathcal{L}^{-1}\left(\frac{1}{k+b}\right)=e^{-bs}\tag{1}$$ leads to: $$ \int_{0}^{+\infty}\frac{J_0(kr)}{k+b}\,dk = \int_{0}^{+\infty}\frac{e^{-bs}}{\sqrt{r^2+s^2}}\,ds \tag{2}$$ where the RHS of $(2)$ is a multiple of the difference between a Bessel $Y_0$ and a Struve $H_0$ function.