I have a pretty ferocious integral to solve, and would be over the moon if I were able to get some sort of analytic expression / insight for it.
$$ I = \int_{r}^{\infty} r_0^{-5/2} W_{-i\alpha'/2, \nu}\left(\frac{-ir_0^2\omega'}{L}\right) W_{-i\alpha/2, \nu}\left(\frac{-ir_0^2\omega}{L}\right) (i^\beta J_{\beta}\left(-i\hat{k}r_0\right)) dr_0$$
All I've really done so far is look at the integrand in Mathematica (plotting Abs(integrand)):
Where $W$ is the $W$ Whittaker function (http://mathworld.wolfram.com/WhittakerFunction.html) and $J$ is the Bessel function of the first kind (http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html )
And also expand the integrand for small $r_0$ to gain some insight as to what happens at small $r$, but this is far too imprecise for the application (particular solution of a differential equation using Sturm-Liouville theory). For the application, the best thing I could do is write $I = I_0 + f(r)$ where $I_0$ is some analytically determined constant. r-dependence is the first datum of importance, and then $\hat{k},\omega',\alpha, \beta, \omega, \alpha'$ dependence. Note that these constants should be positive.
If it helps, note that $\omega' = \omega + \hat{\omega}$ and $\alpha = k^2L/2\omega, \alpha' = (k+\hat{k})^2 L /(2 \omega')$