I have a double integral I am wishing to solve for finding the strength of the van der Waals potential at some distance from a surface. I haven't done any integration in many many years now so I am feeling a bit stupid and stuck on how to actually go about it. The integral specifics aren't important but it is as follows:
$V(r)=\frac{\hbar}{4\pi^3\epsilon_0}\sum_{-\infty}^{\infty}\int_0^\infty dk\left[k^2K_n^{'2}(kr) + (k^2 + n^2/r^2)K_n^2(kr)\right]\times\int_0^\infty d\xi\alpha(i\xi)G_n(i\xi)$
where
$G_n(\omega) = \frac{[\epsilon(\omega)-\epsilon_0]I_n(ka)I_n^{'}(ka)}{\epsilon_0I_n(ka)K_n^{'}(ka) - \epsilon(\omega)I_n^{'}(ka)K_n(ka)}$.
$I$ and $K$ are the Bessel functions.
My question is how can I do this integral if $G_n$ also depends on $k$? I thought maybe doing it as a double integral but it seems to me that the order it is integrated in matters, that I need to do the second integral first? Am I wrong here and can it be integrated just using the usual method for double integration?
Thanks in advance.