For a physics problem that I'm trying to study I would like to exand an eigenproblem in the eigenfunctions of the laplacian over a unit disk with neuman boundry conditions. To do this I need to calculate the following integral:
$I_{mnpq} = \int_0^1 dr \int_0^1 dr' \int_0^{2\pi} d\theta \int_0^{2\pi} d\theta' \frac{J_n(\omega_{nm}r)e^{in\theta} J_p(\omega_{pq}r')e^{-ip\theta'} r r'}{\sqrt{r^2+r'^2 -2rr'\cos(\theta-\theta')}}$
With n,m,p and q integers and $\omega_{nm}$ the $m$th zeropoint of the derivative of $J_n$.
When trying to calculate these integrals I'm always experiencing slow convergence. When I did the same thing with eigenfunctions over a unit square I managed to do an integration variable transformation that solved my slow convergence issues. For this problem I don't really see a smart way to simplify the integrandum. So I was wondering:
Does anyone have a suggestion for a smart transformation of integration variables so that convergence would be faster or that a part of this integral can be done analytically.