I have the following function:
$f(x,y,z) = \frac{1}{\sqrt{x^2 + y^2 + z^2}}+ \sum_{a=-\infty}^{\infty}\sum_{b=-\infty}^{\infty} \frac{1}{\sqrt{x^2 + (y - a)^2 + (z - b)^2}}-\frac{1}{\sqrt{a^2 + b^2}}$
I am summing over all integers $a$ and $b$ but I assume
$|a| + |b| > 0$
This is so the constant term in the sum is never infinity.
I am interested in computing
$\int_{0}^{1}f(x,y,z)dz$
I am not really sure where to start. Does anyone have tips on approaching this sort of integral?
As some background, this function isn't random. Its a doubly periodic Green's function. I have found some resources regarding its asymptotic expansion, but not much on computing its integral.
If I try to integrate term by term, I still end up with an infinite sum that I do not know how to evaluate.