# Triple infinite summation of a 3D Fourier series

I'm trying to evaluate the equation below excluding the case when $$n_x=n_y=n_z=0$$. I know this equation converges everywhere except where x,y, and are all multiples of $$2\pi$$. I've attempted breaking the summation into 3 parts which avoid the unwanted case. Unfortunately Mathematica wasn't able to solve that. I also tried defining a function to use in the place of the fraction with a value of 0 when $$n_x=n_y=n_z=0$$. That hasn't worked either. I've also been trying to break the problem down into smaller pieces and looking at special cases with little success. Any help at all would be appreciated. $$\sum_{n_x=0}^\infty \sum_{n_y=0}^\infty \sum_{n_z=0}^\infty \frac{\cos(n_x x)\cos(n_y y)\cos(n_z z)}{n_x^2+n_y^2+n_z^2}$$

• I think you might get better answers at the math StackExchange. – Roman Jul 18 at 22:32
• As a starting point, the Laplacian of your function is $-\sum_{n_x,n_y,n_z}\cos(n_xx)\cos(n_yy)\cos(n_zz)$, which is a sum of Dirac $\delta$-functions on every grid point that is an integer multiple of $2\pi$ in each coordinate. Now if you can find the anti-Laplacian of this... – Roman Jul 18 at 22:46
• Probably, I should try there as well. The reason I'm doing this is so I can find out the potential of a charge in a torus space. – Laff70 Jul 19 at 3:17
• Or try at the physics StackExchange. This kind of problem has been studied in solid-state physics since Madelung's days. – Roman Jul 19 at 10:33
• I'll x-post it to both of them, thank you. – Laff70 Jul 19 at 16:54