How many dimensions?... one, two, three... or more?...
Kind regards
All right!... but for semplicity sake may be it is better to start with the two-dimensional case ...
The potential function generated by a charge q in the origin is...
(1)
The two-dimension Fourier tranform of V(x,y) by definition is...
(2)
Now if we pass to polar coordinates the (2) becomes...
(3)
Now it is easy to verify that the integral...
(4)
... vanishes everiwhere with the only exception of the case where is so that the final result is...
(5)
Kind regards
I'm still wondering what the Fourier transform is; the delta function can't be the answer since it's the Fourier transform of 1 (or at least of a real constant). The Fourier transform of 1/r in three dimensions has to be some kind of standard transform, hasn't it?