Thanks for your reply.
I'm not sure whether you want me to do the second derivatives of the angular integral w.r.t \overline{r}=qr or something else. If i haven't misunderstood, I'm sorry to tell you that i really don't know what the well known differential equation is...
Would you give...
The result is well known, but i need more details about the integral below
\int \mathrm{d}^2x \frac{1}{|\mathbf{x}|} e^{- \mathrm{i} \mathbf{q} \cdot \mathbf{x}} = \frac{2 \pi}{q}
I've done the Fourier transform of the Coulomb potential in 3D. But failed to get the right answer in 2D...