I'm having some trouble finding the Fourier transform of $$ f(\mathbf{r}) = \dfrac{1}{r} e^{-\alpha r} $$ where $r = \sqrt{x^2 + y^2 + z^2}$ and $\alpha$ is a nonzero integer. The problem gives the hint that you should take $\alpha \rightarrow 0$, but I can't seem to figure out how this helps the integration.
I'm just looking for a hint on how to proceed with this.
Thanks!
EDIT:
Just to clarify, I'm trying to solve this integral: $$ \tilde{f}(\mathbf{k}) = \int f(\mathbf{r})e^{-i\mathbf{k}\cdot \mathbf{r}} d^3 r$$
The solution is $$\tilde{f}(k)=\int_0^{2\pi}d\phi\int_0^\pi d\theta\left[\sin\theta\int_0^\infty\frac{1}{r}\exp -\alpha r\cdot \exp -ikr\cos\theta\cdot r^2 dr\right]\\=2\pi\int_0^\pi d\theta\frac{\sin\theta}{(\alpha+ik\cos\theta)^2}=\frac{2\pi}{ik}\left[\frac{1}{\alpha+ik\cos\theta}\right]_0^\pi=\frac{4\pi}{\alpha^2+k^2}.$$As a sanity check, you can verify $\int_0^\infty |\tilde{f}(k)|^2 k^2 dk=(2\pi)^3\int_0^\infty |f(r)|^2 r^2 dr=\frac{4\pi^3}{\alpha}$.