I believe I can solve this problem, but I would like some help deciphering what my professor is asking me to do. He is having us find the Fourier transform of \begin{equation} r^n e^{-\alpha r} \end{equation} and says "find this in terms of $\alpha$ derivatives of the Fourier transform of part (i)". My result from part (i) is \begin{equation} f(r) = \dfrac{1}{r}e^{-\alpha r} \Rightarrow \tilde{f}(k) = -\dfrac{4\pi}{(\alpha + ik)^2} \end{equation} In that problem, he did explicitly ask us to evaluate part (i) in the limit of $\alpha \rightarrow 0$, which of course is just $\tilde{f}(k) = 4\pi / k^2$.
Any help in understanding what to do would be great. I know how to take the Fourier transforms, but I'm unsure what he is asking us to do with the $\alpha$ derivatives.
Thanks!
If you differentiate your original Fourier transform $$\frac{\partial}{\partial \alpha}\int d^3r\,\frac{e^{-\alpha r}}{r}e^{i\vec{k}\cdot \vec{r}}=\int d^3r\,\frac{\frac{\partial}{\partial\alpha}e^{-\alpha r}}{r}e^{i\vec{k}\cdot \vec{r}}=-\int d^3r\,e^{-\alpha r}e^{i\vec{k}\cdot \vec{r}}=-\mathcal{F}\{e^{-\alpha r}\}=\frac{\partial}{\partial\alpha}\frac{4\pi}{(\alpha+ik)^2}=\ldots$$ And similary for high order derivatives