I would like to be able to derive the following Fourier transform identity: $$ \int_{-\infty}^{+\infty}\frac{d^{D}p}{(2\pi)^{D}}\, |\vec{p}|^{\alpha}e^{i\vec{p}\cdot\vec{r}} = \left(\frac{2}{r}\right)^{\alpha + D}\frac{\Gamma(\frac{\alpha +D}{2})}{(4\pi)^{D/2}\Gamma(-\frac{\alpha}{2})},$$ where $\Gamma(z)$ is the gamma fucntion.
I presume that this equation is symbolic in the sense that the integral on the left-hand side needs to be regulated in some way in order to get the finite result on the right-hand side. I'm really unsure how to proceed though. I tried expressing the integral in spherical polar coordinates, but that doesn't seem to help. I've also tried introducing a regulator of the form $e^{-\epsilon |\vec{p}|^{2}}$, where $\epsilon$ is small, but finite. Unfortunately this hasn't seemed to help either. I must be missing something. Any help would be appreciated.