In textbook electromagnetism we are used to seeing neat, coordinate-free, expressions for the scalar potential from an electric dipole (using Gaussian units)

$$\phi(\mathbf{r}) = \frac{\mathbf{p} \cdot \mathbf{r}}{r^3}$$

and vector potential from a magnetic dipole

$$\mathbf{A}(\mathbf{r}) = \frac{\mathbf{\mu} \times \mathbf{r}}{r^3}$$

My question is: **are there neat expressions for the spatial Fourier transforms of these dipole potentials?**

Doing a quick search, and spending more time than I'd like to admit, I haven't found an equally nice expression for these potentials in the Fourier domain although I would expect that there should be one. I vaguely recall there is an identity such that $\mathcal{F}(\frac{\mathbf{r}}{r^3}) \sim -\frac{\mathbf{k}}{k^2}$ which would easily solve this, but was not able to confirm that after doing some quick math. Moreover, I would naively expect that all multipole potentials should have very straightforward Fourier transforms (maybe in terms of spherical harmonics).