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Although $x\mapsto |x{|}^{-1}$ is not integrable on all of ${\mathbb{R}}^3$, it defines a tempered distribution. For $x\ne 0$, the Gamma function integral givesThe three-dimensional form of the Fourier transform of a Gaussian isTherefore, up to a nonzero constant depending on the Fourier-transform convention,For $\xi \ne 0$, the substitution $u=|\xi {|}^2/(4t)$ turns the last integral intoThus, as asserted by the Fourier transform of inverse distance in three dimensions,as a tempered distribution, where $C^{\prime }\ne 0$.

For integer $\ell \ge 0$ and $r,R>0$, $r_{<}=\min (r,R)$ and $r_{>}=\max (r,R)$. A proof compares the multipole expansion of $1/|\mathbf{r}-\mathbf{R}|$ with its Fourier transform of inverse distance in three dimensions. Applying the Rayleigh plane-wave expansion to the Fourier representation gives coefficient $8{\int }_0^{\infty }{j}_{\ell }(kr)j_{\ell }(kR)dk$ multiplying $Y_{\ell m}(\hat{r})Y_{\ell m}^{\ast }(\hat{R})$; the Coulomb multipole coefficient is $4\pi r_{<}^{\ell }/[(2\ell +1)r_{>}^{\ell +1}]$. Equating the coefficients proves the identity. The powers also match continuously at $r=R$.