Fourier transform of inverse distance in three dimensions
= Fourier transform of inverse distance in three dimensions
{title2=$\mathcal F(|x|^{-1})$}
As a <tempered distribution> on $\mathbb R^3$,
$$
\mathcal F(|x|^{-1})(\xi)=C|\xi|^{-2}
$$
for a nonzero convention-dependent constant $C$. One proof writes
$$
|x|^{-1}=C_1\int_0^\infty t^{-1/2}e^{-t|x|^2}\,dt
$$
and applies the <Fourier transform of a Gaussian> before substituting $u=|\xi|^2/(4t)$.