Fourier transform of the absolute logarithm
= Fourier transform of the absolute logarithm
{c}
{title2=$\widehat{\log|x|}=-\pi\operatorname{Pf}(1/|k|)+C\delta(k)$}
The <distributional derivative> of $\log|x|$ is the <principal-value reciprocal distribution>. Its transform is $-i\pi\operatorname{sgn}k$. Dividing the derivative-transform relation by $ik$ gives the displayed <finite-part inverse-absolute-value distribution>, with an undetermined delta term fixed by the finite-part convention.