Fourier transform of the absolute logarithm ($\widehat{\log |x|}=-\pi \mathrm{Pf}(1/|k|)+C\delta (k)$)

The distributional derivative of $\log |x|$ is the principal-value reciprocal distribution. Its transform is $-i\pi \mathrm{sgn}k$. 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.