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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.

The locally integrable function $\log |x|$ defines a tempered distribution. Its distributional derivative is $\mathrm{PV}(1/x)$: perform integration by parts outside $(-a,a)$, and observe that the boundary contribution $\log a[\phi (a)-\phi (-a)]$ tends to zero. By the Fourier transform of a derivative and the preceding result,Division by $|k|$ needs a finite-part inverse-absolute-value distribution, since $1/|k|$ is not locally integrable at zero. One possible normalization isIt satisfies $k\mathrm{Pf}(1/|k|)=\mathrm{sgn}k$. The general solution of $kT=0$ is a multiple of the Dirac delta distribution: a test function vanishing at zero can be written $k\psi (k)$, so $T$ depends only on its value at zero. Consequently the Fourier transform of the absolute logarithm isChanging the finite-part normalization changes $C$. The printed reciprocal expression is meaningful with this interpretation, not as an ordinary function at $k=0$.

The locally integrable logarithm has distributional derivative $(\log |x|{)}^{\prime }=\mathrm{PV}(1/x)$. By the Fourier transform of a derivative and the previous result,For $k\ne 0$, this gives $-\pi /|k|$. At zero the expression must be extended as a Hadamard finite-part integral, not treated as a locally integrable function. For example, define the finite-part inverse-absolute-value distribution byThen $kT=\mathrm{sgn}k$. Different reference cutoffs change $T$ only by a multiple of the Dirac delta distribution. The general solution of the transform equation is consequentlyIndeed multiplication by $k$ annihilates only a delta term among distributions supported at zero; delta derivatives would generate additional nonzero terms. This supplies the precise interpretation of the paper's formula, and the convention-dependent constant $C$ need not be found.