= Fourier transform of the logarithm of one plus x squared
{c}
{title2=$\mathcal F[\tfrac12\log(1+x^2)]$}
For $u(x)=\tfrac12\log(1+x^2)$, the angular-frequency transform is the <tempered distribution>
$$
\langle\widehat u,\varphi\rangle
=-\pi\int_{\mathbb R}\frac{\varphi(\xi)-\varphi(0)}{|\xi|}e^{-|\xi|}\,d\xi.
$$
The numerator cancels the singularity at zero and makes this integral absolutely convergent. Differentiating $u$ and using $\widehat{(1+x^2)^{-1}}=\pi e^{-|\xi|}$ determines this expression up to a <Dirac delta distribution>. That delta coefficient is zero: testing with the expanding Gaussian $\varphi_n(x)=(2\sqrt\pi)^{-1}e^{-x^2/(4n)}$ makes both the proposed expression and $\langle u,\widehat\varphi_n\rangle$ tend to zero, while $\varphi_n(0)$ stays fixed. Changing the subtraction convention would change the delta coefficient.
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