What is the Fourier transform of $1/|x|$? I looked it up in several tables and calculated it in Mathematica and Matlab. Some tables say that the answer is simply $$\frac{1}{|\omega|}$$ and in other table it is $$-\sqrt{\frac{2}{\pi}}\ln|\omega|$$
and in Mathematica and Matlab (mupad) it is
$$-\sqrt{\frac{2}{\pi}}\ln|\omega|-2\gamma$$
where $\gamma$ is the Euler–Mascheroni constant. Why are there so many answers? Are they all equivalent in some way or two (or all) of them are wrong?

To be fully rigorous, we should concede that $1/|x|$ is not directly a distn on $\mathbb R$. But it does arise as an even tempered distn $u$ such that $x\cdot u=sign(x)$.
Up to a constant, Fourier transform gives ${d\over dx}\hat{u}=PV{1\over x}$. This equation has at least solution $\hat{u}=\log|x|$. The associated homog eqn has only multiples of $1$ as solns, so the eqn’s solns are $\hat{u}=\log|x|+c$ for arbitrary constants $c$ (and with the normalization constant).
The constant $c$ be determined by evaluation against $e^{-\pi x^2}$.
