Fourier transform of fine part of \frac{1}{x}

Question

I'm studying Fourier transform with all its properties. I understood how to find the Fourier Transform of $\log |x|$ and of $\mathrm{sign}(x)$. But I don't know, and I didn't find anything on books and internet about the finite part of $1/x$.

Maybe I have to consider the fact that the derivative of $PV(\frac{1}{x})$ is equal to the finite part of $-\frac{1}{x^2}$? But how can I use this information?

Thanks

Answer 1

If you already know the Fourier transform of $\ln |x|$, then your problem is easy once you notice that in the sense of distributions, $\ln(|x|)' = \mathrm{pv}(\frac{1}{x})$.