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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

LL 3.14
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Skills
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  • Could you give the definition of the finite part of $1/x$? It is not standard terminology, the usual Hadamard finite part is not defined for negative integer powers. Don't you rather mean the Cauchy principal value? – LPZ Jun 20 '23 at 14:11

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

LL 3.14
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  • But i have to do the Fourier transform of Finite Part of $1/x$, not of $PV(1/x)$. – Skills Jun 20 '23 at 13:57
  • What do you call the finite part of $1/x$? The distribution associated to $1/x$ that I know is called the principal value of $1/x$ and defined by $\langle\mathrm{pv}(1/x),\varphi\rangle = \int \frac{\varphi(x)-\varphi(0)}{x}\mathrm d x = \lim_{\epsilon\to 0} \int_{|x|>\epsilon} \frac{\varphi(x)}{x},\mathrm d x$. If you are taling about the finite part of $1/|x|$ you can find it here https://math.stackexchange.com/questions/3723136/the-fourier-transform-of-1-p3/3724502#3724502 (just take $d=1$) – LL 3.14 Jun 20 '23 at 15:47