(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the Fourier transforms of the following distributions: log|x|, d/dx log|x|.

3. The attempt at a solution

I'm starting with the second distribution:

[tex] \langle F(pf(\frac{1}{x})),\phi \rangle = \langle pf(\frac{1}{x}),F(\phi) \rangle [/tex]

where F() is the fourier transform, and pf is a pseudo function. I'm applying the property that lets me move the fourier transform to the test function.

[tex] = \lim_{\substack {\varepsilon \rightarrow 0}} \left[ \int^\infty _\varepsilon \frac{1}{x} \int^\infty _{-\infty} \phi(y) e^{ixy} dy dx + \int^{-\varepsilon} _{-\infty} \frac{1}{x} \int^\infty _{-\infty} \phi(y) e^{ixy} dy dx\right][/tex]

I think the direction I'm supposed to go is to put the exp and 1/x terms together and determine the answer via complex residuals. If that's correct, I could use some help getting there.

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# Homework Help: Fourier transform (log)

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