Fourier Transform of Logarithmic Functions

[tom_rylex]

Homework Statement

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

The Attempt at a Solution

I'm starting with the second distribution:
$$\langle F(pf(\frac{1}{x})),\phi \rangle = \langle pf(\frac{1}{x}),F(\phi) \rangle$$
where F() is the Fourier transform, and pf is a pseudo function. I'm applying the property that let's me move the Fourier transform to the test function.
$$= \lim_{\substack {\varepsilon \rightarrow 0}} \left[ \int^\infty _\varepsilon \frac{1}{x} \int^\infty _{-\infty} \phi(y) e^{ixy} dy dx + \int^{-\varepsilon} _{-\infty} \frac{1}{x} \int^\infty _{-\infty} \phi(y) e^{ixy} dy dx\right]$$

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.

 

[Nino MMP]

I'm also stuck on the first problem, since I can't find any information on Fourier transforming a logarithm.