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Fourier transform (log)

  1. Oct 25, 2008 #1
    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.
     
  2. jcsd
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