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Hard fourier transform

  1. Oct 18, 2011 #1
    1. The problem statement, all variables and given/known data
    Using that the Fourier transform of e[itex]^{|x|}[/itex] is [itex]\frac{2}{\xi^2+1}[/itex]. Compute the Fourier transform of [itex]\frac{x}{(x^2+1)^2}[/itex]


    2. Relevant equations



    3. The attempt at a solution

    My first thought was to try and rewrite the problem in a form I recognized, tried a couple of things but what I though was best was to write:

    [itex]\frac{d}{dx}[/itex] [itex]\frac{-1}{x^2+1}[/itex]

    And transform that to e[itex]^{i*\xi}[/itex]*f([itex]\xi[/itex]). This was wrong. Very wrong actually.

    Anyone have any hints for me?

    thanks!

    edit: Missed some things but should be right this time.
     
    Last edited: Oct 18, 2011
  2. jcsd
  3. Oct 18, 2011 #2
    http://en.wikipedia.org/wiki/Fourier_transform#Functional_relationships Is has nothing to do with convolution! Fourier transform turns DERIVATIVE into MULTIPLICATION with i times argument, so [itex]i \xi[/itex] in this case, so the result should be something like [itex]i \xi e^{|x|}[/itex] (i assume you are supposed to perform the inverse FT, right?
     
  4. Oct 18, 2011 #3
    Hello sussking_leon,

    My bad, the * means multiplication and not convolution. How do you get the [itex]i \xi [/itex] infront of the exponent?
     
  5. Oct 18, 2011 #4
    Check the wiki page, forumla 106 and 107 do the trick. So if FT{e^|x|} = f(v), then FT{x e^|x|} = i d/dv f(v). (Fourier transform non-unitary, angular frequency)
     
  6. Oct 18, 2011 #5
    Ah, I was thinking right at least. Its so stupid, we don't get any of these rules on our exam so without wiki, I would have never solved this exercise. Thanks for the help.
     
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