Compute Fourier Transform of x/(x^2+1)^2 using e^|x|

[saxen]

Homework Statement

Using that the Fourier transform of e^{|x|} is \frac{2}{\xi^2+1}. Compute the Fourier transform of \frac{x}{(x^2+1)^2}

Homework Equations

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:

\frac{d}{dx} \frac{-1}{x^2+1}

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

Anyone have any hints for me?

thanks!

edit: Missed some things but should be right this time.

 

[susskind_leon]

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 i \xi in this case, so the result should be something like i \xi e^{|x|} (i assume you are supposed to perform the inverse FT, right?

 

[saxen]

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 i \xi in this case, so the result should be something like i \xi e^{|x|} (i assume you are supposed to perform the inverse FT, right?

Hello sussking_leon,

My bad, the * means multiplication and not convolution. How do you get the i \xi infront of the exponent?

 

[susskind_leon]

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)

 

[saxen]

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)

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.