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I was given the following exercise: knowing the Fourier transform of $g(x)=\frac{1}{1+x^2}$, compute the Fourier transform of $$f(x)=\frac{x}{(1+x^2)^2}$$

The problem is that maybe I don't know the useful property to solve it. The only way I know to "combine" known Fourier transforms to obtain a new one is convolution but I can't see how $f$ could be a convolution product. I also tried to apply the definition, without success.

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    $\begingroup$ Hint: $g'(x)=\frac{-2x}{(1+x^2)^2}$, so $f(x)=-\frac{1}{2}g'(x)$. And if $F(g)(s)$ is the Fourier transform of $g(t)$, then $F(g'(s))=2\pi isF(g)(s)$. $\endgroup$ – KittyL Dec 26 '16 at 18:53
  • $\begingroup$ You can also derive how to calculate the fourier transform of the derivative of a function. You have to pass the derivative under the integral under the correct assumptions and derive the formula. $\endgroup$ – Harnak Dec 26 '16 at 18:53
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Hint. One may recall that the Fourier transform of the derivative is given by $$ \mathcal{F}(f')(\xi)=2\pi i\xi\cdot\mathcal{F}(f)(\xi). $$ Check that $f$ is linked to the derivative of $g$.

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  • $\begingroup$ Ok got it, thank you. But what if now I have to compute $\frac{x^2}{(1+x^2)^2}$ or $\frac{1}{(1+x^2)^2}$? This is the second part of the exercise and the relation between $g^\prime$ and these is not linear anymore. $\endgroup$ – Paul Dec 26 '16 at 18:59
  • $\begingroup$ Then one might use the convolution product or one might use a partial fraction decomposition of $\frac{4}{(1+x^2)^2}$ that is $-\frac{1}{(x-i)^2}-\frac{i}{(x-i)}-\frac{1}{(x+i)^2}+\frac{i}{(x+i)}.$ $\endgroup$ – Olivier Oloa Dec 26 '16 at 19:03

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