Fourier transform of $x^2(1+x^2)^{-2}$

Question

I have the following

$$ \mathscr{F} \{ x^2 (1+x^2)^{-2} \} ( \xi ) $$

All I could think of was

$$ -\frac{d^2}{d\xi ^2} \mathscr{F} \{ (1+x^2)^{-2} \}(\xi) $$

but I don't know what to do next. I could solve this if the power of the expression was only $-1$.

Do you know how to evaluate integrals using the residue theorem? — joriki, Feb 26 '16 at 01:02
Yes. It is detailed in this response http://math.stackexchange.com/questions/377073/calculate-the-fourier-transform-of-bx-frac1x2-a2 — KSHMR, Feb 26 '16 at 01:07
BTW, one of the answers to that question shows how to do it using the residue theorem. — joriki, Feb 26 '16 at 01:13

score 1 · Accepted Answer · answered Feb 26 '16 at 01:12

1

\begin{align} \mathscr F\left\{ x^2 (1+x^2)^{-2} \right\} ( \xi )&=\mathrm i\frac{\mathrm d}{\mathrm d\xi}\mathscr F\left\{ x(1+x^2)^{-2} \right\} ( \xi )\\&=\frac12\mathrm i\frac{\mathrm d}{\mathrm d\xi}\mathscr F\left\{\frac{\mathrm d}{\mathrm dx}(1+x^2)^{-1} \right\} ( \xi )\\&=\frac12\frac{\mathrm d}{\mathrm d\xi}\xi\mathscr F\left\{(1+x^2)^{-1} \right\} ( \xi )\;. \end{align}

answered Feb 26 '16 at 01:12

joriki

238,052

1

A thousand thanks, friend :) Will mark answer when its possible – KSHMR Feb 26 '16 at 01:15

Fourier transform of $x^2(1+x^2)^{-2}$

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