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Fourier transform of $\frac{y^2-x^2}{(x^2+y^2)^2}$

Discussion in 'Mathematics' started by mat, Oct 8, 2018.

  1. mat

    mat Guest

    As a lecture exercise I am computing several Fourier transforms and want to compute in particular the Fourier transform of $$ g:=\frac{y^2-x^2}{(x^2+y^2)^2}. $$ According to this post the Fourier transform of $(x^2+y^2)^{-2}$ is given by $$ \mathcal F\left[\frac{1}{(x^2+y^2)^2}\right](u,v)=-c_1\frac{u^2+v^2}4\left(-\log\left(\frac{u^2+v^2}4\right)+c_2\right) $$ for some constants $c_1,c_2$.

    Hence (up to a normalization constant) \begin{align} \mathcal F\left[\frac{y^2-x^2}{(x^2+y^2)^2}\right](u,v)&=-D^{(0,2)}\mathcal F\left[\frac{1}{(x^2+y^2)^2}\right](u,v)+D^{(2,0)}\mathcal F\left[\frac{1}{(x^2+y^2)^2}\right](u,v) \\ &= \frac{v^2-u^2}{u^2+v^2} \end{align}

    Mathematica gives me as a result $\frac{ u^2}{u^2+v^2}$ (up to a normalization constant). I see that $\frac{v^2-u^2}{u^2+v^2}=1-\frac{2u^2}{u^2+v^2}$, but where does the $1$ come from, i.e. where is my mistake?

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