I am trying to compute the Fourier transform of $\frac1{|\mathbf{x}|^2+1}$ where $\mathbf{x}\in\mathbb{R}^3$.
Just writing out the integral: $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac1{|\mathbf{x}|^2+1}e^{-2\pi i (\mathbf{x}\cdot\mathbf{\xi})}dx_1dx_2dx_3$.
Mathematica was no help with this integral. I realized though that the function is radial, so that in spherical coordinates $f(\rho,\theta,\phi)=\frac1{r^2+1}=f(\rho)$. I thought this would simplify matters because then the limits of integration are just $\int_0^{\infty}\int_0^{2\pi}\int_0^{\pi}$ and $dx_1dx_2dx_3 \mapsto \rho^2\sin\theta d\theta d\phi d\rho$. However, the dot product in the exponent is messing me up a lot. I think it should be of the form $||\mathbf{x}||||\xi||$ times some trigonometric function of the angle between them, and $||\mathbf{x}||$ is just $\rho$. Because of this, I don't think doing this integral directly will be that much cleaner than Cartesian coordinates either.
However, I was wondering if I could follow the methods of this and leverage the symmetry of my function and get the answer this way.
I already know the Fourier Transform must be radial from the link. However, I am not sure about how to use the dilation part to extract information for my problem here.
Or do you think it would be propitious to write $x_1$ as $\rho\sin\theta\cos\phi$ etc. and just work it out that way?
\lVert
and\rVert
to produce $\lVert$. $\endgroup$