Expression for the Fourier transform of $f(x) = \frac{1}{1 +\|x\|^2}$

Question

I'm having troubles with the Fourier transform of $f(x) = \frac{1}{1 +\|x\|^2} \in L^2(\mathbb{R}^{n})$. For the case $n=1$ I got $\hat{f}(\xi) = \pi e^{-2\pi |\xi|}$ using residues. Does the general case have a nice expression? How is that expression obtained?

Answer 1

It's not in $L^2$ for $n \ge 4$. For $n \le 3$, assume wlog $\xi$ is in the direction of one of the coordinate axes.