Fourier transform of a radial function 4

Question

I want to use the fact that Fourier transform of a radial function is a radial function. I'm trying to find the Fourier transform of $$f(x)=\frac{1}{|x|^2}, x\in R^3$$ I began like this: \begin{align*} \mathcal{F}(f)(\xi)=\int_{R^3}e^{-2\pi i\xi\cdot x}\frac{1}{|x|^2}dx=[\text{ spherical coordinates}]\\= \int_0^{\infty}\int_0^{\pi}\int_0^{2\pi}e^{-2\pi i\xi\cdot (r\sin\theta\cos\phi, r\sin\theta\sin\phi, r\cos\theta)}\frac{1}{r^2}r^2\sin(\phi) d\phi d\theta dr\\=\int_0^{\infty}\int_0^{\pi}\int_0^{2\pi}e^{-2\pi i\xi\cdot (r\sin\theta\cos\phi, r\sin\theta\sin\phi, r\cos\theta)}\sin(\phi) d\phi d\theta dr \end{align*} but I don't know how to continue further. I wanted to find a rotation matrix (a composition of it) so I could use the property above, but without success. Any hints, please?

Answer 1

You may assume $\xi=(0,0,|\xi|)$. This will simplify the integral.

Answer 2

Even though $|x|^{-2}$ is locally integrable in $\mathbb R^3$, the Fourier integral doesn't exist. You have to work with distributions or introduce a regularization of the Fourier integral in some other way in order to define the transform rigorously. In terms of distributions, $$(|x|^{-2}, e^{-2 \pi i w \cdot x}) = \pi |w|^{-1}.$$ Certain results will still hold in terms of ordinary functions as well. For instance, Plancherel's theorem for the inner product holds provided that the other function is sufficiently well-behaved.