= Fourier transform of a reciprocal positive quadratic form
{c}
{title2=$\mathcal F[(x^TGx)^{-1}](\xi)=\dfrac{2\pi^2}{\sqrt{\det G}\sqrt{\xi^TG^{-1}\xi}}$}
For a real symmetric positive-definite three-dimensional <matrix>, the reciprocal <quadratic form> is locally integrable and defines a regular <tempered distribution>. With the Fourier kernel $e^{-ix\cdot\xi}$, radial Abel regularization gives $\mathcal F(|x|^{-2})=2\pi^2/|\xi|$. The linear change of variables $y=G^{1/2}x$ gives the displayed formula. No <principal value> or origin-supported correction is required. The result is homogeneous of degree $-1$ on the frequency side.
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