Fourier transform of a reciprocal positive quadratic form ($\mathcal{F}[(x^{T}Gx{)}^{-1}](\xi )=\frac{2{\pi }^2}{\sqrt{\det G}\sqrt{{\xi }^T{G}^{-1}\xi }}$)

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.

For a complex symmetric three-dimensional $A=G+iB$ with real symmetric $B$ and positive-definite $G$, the reciprocal is a regular tempered distribution since $|x^{T}Ax|\ge x^{T}Gx$. Holomorphic continuation of the real quadratic transform proves the displayed expression. The determinant branch is the analytic determinant square root for accretive symmetric matrices. The other root has positive real part because $\mathrm{Re}({\xi }^T{A}^{-1}\xi )={\xi }^T{G}^{-1/2}(1+C^2{)}^{-1}{G}^{-1/2}\xi >0$ for nonzero real $\xi$. Both the original and transformed singularities are locally integrable, so the continued identity holds as distributions at the origin too.