2D Fourier transform of log function

Question

I am studying the paper found here. Halfway in the paper (Equation 6), the inverse 2D Fourier transform of $1/(k_x^2+k_y^2)$ needs to be determined. Is is stated that this is straightforward, and that the inverse is given by $-\log(r)=-\log(\sqrt{x^2+y^2})$. I am having trouble verifying this; how can this be derived?

Answer 1

If $v(r)=\log r$, then $\Delta v=\delta$ and $\hat {(\Delta v)}=1$, that is $-(\xi^2+\eta^2)\hat {v}(\xi, \eta)=1$. However this yields $-\hat{v}=1/(\xi^2+\eta^2)$ which has no sense since this function is not locally integrable. A precise computation, where a principal value of $1/(\xi^2+\eta^2)$ appears, can be found in Chapter 9 of the book Vladimirov, Equations of mathematical physics.