I am trying to solve a problem, but am really confused on what I'm supposed to prove. The question is:
Prove in $2$D that $\Delta \log(|x|)=2\pi\delta_0$ in the distributional sense.
Now, let $x=\langle\alpha,\beta\rangle$. Then I figured out: $$\langle\Delta\log(|x|),\phi\rangle = \iint_{\mathbb R^2}\log(\sqrt{\alpha^2+\beta^2})(\phi_{\alpha\alpha}+\phi_{\beta\beta})d\alpha d\beta$$ But where am I supposed to go from here? Do I use integration by parts?