Show that \(u(x)=\frac {1}{4\pi |x|}\) is the fundamental solution of the Laplace equation in \(\mathbb {R}^3\), \(-\Delta u=\delta _{0}\) in the sense of distributions, i.e. \[ \int _{\mathbb {R}^3} u(x)\Delta \varphi (x)\,dx=-\varphi (0) \]
for all \(\varphi \) smooth and compactly supported. You should provide a fully detailed proof, not just a formal computation.