Green function of the Laplacian ($G(x,x^{\prime })$)

In three dimensions,Consequently the decaying solution of $-{\nabla }^{2}u=f$ in free space is

Up to normalization and sign convention, a fundamental solution of the Laplace equation is $\Phi (x)=|x{|}^{2-n}$ for $n\ge 3$ and $\Phi (x)=\log |x|$ for $n=2$. It is harmonic away from the origin and satisfies $\Delta \Phi =c_n{\delta }_0$ distributionally.