Green function of the Laplacian
= Green function of the Laplacian
{c}
{title2=$G(x,x')$}
In three dimensions,
$$
-\nabla^2\frac1{4\pi|x-x'|}=\delta^{(3)}(x-x').
$$
Consequently the decaying solution of $-\nabla^2u=f$ in free space is
$$
u(x)=\frac1{4\pi}\int_{\mathbb R^3}\frac{f(x')}{|x-x'|}\,d^3x'.
$$
This is the free-space <Green function> for the negative <Laplace operator>.