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

This is the free-space Green function for the negative Laplace operator.

For three noncompact spatial coordinates of radius $r$ and a circle of circumference $L$, the positive Laplace Green function on the source slice is $G(r,0)={\sum }_{n\in \mathbb{Z}}[4{\pi }^2(r^2+n^2{L}^2){]}^{-1}=(4\pi Lr{)}^{-1}\coth (\pi r/L)$. At $r\ll L$ this is $1/(4{\pi }^2{r}^2)$; at $r\gg L$ it is $1/(4\pi Lr)$ with exponentially small corrections. The long-distance coefficient also follows by replacing the dense image sum with $L^{-1}{\int }_{-\infty }^{\infty }dy/(r^2+y^2)=\pi /(Lr)$. Electric and gravitational potentials inherit this crossover after multiplying by their signed source couplings.

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.