Poisson's equation is a second-order partial differential equation which arises in physical problems such as finding the electric potential of a given charge distribution. Its general form in $n$ dimensions is $$ \nabla^2\phi(\mathbf r)=\rho(\mathbf r) $$ where $\nabla^2$ is the Laplacian and $\rho:D\to\mathbb{R}$ , often called a source function, is a given function on some subset $D$ of $\mathbb{R}^n$ . If $\rho$ is identically zero, the Poisson equation reduces to the Laplace equation.

The Poisson equation is linear, and therefore obeys the superposition principle: if $\nabla^2\phi_1=\rho_1$ and $\nabla^2\phi_2=\rho_2$ , then $\nabla^2(\phi_1+\phi_2)=\rho_1+\rho_2$ . This fact can be used to construct solutions to Poisson's equation from fundamental solutions, or Green's functions, where the source distribution is a delta function.

A very important case is the one in which $n=3$ , $D$ is all of $\mathbb{R}^3$ , and $\phi(\mathbf r)\to 0$ as $\vert\mathbf r\vert\to\infty$ . The general solution is then given by $$ \phi(\mathbf r)=-\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{\rho(\mathbf{r'})}{\vert\mathbf{r}-\mathbf{r'}\vert}\mathrm{d}^3\mathbf{r'}. $$