Poisson's equation

Poisson's equation is a second-order linear partial differential equation

∇2ψ=f\nabla^{2}\psi=f

where ∇2\nabla^{2} is the Laplacian and ff is a complex valued function over a manifold. It is a more general form of Laplace's equation. It can be solved generally using a Green's function:

ψ(r)=−∭f(r′)4π∣r−r′∣dr′\psi(\mathbf{r})=-\iiint \frac{f(\mathbf{r}')}{4\pi|\mathbf{r}-\mathbf{r}'|}d\mathbf{r}'

where the integration happens over all space.