Green's function for the Laplacian

Summary

The Green's function of a domain for the Laplacian is the potential of a unit point charge in the domain. The boundary of the domain is held at potential zero. In a plane domain $\Omega$ with pole $w$, it is a function $g_\Omega(z, w)$ that behaves like $\log(1/\abs{z - w})$ near $w$. It is harmonic elsewhere and vanishes on the boundary. It solves the Poisson equation with zero boundary values: $\Delta_z g_\Omega(z, w) = -2\pi \delta_w$. In free space $\R^3$ the corresponding function is $1/(4\pi\abs{x - y})$, the fundamental solution of the three-variable Laplace equation. Green's functions give integral formulas for the Dirichlet problem and are invariant under conformal maps in the plane.