Green's function

Summary

A Green's function of a linear differential operator $L$ with boundary conditions is a kernel $G(x, s)$ that turns the source term into the solution. The function $y(x) = \int G(x, s) f(s)\, ds$ solves $L y = f$. Formally it is the response of the system to a unit point source at $s$, that is, $L G(\cdot, s) = \delta_s$ with the Dirac delta. It turns a differential equation into an integral formula, and it is basic in boundary value problems, potential theory and physics. The Green's function for the Laplacian is the main case for partial differential equations. See the definition for ordinary differential equations.