Differential Equations

Green's Functions

Solve the problem for a point source, then build every other source out of it by superposition.

Updated May 2026

Green's functions turn on a linearity argument so simple it is easy to miss how much it buys.

The idea

For a linear operator $L$ , define $G$ as the response to a point source:

L G (r, r^{'}) = δ (r - r^{'})

Then for any source $f$ , the solution of $L u = f$ is

u (r) = \int G (r, r^{'}) f (r^{'}) d^{3} r^{'}

Decompose the source into points, add up the responses. The whole method is the superposition principle taken seriously.

Fig. 1

Four point sources of differing strength, each producing the same response shape scaled and shifted. The solid curve is their sum, which is what the integral computes, in the limit of infinitely many infinitesimal sources.

Definition

What G actually is

$G$ is the inverse of $L$ , written as an integral kernel. Solving a differential equation becomes doing an integral, and integration is a far better conditioned operation than differentiation.

The Laplacian

In three dimensions, $\nabla^{2} G = δ^{3} (r - r^{'})$ gives

G = - \frac{1}{4 π ∣ r - r ^{'} ∣}

and the solution of Poisson's equation is Coulomb's law integrated over the charge. Which is to say Coulomb's law is the Green's function of the Laplacian, and always was.

The $1/ r$ is not particular to electrostatics. It is geometry: flux spreads over a sphere whose area grows as $r^{2}$ . In two dimensions the same argument gives $ln r$ , which is why line charges have logarithmic potentials.

Boundaries

The free-space $G$ solves the problem only in infinite space. With boundaries, add any solution of the homogeneous equation that cancels $G$ on the surface:

G_{D} (r, r^{'}) = - \frac{1}{4 π ∣ r - r ^{'} ∣} + H (r, r^{'})

For a grounded plane, $H$ is precisely the image charge from electrostatics. The method of images was a Green's function all along.

Property

Reciprocity

$G (r, r^{'}) = G (r^{'}, r)$ for self-adjoint operators. The response at $A$ to a source at $B$ equals the response at $B$ to the same source at $A$ . This holds for sound in a room, for antennas, and for seismic waves.

Causality

For time-dependent problems the wave operator has two Green's functions: retarded (effect after cause) and advanced (effect before it). Both are mathematically valid; the equations do not choose.

We impose causality by hand, discarding the advanced solution. That choice is not in Maxwell's equations, and where it comes from is not a settled question. Most attempts to explain it end up pointing back at the second law.

Perturbation theory

Write $L = L_{0} + ε V$ where $G_{0}$ is known. Iterating gives

G = G_{0} + G_{0} V G_{0} + G_{0} V G_{0} V G_{0} + \dots

Every term is "propagate freely, scatter, propagate freely, scatter…". Drawing each one gives you Feynman diagrams; the perturbative expansion of quantum field theory is this series with a more elaborate $G_{0}$ .