a Green's function
/ green /
To find how a system responds to a complicated push, first find how it responds to the simplest possible push -- a single sharp poke at one point and instant -- then add up the responses to all the little pokes that make up the real force. That response to a single poke is the Green's function. It is the impulse response of a differential equation.
For a linear differential operator L, the Green's function G(x, x') is the solution of L G(x, x') = delta(x - x'), where delta is the Dirac delta -- a unit source concentrated at x'. Once you have it, the solution of the full inhomogeneous problem L u = f(x) is built by superposition: u(x) = the integral of G(x, x') f(x') dx'. The boundary conditions are baked into G, so this integral automatically satisfies them. Finding G often uses transforms or an eigenfunction expansion: G(x, x') = the sum over n of phi_n(x) phi_n(x') / lambda_n, over the eigenfunctions phi_n and eigenvalues lambda_n of L.
It is the master strategy for linear inhomogeneous problems across physics: the Coulomb potential 1/(4 pi r) is the Green's function of the Laplacian (the field of a point charge), the retarded potential is the Green's function of the wave operator (the field of a point source switched on in time), and in quantum mechanics and field theory the Green's function is the propagator -- the amplitude to go from one point to another -- whose poles give particle masses and whose branch cuts give the continua.
For the 3D Laplacian, laplacian G = -delta(r), the Green's function is G = 1/(4 pi r): the potential of a unit point charge. Smear point charges with density rho and the integral of G rho gives Poisson's solution -- Coulomb's law rebuilt by superposition.
The Coulomb potential is just the Laplacian's Green's function.
A Green's function is not unique until boundary and causality conditions are fixed: the wave equation has both retarded and advanced Green's functions, mathematically equal but physically distinct, and choosing the retarded one is an added input (causality), not something the equation alone decides.