Green's function
/ GREEN /
Suppose you have a linear system — a stretched string, a heated rod, an electric field — and you want to know how it responds to any load at all. Green's idea is to ask one cleaner question first: how does it respond to a single, infinitely concentrated poke at one point? Answer that, and you can build the response to any load by adding up the responses to all its little pieces. The Green's function is that single-poke response, recorded as a function of where you poke and where you observe.
Precisely, for a linear differential operator L and boundary conditions, the Green's function G(x, s) is the solution of L G(x, s) = delta(x - s) that satisfies the boundary conditions, where delta is the Dirac delta — a unit point source placed at s. Once you have G, the solution of the inhomogeneous problem L u = f with those boundary conditions is u(x) = integral of G(x, s) f(s) ds over the domain. In words: weight the response-to-a-point-at-s by how much source f sits at s, and superpose. The Green's function literally inverts the operator: u = L^{-1} f is realized as integration against G.
This converts solving a differential equation into doing an integral, and it does so once and for all — G depends only on the operator and the boundaries, not on the particular f. That is why Green's functions are central in electrostatics (the potential of a point charge), quantum mechanics (the propagator), heat flow, acoustics, and elasticity. The catch: G encodes the boundary conditions, so changing the geometry or the boundary data changes G; the free-space Green's function (no boundaries) is the simplest building block, and the method of images or eigenfunction expansions adapt it to bounded regions.
For L = -d^2/dx^2 on [0, 1] with u(0) = u(1) = 0, the Green's function is G(x, s) = x(1 - s) for x < s and s(1 - x) for x > s. Then u(x) = integral from 0 to 1 of G(x, s) f(s) ds solves -u'' = f, u(0) = u(1) = 0 for any source f — one formula, every load.
The piecewise formula is symmetric in x and s — a hallmark of self-adjoint operators (Maxwell-Betti reciprocity).
A Green's function exists and is unique only when the homogeneous boundary-value problem has just the trivial solution; if zero is an eigenvalue, no ordinary Green's function exists and you must use a modified Green's function constrained by the solvability condition.