the spectral representation of a Green's function
There is a second way to build a Green's function that does not need images or clever geometry: assemble it from the natural modes of vibration of the region. Every nice domain with an operator on it (a drum, a string, a heat-conducting plate) has a set of standing-wave shapes, its eigenfunctions, each ringing at its own frequency, its eigenvalue. The spectral representation expresses the Green's function as a weighted sum over all these modes — the operator's own resonances are its building blocks.
Concretely, suppose the eigenfunctions phi_n satisfy L phi_n = lambda_n phi_n with the boundary condition, chosen orthonormal: integral of phi_m phi_n = 1 if m = n, else 0. Because they form a complete basis, you can expand the delta source itself: delta(x - y) = sum over n of phi_n(x) phi_n(y). Now solving L G(., y) = delta becomes term-by-term trivial — divide each mode by its eigenvalue — giving G(x, y) = sum over n of phi_n(x) phi_n(y) / lambda_n. Read it: the response to a source at y is gotten by exciting every mode in proportion to how strongly the source y couples to it (the phi_n(y) factor) and how easily that mode responds (the 1/lambda_n factor — soft, low-eigenvalue modes dominate). The symmetry G(x, y) = G(y, x) is now obvious from the formula. This is the Mercer / spectral form, and it is exactly the resolvent kernel of the operator.
This representation is powerful precisely where images fail: it works on any domain whose eigenfunction problem you can solve (rectangles via Fourier series, disks via Bessel functions, spheres via spherical harmonics), and it makes the connection between Green's functions and spectral theory explicit. Two honest cautions. First, it breaks down if zero is an eigenvalue (lambda_n = 0): a vanishing denominator signals that the bare Green's function does not exist and you must work modulo that mode — this is the Fredholm alternative, and it is exactly the Neumann-problem compatibility condition. Second, the series converges in a mean-square sense and may converge slowly or non-uniformly near the source, where G is singular.
On the interval [0, pi] with L = -d^2/dx^2 and zero ends, the eigenfunctions are sqrt(2/pi) sin(n x) with eigenvalues n^2. The Green's function is G(x, y) = (2/pi) sum over n of sin(n x) sin(n y) / n^2 — and this series sums to the explicit triangular tent function you also get directly.
The same Green's function, assembled from the operator's eigenmodes.
If zero is an eigenvalue (a soft mode with lambda_n = 0) the term blows up: the ordinary Green's function does not exist and you must project that mode out — this is the Fredholm alternative and the Neumann compatibility condition in disguise.