the method of images

Stand a charge in front of a flat grounded metal sheet and the sheet rearranges its own charges so the field meets it just right. The astonishing shortcut is that, in the region you care about, that whole complicated response of the sheet is reproduced exactly by a single fictitious mirror-image charge placed behind it — as if the sheet were a mirror. The method of images builds Green's functions by this trick: satisfy the boundary condition not by solving for a corrector, but by adding imaginary sources outside the region whose free-space fields cancel correctly on the boundary.

Take the upper half-plane with a Dirichlet condition. The Green's function must have a source at y = (y1, y2) with y2 > 0 and vanish on the line x2 = 0. Reflect y across the boundary to its mirror image y* = (y1, -y2), and place there an opposite source. Then G(x, y) = E(x - y) - E(x - y*): on the boundary line, x is equidistant from y and y*, so the two fundamental solutions are equal and cancel, giving G = 0 exactly as required, while inside the half-plane the image source y* lies outside, contributes a homogeneous (source-free) field, and so serves perfectly as the boundary corrector. For a disk or a ball the right image of a point y is its inversion y* = R^2 y / |y|^2 across the sphere of radius R, with a strength factor, and the same cancellation works.

Images are the most elementary, most visual way to get a Green's function, and they make the structure G = fundamental + corrector concrete. The honest limit is severe: images only work in geometries with enough symmetry for a finite set of reflections or inversions to close up — half-spaces, slabs, wedges of special angles, disks, balls. A general region admits no images; you fall back to solving the corrector's boundary-value problem or to numerical methods. Also, the image is a bookkeeping device living outside your region — there is no real source there.