the representation formula

After all the machinery, here is the payoff: a single formula that hands you the value of the solution at any interior point, built out of the source inside and the data on the boundary. The representation formula is the sentence 'the answer here equals the Green's function summed against the source, plus a boundary term that brings in the boundary data.' It is the explicit solution that the whole point-source program was aiming at.

For the Dirichlet problem -Laplacian u = f in a region with u = g prescribed on the boundary, the representation formula reads u(x) = integral over region of G(x, y) f(y) dy - integral over boundary of g(y) (dG/dn_y)(x, y) dS(y). Read it in two pieces. The first term spreads the source f through the Green's function G — that is the point-source response integrated against the actual source. The second term is the boundary contribution: the prescribed boundary values g are propagated inward by the normal derivative of the Green's function, which for Laplace's equation is precisely the Poisson kernel. The derivation is the third Green's identity: write u via Green's second identity with v = G, and the boundary terms involving the unknown du/dn drop out exactly because G was built to vanish on the boundary. A Neumann problem has the dual version, using the Green's function with vanishing normal derivative.

Why this is the goal: it converts the boundary-value problem into pure integration, separates cleanly the effect of interior sources from the effect of boundary data, and exposes the structure of the answer (which boundary values matter most, how influence decays with distance). On the disk it specializes to the Poisson integral formula. The honest limitation is the usual one: you need the Green's function G explicitly, which restricts the clean formula to special domains; for a general region the formula is true and conceptually central but not a closed-form recipe.