the Green's function
/ GREEN /
Imagine placing a single concentrated source — a point charge, a pinpoint heat source — at one spot inside a region whose boundary is held grounded (potential zero). The potential that results, as a function of where you look, is the Green's function of the region for that source point. It is the fundamental response: once you know how the region answers a single point source, you can build the answer to almost any source or boundary condition by superposing copies of it.
For a region D and a fixed source point z_0 inside it, the Green's function G(z, z_0) is the function of z that is harmonic on D except at z_0, that blows up exactly like -log|z - z_0| as z -> z_0 (the potential of a point source, the singular solution of Laplace's equation), and that vanishes for z on the boundary of D. The structure splits neatly: G(z, z_0) = -log|z - z_0| - h(z), where h is the harmonic function on D with boundary values equal to -log|z - z_0| on the boundary — so finding the Green's function is itself a Dirichlet problem, solved once for the singular part's boundary trace. On the unit disk there is a clean closed form using a Mobius map: G(z, z_0) = -log| (z - z_0) / (1 - (z_0-bar) z) |, where the denominator reflects z_0 to its mirror point outside the disk so the whole thing vanishes on |z| = 1.
The Green's function is the master key of potential theory in a region. It gives a formula for the solution of the Dirichlet problem on D (the normal derivative of G on the boundary is the Poisson kernel of that region), it produces the region's harmonic measure (how much boundary weight a point sees), and it transforms under conformal maps in a beautifully simple way (it is a conformal invariant), which lets you carry it between regions. The caveat: a Green's function with these clean properties exists for nice regions but can fail to exist for very wild ones (for instance the whole plane has no Green's function, tied to the fact that -log|z| has nowhere to vanish at infinity); its existence is essentially equivalent to the solvability of the Dirichlet problem on the region.
On the unit disk with source at the center z_0 = 0, the Green's function is G(z, 0) = -log|z| = -log r: it is harmonic for 0 < r < 1, blows up like -log r at the center, and equals -log 1 = 0 on the boundary circle |z| = 1.
The disk's Green's function with a central source is simply -log r, vanishing on the boundary.
Conventions vary by a sign and a factor of 2 pi between physics and mathematics texts; here G is the positive potential vanishing on the boundary with logarithmic singularity. The plane and other regions on which Laplace's equation is too unconstrained simply have no Green's function.