the logarithmic potential
In three dimensions, the gravitational or electric field of a point falls off like 1/distance, giving the familiar 1/r potential. Flatten the world to two dimensions — think of an infinitely long charged wire seen end-on, or heat from a line source — and the geometry changes: the potential no longer falls off like 1/r but grows like the logarithm of distance. The logarithmic potential is this two-dimensional analogue: the fundamental solution of the Laplacian in the plane.
Concretely, in the plane the fundamental solution of -Laplacian is E(x) = -(1/(2 pi)) log|x|, so that -Laplacian E = delta. Equivalently Laplacian of ((1/(2 pi)) log|x|) = delta. You can sanity-check it away from the origin: log|x| = (1/2) log(x1^2 + x2^2) is harmonic for x not equal to 0 (its Laplacian is zero), and the delta is produced entirely by the singularity at the origin — Gauss's flux of grad E through any small circle around 0 equals 1. The solution of Poisson's equation -Laplacian u = f in the plane is then u(x) = -(1/(2 pi)) integral of log|x - y| f(y) dy: smear the log around the source.
Two features deserve attention. The logarithm changes sign — log|x| is negative inside the unit circle and positive outside — and, crucially, it does not decay at infinity; it grows without bound. That is why two-dimensional potential theory has subtleties the three-dimensional case lacks: there is no nontrivial potential that both solves the equation and stays bounded at infinity, so 'the' potential is fixed only up to an additive constant and a careful treatment of behaviour far away. This same log appears as the real part of log z in complex analysis, tying plane potential theory to holomorphic functions.
A long straight wire carrying uniform charge, viewed in the cross-sectional plane, produces the potential -(1/(2 pi epsilon)) log r at distance r — the logarithmic potential — and its gradient gives the familiar 1/r electric field of a line charge.
Two-dimensional potentials grow like log r, not decay like 1/r.
Unlike the 3D Newtonian potential, the logarithmic potential does not vanish at infinity — it grows — so the 2D Laplacian has no bounded free-space potential and is fixed only up to a constant.