harmonic function
Stretch a soap film across a wire loop, or let heat settle in a metal plate until nothing changes, or map out the electric potential in charge-free space. In every case the resulting field is a harmonic function: a function whose value at any point equals the average of its values on a small circle around that point. It has no local bumps or dips that are not forced by the boundary — it is the smoothest possible interpolation of whatever values are prescribed around the edge.
Precisely, a real function u(x, y) is harmonic on a region if it has continuous second derivatives there and satisfies Laplace's equation, nabla^2 u = d^2u/dx^2 + d^2u/dy^2 = 0. The averaging property is a theorem: the value at the centre of any disk equals the mean over the boundary circle. From this flows the maximum principle — a harmonic function attains its largest and smallest values only on the boundary, never strictly inside — which is why a charge-free region has no interior peaks of potential and a heated plate's hottest interior point cannot exceed its rim.
Complex analysis gives harmonic functions a royal road. The Cauchy-Riemann equations force the real and imaginary parts of every analytic function to be harmonic, so each analytic f(z) supplies two harmonic functions at once. Conversely, in the plane every harmonic function is the real part of some analytic function. This is why two-dimensional electrostatics, steady heat flow, and ideal fluid flow can all be solved with the machinery of analytic functions and conformal mapping rather than by grinding out the partial differential equation directly.
u(x, y) = x^2 - y^2 is harmonic: d^2u/dx^2 = 2 and d^2u/dy^2 = -2, so the sum is zero. It is the real part of the analytic function f(z) = z^2, whose imaginary part v = 2xy is its harmonic conjugate. The level curves x^2 - y^2 = const and 2xy = const meet at right angles everywhere — the signature of a harmonic pair.
Every analytic function delivers a harmonic real part and a harmonic imaginary part for free.
Harmonic is not the same as smooth or as a minimum: a harmonic function is a saddle-like balance, never having a strict interior maximum or minimum. Also, the clean two-dimensional link to analytic functions is special to the plane — in three dimensions harmonic functions exist but are not real parts of complex-analytic functions.