Harmonic function

A function $f$ is called harmonic if it satisfies Laplace's equation:

Harmonic functions are important in many areas of mathematics and physics, including potential theory, electrostatics, fluid dynamics, and heat conduction. Examples include:

• The real and imaginary parts of a holomorphic function in complex analysis.
• The electrostatic potential in a charge-free region.
• The steady-state temperature distribution in a region with no heat sources.
  For an interpretation see Laplacian operator.

If you picture the graph of a function $u(x,y)$ as a flexible rubber sheet fixed along its boundary, then:

• The Laplacian operator $\mathrm{\Delta }u=u_{xx}+u_{yy}$ measures how much the sheet bends up or down relative to its immediate surroundings — a kind of restoring force toward equilibrium.
• When $\mathrm{\Delta }u=0$, there’s no net tendency for the sheet to move: all local forces balance out.

So harmonic functions correspond to surfaces in equilibrium, with no internal “pressure” pushing them up or down. Their graphs are the equilibrium shapes of a stretched membrane fixed along a boundary.