Laplace's equation

$u_{xx}+u_{yy}=0$ -- the defining equation of a harmonic function.

Laplace's equation is the partial differential equation $u_{xx}+u_{yy}=0$ , sometimes written $\nabla^2 u = 0$ using the Laplacian operator $\nabla^2 = \partial_x^2+\partial_y^2$ . A function is harmonic precisely when it satisfies Laplace's equation on the domain in question, so the two ideas -- the equation and the property of being harmonic -- are two sides of the same statement.

Examples

$u(x,y)=e^x\cos y$ satisfies Laplace's equation, since $u_{xx}=e^x\cos y$ and $u_{yy}=-e^x\cos y$ sum to $0$ .
Laplace's equation also governs steady-state temperature and electrostatic potential in physics, part of why harmonic functions matter beyond pure mathematics.