Harmonic function

A real function satisfying Laplace's equation, $u_{xx}+u_{yy}=0$ .

A real-valued function $u(x,y)$ with continuous second partial derivatives is harmonic on a domain if it satisfies Laplace's equation there: $u_{xx}+u_{yy}=0$ at every point. Harmonic functions matter to complex analysis because the real and imaginary parts of any analytic function are automatically harmonic -- a fact that ties the algebra of complex differentiability directly to this equation from real, two-variable calculus.

Examples

$u(x,y)=x^2-y^2$ is harmonic: $u_{xx}=2$ and $u_{yy}=-2$ , which sum to $0$ .
$u(x,y)=x^2+y^2$ is not harmonic: $u_{xx}+u_{yy}=2+2=4\ne 0$ .