# Harmonic Function Theory/Definitions, elementaries

Definition (harmonic function):

A harmonic function is a function ${\displaystyle f\in {\mathcal {C}}^{2}(\mathbb {R} ^{n})}$ such that ${\displaystyle \Delta f=0}$.[note 1]

Proposition (closure properties of harmonic functions):

Harmonic functions are closed under the following operations:

1. The Laplace operator ${\displaystyle \Delta }$ is defined as ${\displaystyle \Delta f:=\sum _{k=1}^{n}{\frac {\partial ^{2}}{\partial x_{k}^{2}}}f}$.