Harmonic function

Definition

On an open subset in the complex numbers

Let $U$ be an open subset in $\mathbb{C}$. A $C^2$-function $f:U\to \mathbb{R}$ is termed a harmonic function if we have:

$\mathrm{\Delta }f:=\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}=0$

where equality holds identically, at all points of $U$.

On an open subset in a real vector space

Let $U$ be an open subset of ${\mathbb{R}}^n$, and $f:U\to \mathbb{R}$ be a $C^2$-function (i.e. $f$ is twice continuously differentiable). Then, we say that $f$ is harmonic if we have:

$\mathrm{\Delta }f:=\sum _{k=1}^n\frac{{\partial }^{2}f}{\partial x_k^2}=0$

where equality must hold at all points of $U$.