harmonic functionHarmonicFunction2002-06-04 11:11:572005-03-25 23:47:20Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Prob}[2]{\mathbb{P}_{#1}\left\{#2\right\}}A twice-differentiable real or complex-valued function $f\colon U\to\mathbb{R}$ or $f\colon U\to\mathbb{C}$, where $U\subseteq\mathbb{R}^n$ is some \PMlinkescapetext{domain}, is called \emph{harmonic} if its Laplacian vanishes on $U$, i.e. if $$\Delta f\equiv 0.$$
Any harmonic function $f\colon\mathbb{R}^n\to\mathbb{R}$ or $f\colon\mathbb{R}^n\to\mathbb{C}$ satisfies Liouville's theorem. Indeed, a holomorphic function \emph{is} harmonic, and a real harmonic function $f\colon U\to\mathbb{R}$, where $U\subseteq\mathbb{R}^2$, is locally the real part of a holomorphic function. In fact, it is enough that a harmonic function $f$ be \PMlinkescapetext{bounded} below (or above) to conclude that it is \PMlinkescapetext{constant}.