maximum principleMaximumPrinciple2002-06-09 04:58:392004-10-23 18:08:20Theorem% this is the default PlanetMath preamble. as your knowledge
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\item[Maximum principle]
Let $f:U\to \Reals$ (where $U\subseteq \Reals^d$) be a harmonic function. Then $f$ attains its extremal values on any compact $K\subseteq U$ on the boundary $\partial K$ of $K$. If $f$ attains an extremal value anywhere in the \emph{interior} of $K$, then it is constant.
\item[Maximal modulus principle]
Let $f:U\to\Complex$ (where $U\subseteq \Complex$) be a holomorphic function. Then $|f|$ attains its maximal value on any compact $K\subseteq U$ on the boundary $\partial K$ of $K$. If $|f|$ attains its maximal value anywhere on the \emph{interior} of $K$, then it is constant.
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