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maximum principle
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(Theorem)
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- 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 interior of $K$ , then it is constant.
- 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 interior of $K$ , then it is constant.
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Cross-references: holomorphic function, interior, boundary, compact, harmonic function
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This is version 2 of maximum principle, born on 2002-06-09, modified 2004-10-23.
Object id is 3078, canonical name is MaximumPrinciple.
Accessed 13124 times total.
Classification:
| AMS MSC: | 30C80 (Functions of a complex variable :: Geometric function theory :: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination) | | | 31A05 (Potential theory :: Two-dimensional theory :: Harmonic, subharmonic, superharmonic functions) | | | 31B05 (Potential theory :: Higher-dimensional theory :: Harmonic, subharmonic, superharmonic functions) | | | 30F15 (Functions of a complex variable :: Riemann surfaces :: Harmonic functions on Riemann surfaces) |
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Pending Errata and Addenda
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