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In mathematics, the maximum modulus principle in complex analysis states that if ''f'' is a holomorphic function, then the modulus |''f'' | cannot exhibit a true local maximum that is properly within the domain of ''f''. In other words, either ''f'' is a constant function, or, for any point ''z''0 inside the domain of ''f'' there exist other points arbitrarily close to ''z''0 at which |''f'' | takes larger values. ==Formal statement== Let ''f'' be a function holomorphic on some connected open subset ''D'' of the complex plane ℂ and taking complex values. If ''z''0 is a point in ''D'' such that : for all ''z'' in a neighborhood of ''z''0, then the function ''f'' is constant on ''D''. By switching to the reciprocal, we can get the minimum modulus principle. It states that if ''f'' is holomorphic within a bounded domain ''D'', continuous up to the boundary of ''D'', and non-zero at all points, then |''f'' (z)| takes its minimum value on the boundary of ''D''. Alternatively, the maximum modulus principle can be viewed as a special case of the open mapping theorem, which states that a nonconstant holomorphic function maps open sets to open sets. If |''f''| attains a local maximum at ''z'', then the image of a sufficiently small open neighborhood of ''z'' cannot be open. Therefore, ''f'' is constant. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Maximum modulus principle」の詳細全文を読む スポンサード リンク
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