page_title: Maximum modulus principle
hierarchical_section_title: Maximum modulus principle
caption_attribution_description: Illustration of en:Maximum modulus principle
context_page_description: In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus |f | cannot exhibit a strict local maximum that is properly within the domain of f.
  In other words, either f is a constant function, or, for any point z₀ inside the domain of f there exist other points arbitrarily close to z₀ at which |f | takes larger values.
context_section_description: In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus |f | cannot exhibit a strict local maximum that is properly within the domain of f. 
  In other words, either f is a constant function, or, for any point z₀ inside the domain of f there exist other points arbitrarily close to z₀ at which |f | takes larger values.
caption: A plot of the modulus of cos(z) (in red) for z in the unit disk centered at the origin (shown in blue). As predicted by the theorem, the maximum of the modulus cannot be inside of the disk (so the highest value on the red surface is somewhere along its edge).
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