The maximum modulus principle states that if a function is holomorphic (complex differentiable) on a connected open set and continuous on its closure, then the maximum value of the modulus of that function occurs on the boundary of the set, not in the interior. This principle helps understand the behavior of holomorphic functions, especially when studying zeros and poles, and is foundational in demonstrating results like Liouville's theorem and the Schwarz lemma, as well as in analyzing entire functions.
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