Liouville's theorem (complex analysis)

Theorem in complex analysis

In complex analysis, Liouville's theorem states that every bounded entire function must be constant. That is, every holomorphic function f {\displaystyle f} for which there exists a positive number M {\displaystyle M} such that | f ( z) | ≤ M {\displaystyle |f(z)|\leq M} for all z ∈ C {\displaystyle z\in \mathbb {C} } is constant. Equivalently, non-constant holomorphic functions on C {\displaystyle \mathbb {C} } have unbounded images.

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