Liouville's theorem (complex analysis)

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

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