: Mathematics: Liouville's theorem

mathematical analysis > complex analysis > Liouville's theorem

Liouville's theorem

In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, every holomorphic function ${\\displaystyle f}$ for which there exists a positive number ${\\displaystyle M}$ such that ${\\displaystyle |f(z)|\\leq M}$ for all ${\\displaystyle z\\in \\mathbb {C} }$ is constant. Equivalently, non-constant holomorphic functions on ${\\displaystyle \\mathbb {C} }$ have unbounded images.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis))

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