Every bounded entire function, meaning a function holomorphic on the whole complex plane, is constant. Named for Joseph Liouville, it gives a short proof of the fundamental theorem of algebra and is a basic fact about the rigidity of holomorphic functions.
Facts
StatementEvery bounded entire function on the complex plane must be constant; equivalently, a non-constant holomorphic function on all of C has an unbounded image. 1 Proof YearAttribution question, not yet a formal Dissent: the theorem is named for Joseph Liouville, but the cited source states it was first proven by Augustin-Louis Cauchy in 1844; Liouville is credited with an independent 1847 proof of a related special case via elliptic function theory. Named as a content.Dissent candidate in this lane's report. Classification
Statement FormCharacterization Theorem 1 Connections
Sources
1. Liouville's Theorem (Complex Analysis) (Wikipedia)
Wikimedia FoundationLead section, opening sentence
Liouville's theorem states that every bounded entire function must be constant.
Lead section, attribution and history sentence on Cauchy's 1844 proof
the theorem was first proven by Cauchy in 1844
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