Editing Liouville's theorem (complex analysis) (section)

{{Short description|Theorem in complex analysis}}
{{About|Liouville's theorem in complex analysis||Liouville's theorem (disambiguation)}}
{{Complex analysis sidebar}}
In [[complex analysis]], '''Liouville's theorem''', named after [[Joseph Liouville]] (although the theorem was first proven by [[Cauchy]] in 1844<ref>{{Eom| title = Liouville theorems
 | author-last1 =  Solomentsev| author-first1 = E.D.| author-last2 = Stepanov| author-first2 = S.A.| author-last3 = Kvasnikov| author-first3 = I.A.| oldid = 52098}}</ref>), states that every [[bounded function|bounded]] [[entire function]] must be [[Constant function|constant]]. That is, every [[holomorphic function]] <math>f</math> for which there exists a positive number <math>M</math> such that <math>|f(z)| \leq M</math> for all <math>z\in\Complex</math> is constant. Equivalently, non-constant holomorphic functions on <math>\Complex</math> have unbounded images.

The theorem is considerably improved by [[Picard theorem|Picard's little theorem]], which says that every entire function whose image omits two or more [[complex number]]s must be constant.