Liouville's theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Name

This result is termed Liouville's theorem in complex analysis. However, the term Liouville's theorem is also used for theorems proved by Liouville in other parts of mathematics.

Statement

Any entire function (holomorphic function on the whole of $\mathbb {C}$) that is bounded (i.e. its image is bounded in absolute value) must be constant.

Facts used

• Cauchy estimates for derivatives