Liouville's theorem (complex analysis)

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.

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