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^[1]), 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.