Liouville's Theorem (complex analysis)

In complex analysis, Liouville's Theorem states that a bounded holomorphic function on the entire complex plane must be constant. It is named after Joseph Liouville. Picard's Little Theorem is a stronger result.

Statement

Let $f : \mathbb{C} \to \mathbb{C}$ be a holomorphic function. Suppose there exists some real number $M \ge 0$ such that $\lvert f(z) \rvert \le M$ for all $z \in \mathbb{C}$ . Then $f$ is a constant function.

Proof

We use Cauchy's Integral Formula.

Pick some $z_0 \in \mathbb{C}$ ; let $C_R$ denote the simple counterclockwise circle of radius $R$ centered at $z_0$ . Then $\lvert f'(z_0) \rvert = \biggl\lvert \frac{1}{2\pi i} \int_C -\frac{f(z)}{(z-z_0)^2}dz \biggr\rvert \le \frac{M}{R^2} .$ Since $f$ is holomorphic on the entire complex plane, $R$ can be arbitrarily large. It follows that $f'(z) = 0$ , for every point $z \in \mathbb{C}$ . Now for any two complex numbers $A$ and $B$ , $f(B) - f(A) = \int_{A}^B f'(z)dz = 0 ,$ so $f$ is constant, as desired. $\blacksquare$

Page

Toolbox

Search

Liouville's Theorem (complex analysis)

Statement

Proof

See also