Complex Analysis/Liouville's Theorem

The Liouville Theorem is a statement about holomorphic functions defined on the entire complex plane ${\displaystyle \mathbb {C} }$.

Statement

Let ${\displaystyle f\colon \mathbb {C} \to \mathbb {C} }$ be holomorphic and bounded. Then ${\displaystyle f}$ is constant.

Proof

For every ${\displaystyle R>0}$ and every ${\displaystyle z\in \mathbb {C} }$, we have by the Cauchy integral formula:

${\displaystyle {\begin{array}{rl}|f'(z)|&=\displaystyle {\frac {1}{2\pi }}\left|\int _{\partial B_{R}(z)}{\frac {f(w)}{(w-z)^{2}}}\,dw\right|\\&\leq \displaystyle {\frac {1}{2\pi }}2\pi R\cdot {\frac {1}{R^{2}}}\cdot \sup _{w\in B_{R}(z)}|f(w)|\\&\leq {\frac {M}{R}}\\&\to 0,\qquad R\to \infty \end{array}}}$

Thus, ${\displaystyle f'=0}$, and therefore ${\displaystyle f}$ is constant.

See Also

• Complex Analysis

• Cauchy's integral formula
• Wiki2Reveal

Page Information

This learning resource can be presented as a Liouville's Theorem - Wiki2Reveal slides