Complex Analysis/Liouville's Theorem

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

Statement

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

Proof

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

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

Thus, $f'=0$, and therefore $f$ is constant.

See Also

• Complex Analysis

• Cauchy's integral formula
• Wiki2Reveal

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• Source: Kurs:Funktionentheorie/Satz von Liouville - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Satz von Liouville
• Date: 12/17/2024