Complex Analysis/Liouville's Theorem

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

Statement

Let $f : ℂ \to ℂ$ be holomorphic and bounded. Then $f$ is constant.

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

\begin{matrix} | f^{'} (z) | & = \frac{1}{2 π} | \int_{\partial B_{R} (z)} \frac{f (w)}{(w - z)^{2}} d w | \\ \leq \frac{1}{2 π} 2 π R \cdot \frac{1}{R^{2}} \cdot \sup_{w \in B_{R} (z)} | f (w) | \\ \leq \frac{M}{R} \\ \to 0, R \to \infty \end{matrix}

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

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