12 - Liouville's Theorem

Liouville's Theorem

Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function

If $f$ is bounded then $f$ is constant

Proof

Suppose that $|f(z)|\le M$ for all $z\in \mathbb{C}$

Let ${\gamma }_R(t)=R{e}^{2\pi it}$ be the circular path centred at origin with radius $R$ then

For $R>|w|$ then using Cauchy’s Integral Formula

$$\begin{array}{rl}|f(w)-f(0)|& =\left|\frac{1}{2\pi i}{\int }_{{\gamma }_R}f(z)\left(\frac{1}{z-w}-\frac{1}{z}\right)dz\right|\\ & =\frac{1}{2\pi }\left|{\int }_{{\gamma }_R}\frac{wf(z)}{z(z-w)}dz\right|\\ & \le \frac{1}{2\pi }\cdot 2\pi R\underset{z:|z|=R}{\sup }\left|\frac{wf(z)}{z(z-w)}\right|\\ & \le R\frac{M|w|}{R(R-|w|)}\\ & =\frac{M|w|}{R-|w|}\end{array}$$

As $R\to \infty$ then $|f(w)-f(0)|=0$ so $f$ is constant