Liouville's Theorem.

Theorem 6.2.1   Let $ f$ be an entire function. If $ f$ is bounded, then $ f$ is a constant function.

Example 6.2.2   Let $ f(z)=e^{iz}$ . The function $ f$ is the composition of two entire functions, therefore is entire.

This function is obviously non constant. Actually, if Im$ (z)<0$ , then $ f(z)$ is not bounded.

Theorem 6.2.3 (Generalization of Liouville's theorem)   Let $ f$ be an entire function verifying the following property: for some integer $ k \geq 0$ , there exist two constants $ \alpha$ and $ beta$ such that $ \forall z \in \mathbb{C}, \vert f(z)\vert \leq \alpha + \beta
\vert z\vert^k$ . Then $ f(z)$ is a polynomial of degree at most $ k$ .

Example 6.2.4   Suppose that $ f$ is entire and that $ \forall z \in \mathbb{C}, \vert f(z)\vert
\leq \alpha + \beta \vert z\vert^{4/3}$ . Then $ f(z)$ is a polynomial of degree at most 1.



Noah Dana-Picard 2007-12-24