Liouville's Theorem

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

Example 6.2.2   Let . The function is the composition of two entire functions, therefore is entire.

This function is obviously non constant. Actually, if Im , then is not bounded.

Theorem 6.2.3 (Generalization of Liouville's theorem)   Let be an entire function verifying the following property: for some integer , there exist two constants and such that . Then is a polynomial of degree at most .

Example 6.2.4   Suppose that is entire and that . Then is a polynomial of degree at most 1.