Entire function
Prove that if is an entire function and then is constant or there exists with and some positive integer such that .
Solution: cannot pe identically 0, therefore, the set of zeros of is a discrete set ( formed by isolated points ). Define on the set where is never a zero for . Check by definition that is also holomorphic on . Therefore, the equality implies on $\Omega$.
Suppose has a zero . Then is bounded in a punctured disk around , which should be a pole for . But in a sufficiently small punctured disk around , and thus is bounded in a neighbourhood of this point, and the singularity is not a pole, but is removable. Contradiction.
Therefore, the only point where can be 0 is at . Suppose now that , where . Then satisfies the same conditions as , but is never . Defining an analogue function for also, we see that which is bounded. By Liouville’s Theorem, is constant. Therefore .

April 29, 2010 at 4:57 pmEntire Function Revisited « Problems