# Thread: complex analysis analytic functions

1. ## complex analysis analytic functions

Prove or disprove: Suppose $\displaystyle f(z)$ and $\displaystyle g(z)$ are analytic functions on an open and connected region $\displaystyle \omega$ and $\displaystyle f(z)g(z)=0 \in \omega$, then either $\displaystyle f(z)=0$ or $\displaystyle g(z)=0$ in $\displaystyle \omega$.
Can I get some help please?

2. otherwise choose any point p that f(p) is not 0. For any natural number n, there must be a point $\displaystyle p_n$ in the open ball B(p,1/n) that $\displaystyle f(p_n)$ is 0, otherwise g will be indentically zero in this open ball which leads to a global zero of g.
So we get a series of points $\displaystyle p_n \rightarrow p$ with $\displaystyle f(p_n)=0$ which implies f(p)=0.