Possible Duplicate:
analytic functions defined on $A\cup D$
Let $f$ and $g$ be holomorphic functions defined on $A\cup D$ , where $A=\{z\in\mathbb{C}:1/2<|z|<1\}$ and $D=\{z\in\mathbb{C}: |z-2|<1\}$. Which of the following are true?
- If $f(z)g(z)=0$ for all $z\in A\cup D$, then either $f(z)=0$ for all $z\in A$ or $g(z)=0$ for all $z\in A$.
- If $f(z)g(z)=0$ for all $z\in D$, then either $f(z)=0$ for all $z\in D$ or $g(z)=0$ for all $z\in D$.
- If $f(z)g(z)=0$ for all $z\in A $, then either $f(z)=0$ for all $z\in A$ or $g(z)=0$ for all $z\in A$.
- If $f(z)g(z)=0$ for all $z\in A\cup D$, then either $f(z)=0$ for all $z\in A∪D $ or $g(z)=0$ for all $z\in A\cup D$.
How should I solve this problem?