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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?

  1. 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 $z\in A$.
  2. 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 $z\in D$.
  3. 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 $z\in A$.
  4. 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?

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