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Possible Duplicate:
analytic functions defined on $A\cup D$

Let $f$, $g$ be analytic function defined on $A\cup D$ where $A = \{z \in \mathbb{C}: \frac{1}{2}<|z|<1\}$ and $D = \{z \in \mathbb{C}: |z-2|<1\}$ Which of the following statements 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 $z \in A\cup D$ or $g(z) =0$ for all $z \in A\cup D$.

I am stuck on this problem. Can anyone help me please?

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