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Let $f$ be an entire function with zeros $z_{1}, z_{2}, \ldots$. Is it true that only finitely many zeros lie in the open unit disc? If so why?

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The unit disc is compact, hence the zeros accumulate there. Now use the Identity Theorem.

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  • $\begingroup$ (To be precise, the closed unit disc is compact.) $\endgroup$ – Hans Lundmark Jan 16 '12 at 21:13
  • $\begingroup$ Thanks Hans! It's important here that the function is holomorphic in a neighborhood of the unit disc. $\endgroup$ – Dirk Jan 17 '12 at 8:04

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