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I am trying to solve the following question:

Prove the uniqueness theorem for functions with poles: If $f$ is holomorphic on $D$ apart from some poles, and there exists $z_n \to z$ in $D$ such that $f(z_n) = 0$ then $f$ is constant.

There are two things I don't understand:

  1. If the function is constant, how can it have poles? By definition, poles are points where the function approaches infinity.
  2. Assuming it is possible, can I prove it by removing small circle around each pole such that $z$ is still in the remaining area and using the regular uniqueness theorem?

Thanks!

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Concerning the first point, read "apart from possibly some poles" or understand "some" to include the possibility of no poles at all.

Regarding the second point, you need to exclude the possibility that $z$ is a pole, then you can use the ordinary identity theorem to conclude that $f$ is constant on the connected component of $D\setminus P$ containing $z$, where $P$ is the (possibly empty) set of poles of $f$. Then - assuming $D$ is connected, otherwise $f$ need not be constant - find that $D\setminus P$ is still connected, hence $f$ is constant on $D\setminus P$, whence $P = \varnothing$.

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