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:
- If the function is constant, how can it have poles? By definition, poles are points where the function approaches infinity.
- 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?