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Let $f(x)$ be an analytic function in $\mathbb{C}\setminus A$, where $A$ is a discrete finite set of branch points. I have some questions.

  1. Given a set of values of $f(x)$ is some domain $U \subset \mathbb{C}$ is it possible to say without performing analytic continuation how many branch points are in $\overline{\mathbb{C}} \setminus \overline{U}$ or do they exist at all? Is there some formula?

  2. I have the same question about total index of branch points lying outside $U$.

  3. Is there a way to compute a genus of Riemann surface of function $f(x)$ given its values only in the domain $U$? If the answer to second question is positive then it can be done via Riemann-Hurwitz formula $$ \text{(genus)} = \frac{1}{2}\text{(total index)}-\text{(number of sheets)}+1. $$

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