Suppose I have a function $Q(z)$ of a complex variable $z\in\mathbb P^1$, possessing square root type branch points at the positions $\left\{z_i\right\}_{i=1}^{2M}$. I know that the Riemann surface $\mathcal M$ on which $Q$ lives has genus $0$.

I have the following questions:

- Aside from the knowledge of the poisitions $\left\{z_i\right\}_{i=1}^{2M}$, which additional informations on $Q$ do I need in order to uniquely define the uniformizing variable $\omega(z)$?
- Does an explicit general formula for $\omega(z)$ exists?
- If so, can the limit $M\rightarrow\infty$ be obtained?

I remember seeing an expression of the type $\omega(z)=\frac{1}{z}+\sum_{k=1} a_k z^k$, but I am not sure the context corresponded to mine. Moreover there was no mention of how to fix the coefficients $a_k$ in terms of the positions of the branch points or informations on $Q$.

Thanks a lot!

Stefano