The following is Theorem 4.10 ("Branched Covering Prnciple") on p. 361 of Bruce P. Palka's An Introduction to Complex Function Theory, Springer, 1991 (corrected second printing of 1995).
Suppose that $f:U\rightarrow\widehat{\mathbb{C}}$ — where $\widehat{\mathbb{C}} = \mathbb{C}\cup\{\infty\}$, and $U$ is a non-empty, open subset of $\widehat{\mathbb{C}}$ — is a meromorphic function, that $z_0$ is a point of $U$, that $f$ is not constant in the component of $U$ that includes $z_0$, and that $f$ takes the value $w_0$ with multiplicity $m$ at $z_0$ ($m \in \{1,2,\dots\}$, $m \neq \infty$). Let $r \in (0,\infty)$ be any number that the following conditions prevail: the closed disk $\overline{\Delta} = \overline{\Delta}(z_0, r)$ is contained in $U$, and the statements $f(z)\neq w_0$, $f(z)\neq\infty$, $f'(z)\neq0$ are true for every $z$ in the set $\overline{\Delta}\setminus\{z_0\}$. Define $s = s(r)$ to be the largest number $\in (0,\infty)$ for which $\Delta(w_0,s)\cap f(K) = \emptyset$, where $K$ is the circle that bounds $\overline{\Delta}$. Then $G = \{z\in\Delta(z_0,r) : f(z) \in \Delta(w_0,s)\}$ is a domain. Moreover, for each point $w$ of the punctured open disk $\Delta^*(w_0,s)$ the set $E_w = \{z\in\Delta(z_0,r) : f(z) = w\}$ consists of exactly $m$ points. Furthermore, at each of these points $f$ assumes the value $w$ with multiplicity one.
Palka doesn't prove this theorem; he relegates the proof to an exercise (Exercise 5.83, p. 373). Palka does prove a similar result earlier in the book, Theorem 3.7 (likewise titled "Branched covering Principle", on p. 344), which differs from Theorem 4.10 only in the following points:
- Theorem 3.7 has $\mathbb{C}$ where Theorem 4.10 has $\widehat{\mathbb{C}}$.
- Theorem 3.7 has "analytic" where Theorem 4.10 has "meromorphic".
In other words, in Theorem 3.7 $f$ is an analytic, complex-valued function of a complex variable.
I imagine that it is possible to prove Theorem 4.10 using Theorem 3.7 by considering the four cases:
- $z_0 \in \mathbb{C}$ and $w_0 \in \mathbb{C}$,
- $z_0 \in \mathbb{C}$ and $w_0 = \infty$,
- $z_0 = \infty$ and $w_0 \in \mathbb{C}$, and
- $z_0 = \infty$ and $w_0 = \infty$.
The first case reduces to Theorem 3.7. As for the other three cases, I have trouble proving them, and would appreciate help. I imagine that case 2 can be reduced to case 1 using the transformation $f^{-1}$, whereas cases 3 and 4 can be reduced to cases 1 and 2, respectively, using the transformation $f(z^{-1})$, but even so, I don't know how to proceed.
If this theorem is proved (not necessarily under the same name) in some other textbook, I'll appreciate a reference.