I was reading Forster's lectures on Riemann surfaces and I came across the following definition and theorem:
Definition:
Suppose $X$ is a Riemann surface, $a\in X$, $\varphi\in \mathcal{O}_a$ is a function germ at $a$. A quadruple $(Y,p,f,b)$ is said to be an analytic continuation of $\varphi$ if the following conditions are satisfied:
$Y$ is a Riemann surface, $p:Y\rightarrow X$ is an unbranched holomorphic map, $f$ is a holomorphic function on $Y$, $p(b)=a$ and $p_*(\varphi)=f_b$.
An analytic continuation $(Y,p,f,b)$ of $\varphi$ is called maximal if for any analytic continuation $(Z,q,g,c)$ of $\varphi$ there exists a fiber-preserving holomorphic map $F:Z\rightarrow Y$ such that $F(c)=b$ and $F_*(f_b)=g_c.$
Theorem:
If $X$ is a Riemann surface and $\varphi\in \mathcal{O}_a$ is function germ at $a\in X$ then $\varphi$ has a maximal analytic continuation.
In the proof of this theorem, $Y$ is taken to be the connected component of $\coprod_{p\in X}\mathcal{O}_p$ containing $\varphi.$ I was wondering if there is a more "explicit" way of defining $Y$ in the case where $\varphi$ is the germ of the complex logarithm $\operatorname{Log}:\mathbb{C}\setminus (-\infty,0]\rightarrow \mathbb{C}$ at a point.
Any help would be appreciated, thank you!