I am a beginner studying Riemann surface. First things first, I learned about the definition of Riemann surface as follows.

**Definition**: A Riemann surface $X$ is defined by the following data:

1. a connected Hausdorff topological space $X$;
2. an open cover $\{U\_\alpha\}\_{\alpha\in A}$ of $X$;
3. for each $\alpha\in A$, a homeomorphism $\phi\_\alpha:U\_\alpha\rightarrow V\_\alpha$ to an open subset $V\_\alpha\subset\mathbb{C}$ in such a way that for each $\alpha,\beta\in A$, if $U\_\alpha \cap U\_\beta\neq \emptyset$, the transition functions
$$\phi\_\beta\circ\phi^{-1}\_\alpha:\phi\_\alpha(U\_\alpha\cap U\_\beta)\rightarrow\phi\_\beta(U\_\alpha\cap U\_\beta),$$
is bi-holomorphic, namely, holomorphic with inverse holomorphic.

We know the famous Riemann surface for $\log z$, which has the following shape
[![enter image description here](https://i.stack.imgur.com/SAXum.png)](https://i.stack.imgur.com/SAXum.png)

How can I describe such surface with the formal definition above? In other words, I need to construct a precise $U\_\alpha$, $\phi\_\alpha$ and transition map to describe the Riemann surface for $\log z$, not using intuitive words such as "gluing", "rotate", "visualized like parking lot" etc...

Thanks very much for your help.