Show that an annulus $A$ is biholomorphic to one of the following Riemann surfaces:
- the punctured disc $ \mathbb{D}^{*} $
- the punctured plane $ \mathbb{C}^{*} $
- a round annulus $ A_{R} = \left\{ \, z \in \mathbb{C} \ \big| \ 1 < |z| < R \, \right\} $