Embedding a Riemann surface which is diffeomorphic to a punctured disc in $\mathbb{C}$ How do we prove that any Riemann surface which is diffeomorphic to a punctured disc can be holomorphically embedded in $\mathbb{C}$? 
The reason I am thinking about this is because I was trying to classify complex structures on a punctured disc. Any hints will be appreciated. Thanks.

Such a Riemann surface $X$ has fundamental group $\Bbb Z$. 
By the uniformisation theorem,
its universal cover $U$ is either $\Bbb C$ or the upper half plane, and $X$
is the quotient of $U$ by an automorphism $\gamma$ of infinite order. Up to inner automorphisms, if $U=\Bbb C$ then $\gamma$ is conjugate to $z\mapsto z+1$, and then $X$ is conformally equivalent to $\Bbb C^*$.
If $U$ is the upper half plane, then we can take $\gamma:z\mapsto z+1$
or $\gamma: z\mapsto \lambda z$ ($\lambda>1$). In the former case we get
a punctured disc. In the latter case we get an annulus.
