Let $\Sigma$ be a Riemann surface, $x \in \Sigma$ and let $z: U \rightarrow D \subset \mathbb{C}$ be a local coordinate system centered at $x$. For every $k \in \mathbb{Z}$, a holomorphic line bundle $L_{kx}$ on $\Sigma$ is defined by gluing the trivial vector bundles $(\Sigma \backslash \{x\}) \times \mathbb{C} \rightarrow \Sigma$ and $U \times \mathbb{C} \rightarrow U$ via the holomorphic transition function $$(U \backslash \{x\}) \times \mathbb{C} \rightarrow (U \backslash \{x\}) \times \mathbb{C}, \ (p, v) \mapsto (p, z(p)^k v)$$

Can someone give me some hints how to prove the following claims ?

The space of global holomorphic sections of $L_{kx}$ is isomorphic to the space of meromorphic functions on $\Sigma$, holomorphic outside of $x$ and has a pole of order at most $k$ if $k \geq 0$ and a zero of order $k$ if $k < 0$.

Thanks for your help.


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