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I am reading Hitchens text on integral systems

A Riemann surface is a one-dimensional complex manifold with a maximal set of coordinate charts $\{U_\alpha,\varphi_\alpha\}_{\alpha\in I}$ where $\varphi_\alpha:U_\alpha\to \Bbb C$ such that $\varphi_\beta\circ \varphi_\alpha:\varphi_\alpha(U_\alpha\cap U_\beta)\to \varphi_\beta(U_\alpha\cap U_\beta)$ is an invertible holomorphic map for every pair $\alpha,\beta\in I$

My text then says:

Thus a neighbourhood of any point can be parametrized by a complex number $z$ and on any overlapping neighbourhood with parameter $w$, $w(z)$ is a holomorphic function of one variable.

What does this second paragraph actually mean? Let $M$ be a Riemann surface. Surely if I consider any point $p\in M$ I can find a chart $\{U_\alpha,\varphi_\alpha)$ where $p\in U_\alpha$, and then can consider $p$ in local coordinates via $\varphi_\alpha(p)$. I could then call $\varphi_\alpha(p)=z$ and then consider $\varphi_\beta(p)=w$ and abuse notation and write $w(z)$ for the transition. Is that what they intend (in which case I personally think this is stupidly cumbersome, so I doubt it)?

Later they write:

Let us use the coordinate $z$ corresponding to the coordinate chart $\varphi_\gamma$

and they start taking polynomials in $z$. What exactly is going on?

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  • $\begingroup$ I think what they mean is $w(z):\phi_{\alpha}(U_{\alpha}\cap U_{\beta})\to \phi_{\beta}(U_{\beta}\cap U_{\alpha})$ given by $z\mapsto w$ $\endgroup$ – daruma Aug 1 '18 at 10:13

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