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Motivation and Background: I'm reading Weyl's text The Concept of a Riemann Surface and I'm having a bit of difficulty. (I can't find an online version which is not a google book, so if the following terms are non-standard, could someone point me to the more standard terms?) I will italicize relevant terms for easy reading.

He begins by noting that a function element is a power series at some point that converges in some disk of positive radius, and that an analytic function is the collection of all of the function elements gained by analytic continuation. So far, so good. He then notes we should further generalize our function elements so that we can include things like poles and branch cuts, and eventually we will get to the concept of an analytic form.

Following this, he notes that we can take a power series $\sum_{i=0}^{\infty} a_{i}(z-a)^{i}$ and introduce some $t$ such that $z = a + t$ so that we now have $\sum_{i=0}^{\infty} a_{i}t^{i}$. He then notes, "If we abandon the distinguished role played by $z$ and also allow a finite number of negative powers of $t$, we obtain a more general formulation." He then lets $z = P(t)$ and $u = Q(t)$ be any two series with only a finite number of negative powers of $t$ which, for a sufficiently small neighborhood of the origin, both converge and no two different values of $t$ in this neighborhood give the same pair of values $(z,u)$. This pair now defines a function element.

Main Question: I do not see how these two series can represent a function element as we defined before. I also cannot see the significance of having no two different values of $t$ giving the same pair $(z,u)$. I'm not exactly sure what makes this definition more general than the previous one, and I'm not exactly sure why we cannot introduce branch cuts by using the same analytic continuation techniques as before. I'm also not sure why replacing "$z-a$" with $t$ allows us to "abandon the distinguished role played by $z$.*

If someone could lead me in the right direction, I'd appreciate it!

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It appears to me that Weyl is trying to slowly build up to the idea of a Riemann surface. The pair $(P, Q)$ are a meromorphic parametrisation of a surface in $\mathbb{C}^2$. The connection with the earlier definition is obtained by taking $z = a + t$, and $u$ to be the power series, of course. – Zhen Lin Aug 7 '11 at 10:15
up vote 3 down vote accepted

Trying to learn about Riemann Surfaces from Weyl's book is a horrible idea. I know because I tried it myself. It's a good piece of history, but pedagogically terrible, and the definitions and terminology are very outdated. There are many good modern books on the topic. I would specifically recommend Miranda's "Algebraic Curves and Riemann Surfaces."

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As far as I am aware, the term function element is not in common use anymore. The notion of germs of holomorphic functions might be related.

Formally germs of holomorphic functions at a point $P$ (on either a Riemann surface, or some more genral context such as a complex manifold or complex analytic space) can be obtained from bituples $(f,U)$ of a neighbourhood $U$ of $P$ and a holomorphic function $f\in\mathcal O(U),$ where $(f,U)$ and $(g,V)$ define the same germ if and only if $f$ and $g$ agree on some neighbourhood $W\subseteq U\cap V$ of $P.$ The germ of $f\in\mathcal O(U)$ at a point $P\in U$ is often denoted $\gamma_P(f).$ If you want to be a bit more sophisticated, you can view germs of holomorphic functions at a point $P$ as elements of $\mathcal O_{P},$ the stalk at $P$ of the sheaf of holomorphic functions on the Riemann surface.

Germs naturally correspond to power serie (especially, a germ contains information about the local behaviour of a holomorphic function around a point, not merely its value at it). Given a germ $\xi\in\mathcal O_P,$ we may wonder what is the largest subset $U$ od the Riemann surface in question, such that there exists some $f\in\mathcal O(U)$ for which $\gamma_P(f)=\xi.$ There is no per se guarantee for such a function $f$ to exist nor, if it does, to be unique. Thus germs are the natural starting point for questions regarding analytic continuation. Intuitively, the germ $\gamma_P(f)$ is obtained by forgetting the analytic continuation which defines the function $f$ away from the point $P.$

I do not know if that was in any way helpful, but I suspect perhaps the notion of a germ grew out of Weyl's "function element".

As far as learning Riemann surface theory goes, I agree with Potato that learning from Weyl directly might not be the best idea. For modern textbook on the theory, I would like to recommend Forster's "Lectures on Riemann Surfaces" and the more analytically minded Varolin's "Riemann Surfaces By Way of Complex Geometry".

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