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!