About the Riemann surface associated to an analytic germ I've taken a small course in Riemann surfaces, and there is one part that I still don't understand (and I've been unable to find a reference that explains this rigorously and in detail).
It is about the construction of a Riemann surface associated to an analytic germ.
By analytic germ we mean a couple $(z_0, (a_n)_{n \in \mathbb{N}})$ where $z_0$ and the $a_i$ are in the complex plane, and the series $\sum a_n (z-z_0)^n$ has strictly positive (but typically finite) radius of convergence.
Then (without getting into technical details) the Riemann surface associated to such a germ is the connected component of that germ in the space of all analytic germs, equipped with a certain topology. I get that it defines a Riemann surface.
I also understand "intuitively" what it does on the classic examples : for example, the Riemann surface associated to $(1,\log)$ is a sort of "spiral surface", since when we turn around zero, we add (if turning counter-clockwise) a $2 i \pi$ to the principal determination (so there are countably many sheaves).
In the case of the germ of the square root at $z_0=1$, it is a two-sheaves surface.
However I have no idea how one would go to determine this properly. In this example, I know that this surface is biholomorphic to $\mathbb{C}$ in the case of the log but I have no idea how to prove that.
Can anyone either give me a detailed reference or a sketch of the proof (but with all the key arguments) ? 
 A: What you are describing is more or less Weierstrass's construction of the Riemann surface of a multi-valued function. It is a very intuitive construction, but, as you point out, not convenient to do concrete calculations with. If you want to explicitly describe the Riemann surface obtained by analytic continuation of a given germ, there is not much choice other than to go back to the way in which the particular germ was obtained -  the behavior of a power series under analytic continuation can be very complicated and can't readily be predicted simply by looking at the coefficients $\{a_n\}$.
For example, if you want to show that $\log z$ increases by $2\pi i$ with every counter-clockwise turn of $z$ around the origin, it is much better to go back to the definition of $\log z$ than to start from the power series expansion of its principal branch around $z=1$, although a brute-force calculation is certainly possible. What is $\log z$? By definition, $\log z = \int_1^z dt/t$. Of course, this is only well-defined up to the choice of path from $1$ to $z$, which results in an ambiguity of $2\pi i n$ in the value of the function. What we see, in fact, is that $\log z$ is well-defined as a function on the universal cover of the punctured plane, as soon as we specify its value at any given point. This universal cover $E$ can be intuitively thought of as a helicoid, as you point out. And, as you say, it is biholomorphic to $\mathbb C$. The function $\log$, well-defined on $E$, is the isomorphism you are looking for.
A: The problem is that this is AFAIK treated in older texts, those which are still influenced by the first book by Weyl ("The Concept of a Riemann Surface").
You coud try e.g.


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*George Springer: "Introduction to Riemann Surfaces", chapter 3: "Riemann Surface of an Analytic Function"

