Question

It seems that there exists ring structure on all multi-valued holomorphic functions on a punctured disc. Can someone explain the formal definition of multi-valued holomorphic function?

I only know some examples which are multi-valued, for example $x^\lambda$, $log x$.

Edit: My motivation to ask this question, is to understand the mutilvalued holomorphic solution of system of analytic differential equations. For example the solution of $x\partial_x f=\lambda f$ is $x^\lambda$

Answer 1

If $U$ is an open (connected...) subset in $\mathbb C$, a multivalued holomorphic function on $U$ is an holomorphic function on $\tilde U$, the universal cover of $U$, endowed with its natural structure of complex curve; if $\pi:\tilde U\to U$ is the covering map, then univalued holomorphic functions are, in this picture, those of the form $f\circ\pi$ with $f:U\to\mathbb C$ an holomorphic function.

In particular, such multivalued holomorphic functions more or less obviously form a ring.

Answer 2

If you choose a basepoint of the disc and a local section there, you can employ analytic continuation to interpret a multivalued holomorphic function as a holomorphic function on the universal cover of the disc, i.e., (some open subset of) the complex upper half plane. If you have a formula in terms of the coordinate $x$, you can replace all appearances of $x$ with $e^{2 \pi i z}$. For example, $x^\lambda$ on the disc becomes $e^{2 \pi i \lambda z}$ on the upper half plane. You will end up with an ambiguity of lift given by maps $z \mapsto z+n$ for integers $n$.

Edit: I just realized that I hadn't answered the title question. We define a multivalued holomorphic function on a pointed connected complex manifold $(X,x)$ (such as an open punctured disc with a distinguished point) to be a regular function on the pointed universal covering space $(\widetilde{X},\widetilde{x})$. The set of multivalued functions then forms an augmented commutative ring under pointwise operations. This definition generalizes to pointed connected complex analytic spaces.

You might ask: What if I don't want a basepoint? In that case, "universal covers" are not canonical (as Theo mentioned in the comments to Mariano's answer), and even if you fix a cover $\widetilde{X} \to X$, there is no canonical choice of lift of a locally defined function to $\widetilde{X}$. Often in practice, you will not see a basepoint specified, but you might find people using the equivalent technique of choosing a preferred simply connected neighborhood (e.g., the complement of negative reals in $\mathbb{C} \setminus \{0\}$ when working with logarithms). In this case, it is essential that you specify a distinguished preimage of that neighborhood in $\widetilde{X}$ in order to get a well-defined ring structure on locally defined functions.

Answer 3

It seems this notion causes some confusion not only on me. Let me give my own definition, by taking into account with yours.

Let $X$ be an analytic manifold, and let $\pi:\tilde{X}\mapsto X$ be some universal covering of $X$.　A multivalued holomorphic function on $X$ is a holomorphic function on $U$, where $U$ is some contractible open subset of $X$, if the function $\pi\circ f$ on some component $\mathcal{C}$ of $\pi^{-1}(U)$ can be analytically extended to $\tilde{X}$. Denote this function by $\tilde{f} _\mathcal{C}$

Two multivalued holomorphic functions $(f,U)$ and $(g,V)$ are equivalent, if there exists some component $\mathcal{C}$ of $\pi^{-1}(U)$ and $\mathcal{D}$ of $\pi^{-1}(V)$, such that $\tilde{f}_\mathcal{C} = \tilde{g}_\mathcal{D}$.

This definition doesn't depend on the choice of universal covering.

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Edit:

Here is a more rigorous definition：

Definition. A multivalued holomorphic function on $X$ based at $x$ is a function $f$ in $\mathcal{O}_x$, such that one component of $\pi^* f$ can be analytically extended to $\tilde{X}$, where $\pi: \tilde{X}\mapsto X$ is some universal covering.

Denote the ring of multivalued holomorphic functions on $X$ based at $x$ as $\tilde{\mathcal{O}} _x$, which is a subring of $\mathcal{O}_x$. Any loop based at $x$ gives an action on $\tilde{\mathcal{O}} _x$, which is essentially the monodromy.

Claim. $\tilde{\mathcal{O}} _x$ is locally free $\mathcal{O}(X)$-module with rank the order of fundemental group.

It is essentially equivalent to the defintion that a mutilvalued fundtion is defined as function on covering space. But i feel it is more intuitive and doesen't depend on the choice of covering space.

Under this definition, I can make sense of the solution of mulivalued function of some analytic differential equation, which is actually my original motivation to ask this question.