Question: I have been looking at the definition of a 'branch' of a complex function. The typical definition I am getting is as follows:

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> ...branches $f\_1(z)$, $f\_2(z)$, $...$ of $f(z)$ each defined as a single-valued continuous function throughout its range of definition. Each branch assumes one set of the function values of $f(z)$. (Korn & Korn, 2013; pg 193)
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(a similar definition is given [here](http://mathworld.wolfram.com/Branch.html))

The problem with these definition is that there is an ambiguity of what we call a branch, and by these definitions we should have an infinite number. e.g. we could have a branch defined for $0\le \theta \lt 2\pi$ and another for $0.001\le \theta \lt 2\pi+0.001$. 

My question is; is their a name for the set of branches $f\_i$ such that they don't overlap in this way?

Answer: 
Let $f:U\to \mathbb C$ and $g:V\to\mathbb C$ be analytic on connected open sets $U,V$. They are branches of the same function if there is a sequence of open connected subsets, $U=W\_1,W\_2,\dots,V=W\_n$ and analytic functions $f\_i:W\_i\to \mathbb C$ such that $f\_1=f,f\_n=g$ and $W\_{i}\cap W\_{i+1}$ is non-empty, and $f\_{i}(x)=f\_{i+1}(x)$ for all $x\in W\_{i}\cap W\_{i+1}$.

So, it can be seen as the transitive closure of the relationship on analytic functions $f,g$ defined by:

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> $fRg$ iff $\mathrm{domain }(f) \cap\mathrm{domain}(g)\neq \emptyset$ and $f(x)=g(x)$ for all $x\in \mathrm{domain }(f) \cap\mathrm{domain}(g)$.
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You can use this set to define something called the Riemann manifold of a function, $f$.

Given analytic $f$, we take $X$ to be the set of all pairs $(g,x)$ where $g$ is a branch of $f$ and $x\in\mathrm{domain}(g)$. We define a relationship $(g\_1,x)\sim (g\_2,y)$ if $x=y$ and $g\_1(z)=g\_2(z)$ for all $x$ where both are defined.

Then $M=X/\sim$ is the Riemann manifold of $f$. 

Then there is a natural map $M\to\mathbb C$ defined as $(g,x)\mapsto x$, and the function $(g,x)\mapsto g(x)$ which is the "lift" of $f$ from $\mathbb C$ to $M$.
Echoing @Andrew D. Hwang's good answer, I'd agree that a big part of the common problem in parsing discussion about "branches" is that the vocabulary to talk about many of these basic complex analysis notions is out-of-date and/or vague, even though the *ideas* go back at least to Riemann, c. 1850's.

Somewhat contrary to common textbook portrayals of "the principal branch of ...", this is very much just *convention*, and does not often express any genuine mathematical fact. As in the other answers, as one can say in various ways, one has some sort of *local* description of some holomorphic function(s), ambiguous up to ... sign? permutation of solutions of algebraic equation? linear combinations of solutions of ODE's... ? but/and we can specify *local* "solutions/sections/whatever" on sufficiently small neighborhoods away from some "bad points" ("ramified points"? "branch points"?) The question one may wish to address is about sticking together *local* solutions to give a (relatively) *global* solution on as large an open set as possible.

Naturally, already in easy examples, there is *not* a unique maximal such, despite a sort of traditional impulse to try to make it be so.

What we're doing is looking for "sections" of a *sheaf* (the aggregate of the local pieces and possible glue-ings together whether they match) on varying open subsets of $\mathbb C$. The seeming ambiguities are not really avoidable, since for a given open, the set/vectorspace/whatever of possible "sections on that open" is not often a single thing.

(Indeed, the various discrepancies about things fitting together produce various "cohomology theories": DeRham, Cech, and Grothendieck's understanding that many of these are derived functors of the "sections" functor, as in his Tohoku J. paper from long ago...)

Operationally, one point is to not (foolhardily) attempt to over-simplify things that do not admit much further simplification. There are complexities there that are genuine, not just issues with notation or terminology.