Definition of a branch of a complex function?

Question

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

...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)

(a similar definition is given here)

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 1

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:

$fRg$ iff $\mathrm{domain }(f) \cap\mathrm{domain}(g)\neq \emptyset$ and $f(x)=g(x)$ for all $x\in \mathrm{domain }(f) \cap\mathrm{domain}(g)$.

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 $M$.

Answer 2

"Branch" is a slightly tricky term to pin down. Literally, a branch of a function is just a holomorphic function in some non-empty open subset of the complex plane, or a continuous extension to the closure.

That alone, however, misses a substantial cultural subtext: If $f$ is holomorphic, a branch of inverse is a left inverse defined on some open subset $V = f(U)$ of the image of $f$, namely, a holomorphic bijection $g:V \to U$ such that $(g \circ f)(z) = z$ for all $z$ in $U$.

For instance:

• A branch of logarithm is a left inverse of the complex exponential. Customarily, one fixes an even integer $k$ and takes $U$ to be the set of $z$ whose imaginary part is between $(k - 1)\pi i$ and $(k + 1)\pi i$, and $V$ to be the slit plane, with the non-positive reals removed.

• A branch of square root is a left inverse of the complex squaring function. Customarily, one takes $U$ to be either the right or left open half-plane, and $V$ to be the slit plane.

A bit more generally, if $P(z, w) = 0$ is an analytic relation, a branch of $w$ is a holomorphic function $f$ defined in some open set $U$ satisfying $P(z, f(z)) = 0$ for all $z$ in $U$.

For instance:

• If $P(z, w) = z - \exp w$, then a branch of $w$ is a branch of $\log$.

• If $P(z, w) = z - w^{2}$, a branch of $w$ is a branch of square root.

• If $P(z, w) = z^{2} + w^{2} - 1$, a branch of $w = \sqrt{1 - z^{2}}$ is an open set $U \subset \mathbf{C}$ not containing $\pm 1$ and a holomorphic function $f:U \to \mathbf{C}$ satisfying $z^{2} + f(z)^{2} - 1 = 0$ for all $z$ in $U$.

Answer 3

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.