Why is the word "branch" used in complex analysis?

Question

In complex analysis the term branch is used to designate a holomorphic function defined on a region. The most prominent example is the principal logarithm, which is a branch of the logarithm.

Does anybody know the history of why the word branch used?

I'm guessing branch, tree, decisions, choice.

Answer 1

Riemann introduced the term in his paper "Theory of Abelian Functions" ("Theorie der Abel'schen Functionen", Journal für die reine und angewandte Mathematik, vol. 54. pp.101–155. 1857). After discussing the analytic continuation of $\log(x-a)$ around the point $a$, with the change of value $2\pi i$, he writes:

It is convenient to designate the different continuations of one function into the same part of the $z$-plane, branches of this function, and to call a point, around which a branch of a function continues into another, a branch point of the function. Where no branch takes place, the function is called single-valued or monodromic.

Zur bequemeren Bezeichnung dieser Verhältnisse sollen die verschiedenen Fortsetzungen einer Function für denselben Theil der $z$-Ebene Zweige dieser Function genannt werden und ein Punkt, um welchen sich ein Zweig einer Function in einen andern fortsetzt, eine Verzweigungsstelle dieser Function; wo keine Verzweigung stattfindet, heisst die Function einändrig oder monodrom.

In his doctoral thesis, there is a single occurrence of the term "branch" ("Zweig"), in the compound word "branch point" ("Zweigepunkt"). The term is not defined. However, the concept occurs earlier, where it is called a "winding point" ("Windungspunkt"). Riemann has already introduced his notion of a multiple-sheeted covering of a part of the complex plane. He discusses the situation where one returns to the same sheet of the covering surface after $m$ circuits around a point $\sigma$. He then writes:

The point circling $\sigma$ then returns to the same surface segment after every $m$ circuits and is limited to $m$ of the surface segments lying over each other, which are joined by a single point above $\sigma$. We call this point a winding point of order $(m-1)$ of the surface $T$.

Der um $\sigma$ sich bewegende Punkt kommt alsdann nach je $m$ Umläufen in denselben Flächentheil zurück und ist auf $m$ der auf einander liegenden Flächentheile eingeschränkt, welche sich über $\sigma$ zu einem einzigen Punkte vereinigen. Wir nennen diesen Punkt einen Windungspunkt $(m-1)$ter Ordnung der Fläche $T$.

While Riemann doesn't provide an explanation for the term "branch", I think the term "branch point" has a pretty intuitive signficance: several parts of the surface "branch out" from the branch point.