As far as I know, it is considered to be a "fact" that by the Generalized Binomial Theorem, the complex function $\sqrt{1 + z}$ has the following Taylor expansion at $z = 0$: $$\sqrt{1 + z} = \sum_{n \geq 0} {1/2 \choose n} z^n.$$ This is the usual answer obtained using a computer algebra system, and used in combinatorics as a generating function for $\sqrt{1 + z}$ (in analytic combinatorics, generating functions are often viewed as analytic objects, so the connection with complex analysis seems relevant to me).
However, one point seems slightly irritating to me: $\sqrt{1+z}$ is a multifunction, and the Taylor expansion above is actually only a Taylor expansion for one particular branch of this multifunction. This makes me to feel slightly uncomfortable when reading about the Taylor expansion for $\sqrt{1 + z}$.
I understand that only the series above is of combinatorial significance, as the coefficients need to be nonnegative integers. However, things seem to get more complicated for other algebraic functions, or possibly for functions involving irrational powers (which, if I understand the underlying mathematics properly, should have infinitely many branches). For instance, the book of Flajolet and Sedgewick makes extensive use of multifunctions (in particular, their "standard scale" is entirely made up of multifunctions) and I simply cannot see why it is OK to use the series for one of their branches only.
For this reason, I would like to ask if there is any unifying formal argument, which makes writing about the Taylor expansions of complex multifunctions rigorous, at least in the combinatorial context. Thank you in advance.