Branch point theorem

Statement

Suppose $U\subset \mathbb {C}$ is an open subset, $z_{0}\in U$ is a point and $f:U\to \mathbb {C}$ is a holomorphic function. Suppose $n$ is the order of zero of the function $z\mapsto f(z)-f(z_{0})$ at $z_{0}$: in other words, the smallest positive $n$ such that $f^{(n)}(z_{0})\neq 0$. Assume $n$ is finite, i.e. $f$ is not constant around $z_{0}$.

Then, there exist radii $r_{1},r_{2}>0$ such that:

For any $w$ such that $|w-f(z_{0})|<r_{2}$, the set:

$\{z\in U\mid |z-z_{0}|<r_{1},f(z)=w\}$

has cardinality exactly $n$. In other words, $f$ is a $n$-to-one map around $z_{0}$.

Related facts

• Open mapping theorem, which can be viewed as a corollary of the branch point theorem.