Branch point theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

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 $f^{(n)}(z_0)\ne 0$. 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.