Rouche's theorem

Statement

Let $U\subset \mathbb{C}$ be an open subset, and let $f,g$ be holomorphic functions on $U$. Suppose $V$ is a subset of $U$ such that the boundary of $V$ lies completely inside $U$, and is piecewise $C^1$. Further, suppose we have:

$|g(z)|<|f(z)|\mathrm{\forall }z\in \partial V$

Then, $f$ and $f+g$ have the same number of zeros (counted with multiplicity) in $V$.

Facts used

• Homotopy invariance formulation of Cauchy's theorem
• Argument principle