Rouché's theorem

Rouché's theorem is a statement about the location of the zeros of holomorphic functions, often used to estimate the number of zeros.

Statement

Let $U\subseteq \mathbb {C}$ be open, and let $\Gamma$ be a cycle in $U$, which is null-homologous in $U$ and winds around every point in its interior exactly once, i.e., $n(\Gamma ,z)\in {0,1}$ for each $z\in U$. Let $f,g\colon U\to \mathbb {C}$ be holomorphic functions such that

$|f(z)-g(z)|<|f(z)|,\qquad z\in \mathrm {trace} (\Gamma )$

holds. Then $f$ and $g$ have the same number of zeros (counted with multiplicity) in $I_{\Gamma }=\{z\in \mathbb {C} :n(\Gamma ,z)=1\}$.

Proof

For each $\lambda \in [0,1]$, consider the function $f_{\lambda }:=(1-\lambda )f+\lambda g=f+\lambda (g-f)$. Since

$|\lambda (g(z)-f(z))|=|\lambda ||g(z)-f(z)|<|f(z)|,\qquad z\in \mathrm {trace} (\Gamma )$,

$f_{\lambda }$ has no zeros on $\mathrm {trace} (\Gamma )$. Since $f_{\lambda }$ is holomorphic on $I_{\Gamma }$, it follows from the Zero and Pole counting integral that the number of zeros of $f_{\lambda }$ in $I_{\Gamma }$ is

${\frac {1}{2\pi i}}\int _{\Gamma }{\frac {f_{\lambda }'(z)}{f_{\lambda }(z)}},dz={\frac {1}{2\pi i}}\int _{\Gamma }{\frac {(1-\lambda )f'(z)+\lambda g'(z)}{(1-\lambda )f(z)+\lambda g(z)}},dz$.

This means it depends continuously on $\lambda$. A continuous $\mathbb {Z}$-valued function on $[0,1]$ is constant, so $f_{0}=f$ and $f_{1}=g$ have the same number of zeros in $I_{\Gamma }$.

Application

An application of Rouché's theorem is a proof of the Fundamental Theorem of Algebra: Let $p(z)=\sum _{k=0}^{n}a_{k}z^{k}\in \mathbb {C} [z]$ be a polynomial with $n>0$ and $a_{n}\neq 0$. The idea of the proof is to compare $p$ with $q(z)=a_{n}z^{n}$ (the number of zeros of $q$ is known). It holds that

$|p(z)-q(z)|=\left|\sum _{k=0}^{n-1}a_{k}z^{k}\right|\leq \sum _{k=0}^{n-1}|a_{k}||z|^{k}<|a_{n}||z^{n}|=|q(z)|$

for $|z|\geq R$ and a sufficiently large $R$. Hence, $p$ and $q$ have the same number of zeros, namely $n$, in $D_{R}(0)$.

See also

• Zero and Pole counting integral
• Convex Combination

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• Source: Satz von Rouché - URL:

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• Date: 01/07/2024