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

$|f(z)-g(z)|<|f(z)|,z\in \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(\mathrm{\Gamma })$

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

Proof

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

$|\lambda (g(z)-f(z))|=|\lambda ||g(z)-f(z)|<|f(z)|,z\in \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(\mathrm{\Gamma })$,

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

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

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 $I_{\mathrm{\Gamma }}$.

Application

An application of Rouché's theorem is a proof of the Fundamental Theorem of Algebra: Let $p(z)=\sum _{k=0}^n{a}_k{z}^k\in ℂ[z]$ be a polynomial with $n>0$ and $a_n\ne 0$. 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

$|p(z)-q(z)|=\left|\sum _{k=0}^{n-1}{a}_k{z}^k\right|\le \sum _{k=0}^{n-1}|a_k||z{|}^k<|a_n||z^n|=|q(z)|$

for $|z|\ge R$ 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

Page Information

You can display this page as Wiki2Reveal slides

Wiki2Reveal

The Wiki2Reveal slides were created for the Complex Analysis ' and the Link for the Wiki2Reveal Slides was created with the link generator.

• This page is designed as a PanDocElectron-SLIDE document type.
• Source: Wikiversity https://en.wikiversity.org/wiki/Rouch%C3%A9's%20theorem
• see Wiki2Reveal for the functionality of Wiki2Reveal.

Translation and Version Control

This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

• Source: Satz von Rouché - URL:

https://de.wikiversity.org/wiki/Satz_von_Rouché

• Date: 01/07/2024

de:Satz von Rouché