Rouche’s Theorem and Applications

This blog post is on Rouche’s Theorem and some applications, namely counting the number of zeroes in an annulus, and the fundamental theorem of algebra.

Rouche’s Theorem: Let $f(z)$, $g(z)$ be holomorphic inside and on a simple closed contour $K$, such that $|g(z)|<|f(z)|$ on $K$. Then $f$ and $f+g$ have the same number of zeroes (counting multiplicities) inside $K$.

Rouche’s Theorem is useful for scenarios like this: Determine the number of zeroes, counting multiplicities, of the polynomial $f(z)=2z^5-6z^2-z+1=0$ in the annulus $1\leq |z|\leq 2$.

Solution:

Let $K_1$ be the unit circle $|z|=1$. We have

\begin{aligned}|2z^5-z+1|&\leq |2z^5|+|z|+|1|\\ &=2+1+1\\ &=4\\ &<6\\ &=|-6z^2| \end{aligned}

on $K_1$.

Since $-6z^2$ has 2 zeroes in $K_1$, therefore $f$ has 2 zeroes inside $K_1$, by Rouche’s Theorem.

Let $K_2$ be the circle $|z|=2$

\begin{aligned} |-6z^2-z+1|&\leq |-6z^2|+|-z|+|1|\\ &=6(2^2)+2+1\\ &=27\\ &<64\\ &=|2z^5| \end{aligned}

on $K_2$. Therefore $f$ has 5 zeroes inside $K_2$.

Therefore $f$ has 5-2=3 zeroes inside the annulus.

We do a computer check using Wolfram Alpha (http://www.wolframalpha.com/input/?i=2z%5E5-6z%5E2-z%2B1%3D0). The moduli of the five roots are (to 3 significant figures): 0.489, 0.335, 1.46, 1.45, 1.45. This confirms that 3 of the zeroes are in the given annulus.

Fundamental Theorem of Algebra Using Rouche’s Theorem

Rouche’s Theorem provides a rather short proof of the Fundamental Theorem of Algebra: Every degree n polynomial with complex coefficients has exactly n roots, counting multiplicities.

Proof: Let $f(z)=a_0+a_1z+a_2z^2+\dots+a_nz^n$. Chose $R\gg 1$ sufficiently large so that on the circle $|z|=R$,

\begin{aligned} |a_0+a_1z+a_2z^2+\dots+a_{n-1}z^{n-1}|&\leq|a_0|+|a_1|R+|a_2|R^2+\dots+|a_{n-1}|R^{n-1}\\ &<(\sum_{i=0}^{n-1}|a_i|)R^{n-1}\\ &<|a_n|R^n\\ &=|a_nz^n| \end{aligned}

Since $a_nz^n$ has $n$ roots inside the circle, $f$ also has $n$ roots in the circle, by Rouche’s Theorem. Since $R$ can be arbitrarily large, this proves the Fundamental Theorem of Algebra.