Argument Principle and Rouche's Theorem

The argument principle relates the number of zeros and poles of a meromorphic function inside a contour to a contour integral. Rouche's theorem, a consequence, is a powerful tool for locating zeros of holomorphic functions.

The Argument Principle

Theorem6.4Argument principle

Let $f$ be meromorphic inside and on a simple closed contour $\gamma$ (positively oriented), with zeros $z_1, \ldots, z_p$ of orders $m_1, \ldots, m_p$ and poles $w_1, \ldots, w_q$ of orders $n_1, \ldots, n_q$ (none on $\gamma$ ). Then

$\frac{1}{2\pi i}\oint_\gamma \frac{f'(z)}{f(z)}\,dz = \sum_{j=1}^p m_j - \sum_{k=1}^q n_k = N - P$

where $N$ is the number of zeros and $P$ the number of poles, counted with multiplicity.

RemarkGeometric interpretation

The integral $\frac{1}{2\pi i}\oint_\gamma \frac{f'}{f}\,dz$ equals the winding number of the image curve $f \circ \gamma$ around the origin. As $z$ traverses $\gamma$ , the argument of $f(z)$ changes by $2\pi(N - P)$ . This is why the result is called the "argument principle."

ExampleCounting zeros

How many zeros does $f(z) = z^5 + 3z + 1$ have inside $|z| = 1$ ?

On $|z| = 1$ : $|3z| = 3 > |z^5 + 1| \leq 2$ . By Rouche's theorem (below), $f$ and $3z$ have the same number of zeros inside $|z| = 1$ , which is $1$ .

Alternatively, compute $\frac{1}{2\pi i}\oint_{|z|=1}\frac{5z^4+3}{z^5+3z+1}\,dz$ numerically.

Rouche's Theorem

Theorem6.5Rouche's theorem

Let $f$ and $g$ be holomorphic inside and on a simple closed contour $\gamma$ . If $|f(z) - g(z)| < |f(z)|$ for all $z \in \gamma$ , then $f$ and $g$ have the same number of zeros (counted with multiplicity) inside $\gamma$ .

Proof

The condition ensures $g/f = 1 + (g-f)/f$ has image inside the disk $|w - 1| < 1$ on $\gamma$ . Therefore $g/f$ has winding number $0$ around the origin on $\gamma$ , so

$\frac{1}{2\pi i}\oint_\gamma \frac{(g/f)'}{g/f}\,dz = 0.$

This equals (zeros of $g$ minus poles of $g/f$ ) = (zeros of $g$ ) - (zeros of $f$ ) inside $\gamma$ . $\blacksquare$

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Applications of Rouche's Theorem

ExampleProof of the Fundamental Theorem of Algebra via Rouche

Let $p(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_0$ . On $|z| = R$ with $R$ large enough:

$|p(z) - z^n| = |a_{n-1}z^{n-1} + \cdots + a_0| \leq (|a_{n-1}|R^{n-1} + \cdots + |a_0|) < R^n = |z^n|$

for sufficiently large $R$ . By Rouche's theorem, $p$ and $z^n$ have the same number of zeros inside $|z| = R$ , namely $n$ .

ExampleStability of zeros under perturbation

The polynomial $p(z) = z^4 + 6z + 3$ has exactly one zero in $|z| < 1$ . On $|z| = 1$ , compare with $g(z) = 6z$ :

$|p(z) - 6z| = |z^4 + 3| \leq |z^4| + 3 = 4 < 6 = |6z|.$

So $p$ has the same number of zeros as $6z$ in $|z| < 1$ , which is exactly one.

On $|z| = 2$ , compare with $h(z) = z^4$ : $|p(z) - z^4| = |6z + 3| \leq 15 < 16 = |z^4|$ . So $p$ has $4$ zeros in $|z| < 2$ . Therefore $p$ has $3$ zeros in the annulus $1 \leq |z| < 2$ .

Counting Formula

Definition6.4Counting zeros and poles

For a meromorphic function $f$ and any function $g$ holomorphic near the zeros and poles of $f$ inside $\gamma$ :

$\frac{1}{2\pi i}\oint_\gamma g(z)\frac{f'(z)}{f(z)}\,dz = \sum_{j=1}^p m_j g(z_j) - \sum_{k=1}^q n_k g(w_k)$

where $z_j$ are zeros and $w_k$ are poles. Taking $g(z) = z$ gives the sum of the zeros minus the sum of the poles (with multiplicity).

RemarkApplications to locating zeros

The counting formula with $g(z) = z$ computes the center of mass of zeros. Combined with Rouche's theorem, this can precisely locate zeros of polynomials and entire functions, making it essential in numerical analysis and control theory.