Proof of the Argument Principle

The argument principle connects the contour integral of the logarithmic derivative $f'/f$ to the number of zeros and poles of $f$ enclosed by the contour.

Statement

Theorem6.10Argument principle (restated)

Let $f$ be meromorphic inside and on a simple closed positively oriented contour $\gamma$ , with no zeros or poles on $\gamma$ . Then

$\frac{1}{2\pi i}\oint_\gamma \frac{f'(z)}{f(z)}\,dz = Z - P$

where $Z$ is the total number of zeros and $P$ the total number of poles of $f$ inside $\gamma$ , each counted with multiplicity.

Proof

Step 1: Local analysis near a zero.

If $f$ has a zero of order $m$ at $z_0$ , write $f(z) = (z-z_0)^m h(z)$ with $h(z_0) \neq 0$ and $h$ holomorphic. Then

$\frac{f'(z)}{f(z)} = \frac{m(z-z_0)^{m-1}h(z) + (z-z_0)^m h'(z)}{(z-z_0)^m h(z)} = \frac{m}{z-z_0} + \frac{h'(z)}{h(z)}.$

The term $h'/h$ is holomorphic near $z_0$ , so $\text{Res}(f'/f, z_0) = m$ .

Step 2: Local analysis near a pole.

If $f$ has a pole of order $n$ at $w_0$ , write $f(z) = (z-w_0)^{-n}k(z)$ with $k(w_0) \neq 0$ . Then

$\frac{f'(z)}{f(z)} = \frac{-n}{z-w_0} + \frac{k'(z)}{k(z)}.$

So $\text{Res}(f'/f, w_0) = -n$ .

Step 3: Apply the residue theorem.

The function $f'/f$ is holomorphic inside $\gamma$ except at the zeros and poles of $f$ . By the residue theorem:

$\frac{1}{2\pi i}\oint_\gamma \frac{f'(z)}{f(z)}\,dz = \sum_{\text{zeros}} m_j - \sum_{\text{poles}} n_k = Z - P. \qquad \blacksquare$

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Geometric Interpretation

RemarkWinding number interpretation

The integral $\frac{1}{2\pi i}\oint_\gamma \frac{f'}{f}\,dz$ equals the winding number of the curve $f(\gamma)$ around the origin. As $z$ traverses $\gamma$ once, $\arg f(z)$ changes by $2\pi(Z - P)$ .

This can be visualized: draw the image curve $w = f(z)$ as $z$ traces $\gamma$ . Count how many times this curve winds around $w = 0$ ; the answer is $Z - P$ .

Applications

ExampleCounting zeros of $e^z - 3z$

How many solutions does $e^z = 3z$ have in $|z| < 1$ ?

Set $f(z) = e^z - 3z$ . On $|z| = 1$ : $|e^z| \leq e^1 \approx 2.718$ and $|3z| = 3 > e$ . By Rouche with $g(z) = -3z$ dominating $e^z$ : the number of zeros of $e^z - 3z$ in $|z| < 1$ equals the number of zeros of $-3z$ , which is $1$ .

ExampleConnection to control theory

In control theory, the Nyquist stability criterion is a direct application of the argument principle. For a feedback system with open-loop transfer function $G(s)$ , the closed-loop system $1 + G(s)$ is stable if and only if the Nyquist plot of $G(s)$ (the image of the imaginary axis under $G$ ) winds around $-1$ exactly $P$ times counterclockwise, where $P$ is the number of unstable open-loop poles.

RemarkGeneralized argument principle

For any function $g$ holomorphic near the zeros and poles of $f$ :

$\frac{1}{2\pi i}\oint_\gamma g(z)\frac{f'(z)}{f(z)}\,dz = \sum_{\text{zeros}} m_j g(z_j) - \sum_{\text{poles}} n_k g(w_k).$

Setting $g \equiv 1$ recovers the basic argument principle. Setting $g(z) = z$ gives the sum of the zeros minus the sum of the poles.