Argument Principle & Rouché

Count zeros minus poles by contour integral; locate zeros with Rouché.

The idea

Theorem (Argument principle).

Let $\gamma$ be a positively oriented simple closed contour and let $f$ be meromorphic on an open set containing $\gamma$ and its interior — holomorphic there apart from isolated poles — with no zero and no pole of $f$ on $\gamma$ itself. Then $\frac{1}{2\pi i}\oint_{\gamma} \frac{f'(z)}{f(z)}\,dz = Z - P,$ where $Z$ and $P$ are the numbers of zeros and of poles of $f$ inside $\gamma$, each counted with its multiplicity.

The argument principle counts the zeros and poles of a function inside a closed contour by a contour integral, without locating any of them. Since $f'/f$ is the derivative of $\log f$, and the $\log|f|$ part returns to its starting value after a loop, the integral records the total change in the argument of $f$: the number of times the image curve $f(\gamma)$ winds around the origin. This reading gives the principle its name, and it is why poles enter with a minus sign: circling a pole turns the argument of $f$ the opposite way from circling a zero.

Theorem (Rouché's theorem).

Let $f$ and $g$ be holomorphic on an open set containing a simple closed contour $\gamma$ and its interior. If $|f(z) - g(z)| < |g(z)|$ at every point of $\gamma$, then $f$ and $g$ have the same number of zeros inside $\gamma$, counted with multiplicity.

Rouché's theorem turns the count into a tool: a perturbation smaller than $|g|$ on the contour cannot drag the image curve across the origin, so it cannot change the winding.

Ways to work on it

• Walkthrough. The argument principle: a contour integral of f'/f counts zeros minus poles.
• Practice. Count zeros in the unit disk with Rouché's theorem.
• Hardest. Count zeros in an annulus by a two-circle Rouché argument.

Not sure where to start? Take the ten-question placement test.