Argument principle - Knowlpedia

Statement

Let $f$ be meromorphic on a domain containing a positively oriented simple closed contour $\gamma$ and its interior, with no zeros or poles on $\gamma$ . If $N$ and $P$ are the numbers of zeros and poles inside $\gamma$ , counted with multiplicity, then

\frac{1}{2\pi i}\int_\gamma\frac{f'(z)}{f(z)}\,dz=N-P.

Winding interpretation

The integral equals the winding number of the image contour $f\circ\gamma$ about $0$ . As $\gamma$ is traversed, the net change in a continuous choice of $\arg f$ is $2\pi(N-P)$ .

Residue proof

At a zero or pole $a$ , the logarithmic derivative and local factorization give

\operatorname{Res}\!\left(\frac{f'}f,a\right) =\operatorname{ord}_a(f).

The result follows directly from the residue theorem. Rouché's theorem and many root-counting methods are refinements of this mechanism.

References

Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw–Hill, 1979. Relevant: Chapter 5, §2.