Argument Principle

If $f(z)$ is Meromorphic in a region $R$ enclosed by a curve $\gamma$, let $N$ be the number of Complex Roots of $f(z)$ in $\gamma$, and $P$ be the number of Poles in $\gamma$, then

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

Defining $w \equiv f(z)$ and $\sigma\equiv f(\gamma)$ gives

$$N-P = {1\over 2\pi i} \int_\sigma {dw\over w}.$$

See also Variation of Argument

References

Duren, P.; Hengartner, W.; and Laugessen, R. S. ``The Argument Principle for Harmonic Functions.'' Math. Mag. 103, 411-415, 1996.