 Solution

For a closed piecewise smooth contour $\Gamma$ in a domain, null-homologous there and avoiding the poles of a meromorphic function $F$, the residue theorem iswhere $n(\Gamma ,a)=(2\pi i{)}^{-1}{\int }_{\Gamma }dz/(z-a)$ is the winding number and the sum runs over enclosed poles with their indices. Suppose now $f$ is meromorphic and has neither zeros nor poles on $\Gamma$. At a zero or pole $a$, write $f(z)=(z-a{)}^{m}h(z)$ with $h$ holomorphic and nonzero, where $m>0$ for a zero and $m<0$ for a pole. Thenso its residue is $m$. Substituting into the residue theorem and changing variables along the image contour gives the argument principle:For a positively oriented simple boundary, this is the number of zeros minus poles inside, counted with multiplicity. The winding-number form also applies to self-intersecting image paths.