counting zeros (Rouché's theorem)

hbghlyj

H. Priestley Complex Analysis Exercise 18.10
Let $f$ be holomorphic inside and on $\gamma(a;r)$ and assume that $f(z)\ne0$ for $z\in\gamma(a;r)^*$. Find, in terms of the zeros of $f$,
\[\frac{1}{2\pi i}\int _{\gamma \left(a;r\right)}\frac{f'\left(z\right)}{f\left(z\right)z^m}dz\]

hbghlyj

By Argument principle, for $f,g\in H(G)$, we have$$\oint_\gamma g(z)\frac{f'(z)}{f(z)}~\mathrm dz=\sum_kg(z_k)n(\gamma,z_k)$$
Let $g(z)=\frac1{z^m}$, we get\[\frac{1}{2\pi i}\int_{\gamma(a;r)}\frac{f'\left(z\right)}{f\left(z\right)z^m}dz=\sum_k\frac1{z_k^m}\]