Suppose that $f$ is holomorphic in a disk $D$ without zeroes on the boundary $\partial D$. Denote $S_k(f) = \frac{1}{2\pi i}\int\limits_{\partial D} \frac{f'}{f}z^k dz$. How can I show that $S_k(f) = \sum\limits_i \alpha_i^k$, where $\alpha_i$ are zeroes of $f$ counted with multiplicites?
It looks like a corollary from Cauchy's integral formula.