I'm writing a thesis on the topological degree (mapping or Brower degree), and I'm having trouble with the equation, that links the mapping degree to the winding number in $\mathbb{R}^2$(marked below by '?').
$$v_{f \circ \gamma}(0)=\frac{1}{2 \pi i} \int_\gamma \frac{f'(z)}{f(z)} dz \stackrel{?}{=} \sum_{i=1}^k v_\gamma (z_i) \alpha_i=\operatorname{deg}(f,U_1(0),0)$$
Here $z_i$ are the roots of $f$, $\alpha_i$ their multiplicity, $\gamma$ is the the unit circle.
My question is: Why is the winding number of $f \circ \gamma$ at $0$ the same as the sum of the winding numbers of $\gamma$ around the zeropoints of $f$ times their multiplicities?