# connection between winding number and topological degree

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?

• I see no question here. I believe you're having trouble, which you state, but ... what do you actually want? – John Hughes Jul 1 '18 at 10:00
• Right, i couldn't figure out how to put the "?" above the "=". My question is, why the integral (the winding number) is the same as the sum of the winding numbers at the zeropoints of f – akwa Jul 1 '18 at 10:02
• "\stackrel{?}{=}" :) – Andrew Jul 1 '18 at 10:12
• @user573378 Please include the question in words into your post. The comments are mostly for recommendations on how to modify your post to make it clearer. – M. Winter Jul 1 '18 at 10:16
• Isn't it the residue theorem ? Or in this specific case also known as "the argument principle" (at least in french) ? – Max Jul 1 '18 at 10:37