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I'm learning about winding numbers in the subject of topological insulators and I'm having problem proving something that will sort of, connect the winding number as it is defined in physics with how it is usually understood in mathematics (I think).

So, given a Hamiltonian in momentum space, $$\mathcal{H}=\begin{pmatrix} 0 & \mathcal{H}_{AB}(k)\\ \mathcal{H}_{AB}^\dagger(k) & 0 \end{pmatrix},$$ the winding number (in this physical model) is often defined as the number of revolutions of det$\mathcal{H}_{AB}(k)$ about zero. Where the determinant is some complex number, $$\text{det}\mathcal{H}_{AB}(k)=f(k)=R(k)e^{i\phi(k)}.$$ The number of revolutions is, $$\nu=-\frac{1}{2\pi}\big[\phi(\frac{\pi}{r})-\phi(-\frac{\pi}{r})\big],$$ which allows the winding number, $\nu$, to be rewritten as, $$\nu = -\frac{1}{2\pi}\int^\pi_{-\pi}\frac{d\phi(k)}{dk}dk.$$ I don't want to go into too much because it is really not related to the actual question but I can bring it to this form, $$\nu=-\frac{1}{2\pi i}\int^\pi_{-\pi} \frac{d}{dk} \ln f(k) dk=\boxed{-\frac{1}{2\pi i}\int_{-\pi}^\pi\frac{f'(k)}{f(k)}dk.} \;\;\;(1)$$ From this, I would like to prove that this can be understood with the argument principle, \begin{equation} \boxed{\frac{1}{2\pi i}\oint_C\frac{f'(z)}{f(z)}dz= (N_f-P_f).} \;\;\;(2) \end{equation} Now, I'm not sure if it is even possible, but I would like to derive (2) from (1). But I can't change the argument of $f(k)$ to be $z$ without changing the function. If I just change variables then $z=e^{ik\phi(k)}$ then $dz=-ir\phi'(k)e^{ik\phi(k)}dk$. This factor would surely be included and what remains would not be $f'/f$.

Question: Can this winding number be rewritten to be understood in terms of enclosed poles and zeros, or is it in a fundamental way different than the winding numbers of mathematics. Can I rewrite (1) to become (2)?

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