Checking the value of an integral for all points in a neighbourhood

Question

I have this problem that says:

We have a bijective, holomorphic map $f$ on the region $\Omega$ and let $\overline{B(\alpha,\rho)}$ be a closed disk in $\Omega$. We are asked to check that

$$2\pi i f^{-1}=\oint_{S^{1}(\alpha, \rho)} dz[zf'(z)(f(z)-\nu)^{-1}]$$

for every $\nu \in f(B(\alpha, \rho))$.

Seeing that $2\pi i$ at the beginning of the expression, it makes me think that you have to use the residue theorem, but if $f^{-1}$ means the inverse function, well, we don't know it... and the thing that's on the right doesn't seem like the sum of residues either.

Thank you in advance.

Answer 1

Hint: You may write $\nu = f(z_0)$. Then you could e.g. consider what happens if you integrate the following function along the given contour: $$ \frac{z-z_0}{f(z)-f(z_0)} f'(z)$$