Let's say $C$ is a simple closed curve in the complex plane and $f(z)$ is holomorphic and doesn't vanish on $C$.
According to wikipedia, one can make the following change of variables: $$\omega = f\left( z \right)$$
and get: $$\oint\limits_C {{{f'\left( z \right)} \over {f\left( z \right)}}dz} = \oint\limits_{f\left( C \right)} {{1 \over \omega }d\omega } $$
I would like to inquire about what theorem was used to justify this change.
I've searched online and I've found several mentions of the change of variables theorem for complex integrals but in all of them (for example here) it was required that $\omega = f\left( z \right)$ will be a biholorphic mapping.
So my question is, how can the change of variables be rigorously justified?
Let $[a,b]$ be the domain of $C$. Then\begin{align}\oint_{f(C)}\frac1\omega\,\mathrm d\omega&=\int_a^b\frac{(f\circ C)'(t)}{(f\circ C)(t)}\,\mathrm dt\\&=\int_a^b\frac{f'\bigl(C(t)\bigr)C'(t)}{f\bigl(C(t)\bigr)}\,\mathrm dt\\&=\oint_C\frac{f'(z)}{f(z)}\,\mathrm dz.\end{align}