Let $\gamma : I \rightarrow \mathbb C$ be a path. Let $g: \mathbb C \rightarrow \mathbb C$ be a biholomorphic map. Let $f$ be a holomorphic function. Consider the integral
$ \int_{g\circ \gamma} f(g(z)) \mathrm d|g(z)| $.
What is a suitable change of variable formula in this case? My difficulty with the normal is with the absolute value sign with $|dz|$, which I interpret in the following way. Set $z = x+ iy$ with $x,y$ real,
$|dz| = \left( \left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2\right)^{1/2} dt$
where $t \in I$ parametrizes the path $\gamma$. I do not know how to derive a suitable change of variables formula with this setup and would be grateful for a reference with derivation.