TITLE: Change of variables in line integral with abs. value
QUESTION [6 upvotes]: 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.

REPLY [0 votes]: $d|z|=|z+dz|-|z|=\sqrt{(x+dx)^2+(y+dy)^2}-\sqrt{x^2+y^2}=\frac{xdx+ydy}{\sqrt{x^2+y^2}}+o(dx,dy)\approx\frac{xdx+ydy}{\sqrt{x^2+y^2}}$
I don't think you can do some happy variable change in your problem, but maybe this helps
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