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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.

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If $\gamma$ is a path, then the composition $g\circ \gamma$ is also a path, right? $x$ and $y$ in your equation are the real and imaginary parts of the path. – Braindead Aug 3 '11 at 16:00

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