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Suppose $f$ is an analytic function defined on the unit disk, $D$. I want to evaluate

$\int_{D} f(\omega) dA(\omega)$

using a change of variable. Suppose $\phi$ is a conformal map of the $D$ onto itself.

Does

$\int_{D} f(\omega) dA(\omega) = \int_{D} f(\phi(z)) |\phi^{'}(z)|^{2} dA(z) $?,

where $\phi^{'}$ is the derivative of $\phi$.

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1 Answer 1

up vote 3 down vote accepted

Because $|\phi'(z)|^2$ is the determinant of the Jacobian of $\phi$, this follows from the substitution formula for open sets in $\mathbb{R}^n$, as seen for example in this Wikipedia article, which contains further references for the more general result. Technically, to apply the result directly you would break up each side into real and imaginary parts.

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