I want to evaluate the integral $\int_{\mathbb{C}}f(|z|^2) dz d\overline z$. $f(|z|) = \exp(-|z|^2)$. I have never had to evaluate an integral of this type before, and nothing in my undergraduate or graduate literature mentions how to solve it. I tried to use an analogous method as change of variables from $\mathbb{R}^2$ to polar coordianates, the problem is that when I try and compute the Jacobian determinant I get the wrong expression. I consider the transformation $T(r,\theta) = (r e^{i\theta},r e^{-i\theta})$, and I try $$ |\det T'(r,\theta)| = \left| \begin{matrix} \frac{d z}{d r} & \frac{d z}{d \theta}\\ \frac{d \overline z}{d r} & \frac{d \overline z}{d \theta} \end{matrix} \right|= \left| \begin{matrix} e^{i \theta} & re^{i \theta} \\ e^{-i \theta} & -r e^{-i \theta} \end{matrix} \right| = |-ir - ir| = 2r $$ So I get the wrong answer $$ \int_{\mathbb{C}}f(|z|^2) dz d\overline z \stackrel{?}{=} \int_{\mathbb{R}^2}f(r^2)|\det T'(r,\theta)| dr d \theta = \int_0^{2 \pi} \int_0^\infty f(r^2) 2 r dr d\theta = 4 \pi \int_0^\infty f(r^2) r d\theta, $$ which is wrong according to my professor. My prof claims that the prefactor should be $2 \pi$, so I got a factor of 2 too many.
Can somebody show us how to do this correctly, and give some brief justification?