I have a question in mind and I would appreciate your help. Usually in complex analysis we consider integrals of the form $\int_\gamma f(z) dz$ where $\gamma $ is a contour and $f:\mathbb{C}\rightarrow\mathbb{C}$.
What if $g:\mathbb{C}^2\rightarrow\mathbb{C}$, and we want to find the integral $\int_U g(z_1,z_2) \;dz_1dz_2$ where $U\subset\mathbb{C}^2$ is an open subset? How can we evaluate this integral? Can we apply for example the change of variables formula in the same way we apply it for $\mathbb{R^n}$?
For example: $g(z_1, z_2)=\frac{1}{z_1 (z_1+z_2)}.$
Thank you!