complex analysis - Calculating integral using Cauchy integral formula in two variables

I want to compute the integral: $\iint_{\partial_0P}\frac{1}{1-4zw}dzdw$ (or any similar integral) using Cauchy integral formula for two complex variables over polydiscs. The distinguished boundary is given by: $\partial_0P={\{(z,w):|z|=1, |w|=1}\}$.

Nowhere online have I found an example of how to calculate such an integral. Would be really grateful if someone could show me how to do it, or give a link to a text with examples.

Answer

Usually, one evaluates such integrals by iterated integration. Sometimes it is considerably simpler to evaluate in a specific order, but here the situation is completely symmetric with respect to $z$ and $w$, so the order is irrelevant not only for the result but also for the way there. We evaluate the inner integral per Cauchy's integral formula/the residue theorem:

$$\int_{\lvert z\rvert = 1} \frac{dz}{1-4zw} = -\frac{1}{4w}\int_{\lvert z\rvert = 1} \frac{dz}{z-\frac{1}{4w}} = \frac{\pi}{2iw},$$

since we have $\lvert w\rvert = 1$ and hence the singularity $\frac{1}{4w}$ is enclosed by the unit circle. The outer integral becomes

$$\frac{\pi}{2i}\int_{\lvert w\rvert = 1} \frac{dw}{w} = \pi^2.$$