Let $z_1,z_2 \in \mathbb{C}$ If a function $f(z_1,z_2)$ is holomorphic in some domain $\mathcal{D}$, how to evaluate the Cauchy integral
$$\iint_{\mathcal{D}}\frac{f(z_1,z_2)}{(v_1-z_1)(v_2-z_2)}dz_1dz_2$$
For normal Cauchy integral we can use residue theorem, so are there any similar theorems for multivariate case?
I am not sure if I know what you mean, but with the conditions of the Cauchy integral,
holomorphic in both arguments for $z_1\in U$ and $z_2\in V$ and with $\overline{\Delta_1}\subset{U}$ and $\overline{\Delta_2}\subset{V}$ using
$D:= \delta\Delta_1 \times \delta\Delta_2 \subset U\times V$, $\enspace$ I don't see a problem to write :
$\displaystyle \iint_{\delta\Delta_1 \times \delta\Delta_2}\frac{f(z_1,z_2)}{(v_1-z_1)(v_2-z_2)}dz_1dz_2=i2\pi\int_{\delta\Delta_2 }\frac{f(v_1,z_2)}{v_2-z_2}dz_2=-(2\pi)^2 f(v_1,v_2)$