A Polydisc of center $z^o=(z_1^o,\dots,z_n^o)\in\Bbb C^n$ and multiradius $r=(r_1,\dots,r_n)\in(\Bbb R^+)^n$ is defined as $$ P_{z^o,r}:=\prod_{j=1}^n\Delta_{z_j^o,r_j} $$ where $\Delta_{z_j^o,r_j}:=\{z_j\in\Bbb C\;:\;|z_j-z_j^o|<r_j\}$, for $j=1,\dots,n$.
Then we define $$ \partial_0 P:=\{z=(z_1,\dots,z_n)\in\Bbb C^n\;:\;|z_j-z_j^o|=r_j\;,\; j=1,\dots,n\} $$
Let now $f:\Bbb C^n\to\Bbb C$ be continous on $\bar P$, where $P$ is a polydisc; suppose $f$ then holomorphic on every $z_j$, when the other $n-1$ variables are fixed. Then in my book I read that Cauchy formula becomes $$ f(z)=(2\pi i)^{-n}\int_{\partial_0 P}\frac{f(\zeta)}{(\zeta_1-z_1)\cdots(\zeta_n-z_n)}\,d\zeta_1\wedge\cdots\wedge d\zeta_n $$ for every $z\in P$.
What $d\zeta_1\wedge\cdots\wedge d\zeta_n$ mean? Is the usual $d\zeta_1\cdots d\zeta_n$, or does it deal with differential forms/external products and similar stuff?
I would like to know what are the tools to study that will allow me to understand and work with the kind of integrals as above.
Can someone give me some examples of computation of such an integral? I really don't know where to start.
Thanks a lot.
Since $\partial_0P$ is a product of $n$ circles, you can just think of this, indeed, as an $n$-fold iterated integral such as you do with the usual one-dimensional Cauchy Integral Formula. That is, parametrizing each of the circles in the usual way as $\zeta_j = z_j+r_je^{i\theta_j}$, $j=1,\dots,n$, you can rewrite the integral as the iterated integral $$\frac1{(2\pi i)^n}\int_0^{2\pi}\dots\int_0^{2\pi} f\big(z_1+r_1e^{i\theta_1},\dots,z_n+r_ne^{i\theta_n}\big) d\theta_1\dots d\theta_n.$$
(That being said, if you're going to study several complex variables or complex geometry seriously, you absolutely need to learn differential forms.)