The Cauchy integral formula of complex functions of several complex variables I am reading Daniel Huybrechts' Complex Geometry. In chapter 1 he generalized the Cauchy integral formula as follow: 
$f(z)=(2\pi i)^{-n}\int_{T^{n}}\frac{f(m_{1},m_{2},\dots,m_{n})}{(m_{1}-z_{1})(m_{2}-z_{2})\dots(m_{n}-z_{n})}dm_{1}dm_{2}\dots dm_{n}$.
I am curious about what is the definition of this multi-dimensional complex integral? Indeed, I have check several textbooks on this topic, but none of them give a definition. And I can only intepret it as the iterated line intergral of single variable complex function.
 A: The $k$th coordinate projection can be considered as a function $m_k:\mathbb{C}^n\to\mathbb{C}$, and then $dm_k$ is just the differential of this complex-valued function.  Explicitly, if the real and imaginary parts of the $k$th coordinate are $x_k$ and $y_k$, $dm_k$ is the complex-valued $1$-form $dx_k+idy_k$.  Then $dm_1\dots dm_n$ is the complex-valued $n$-form which is the wedge product of all these $1$-forms, and so the integral in question is given by pulling this $n$-form back to the $n$-dimensional submanifold $T^n\subset\mathbb{C}^n$.
(When $n=1$, this coincides with the usual definition of path integrals with respect to $dz$, since pulling the $1$-form $dz$ along a path $t\mapsto \gamma(t)$ will give you $\gamma'(t)dt$.  As a result, in your example where you are integrating over $T^n$, the $n$-dimensional integral can also be obtained by iterating such $1$-dimensional integrals, by Fubini's theorem.  But the definition above using $n$-forms makes sense more generally for integrating on arbitrary smooth $n$-dimensional submanifolds of $\mathbb{C}^n$, even ones that are not just a product of paths on each coordinate so that there is not any obvious way to turn them into iterated integrals.)
