TITLE: Multidimensional complex integral of a holomorphic function with no poles
QUESTION [5 upvotes]: I am looking for a generalization of the Cauchy integral theorem. I know that there are generalizations of the Cauchy integral formula (eg the Bochner-Martinelli formula), but I do not know if this simplifies as I would hope. In single-variable complex analysis, we have the Cauchy integral theorem:
$$
\oint_{\gamma}f(z) = 0
$$
if $f(z)$ is a holomorphic function with no poles within the region that $\gamma$ encloses.
If $f(z_1\cdots,z_n)$ is a holomorphic function in $n$ complex variables with no poles within a domain $D$, is it true that
$$
\oint_{\partial D}f(z_1,\cdots,z_n) = 0
$$

REPLY [2 votes]: Following up on @mrf's answer: Another way to go is to generalize the Residue Theorem. If $f$ is meromorphic in a neighborhood of $a\in\Bbb C$, you can write 
$$\frac1{2\pi i}\int_{|z|=r} \frac{df}{f} = \text{Res}(f,a).$$
Generalizing to $n$ dimensions, you can consider $n+1$ holomorphic functions $f_1,\dots,f_n,g$ in a neighborhood of $a\in \Bbb C^n$ with $\{a\} = f_1^{-1}(0)\cap\dots\cap f_n^{-1}(0)$. Then, for small $r_1,\dots,r_n>0$, let $\gamma = f_1^{-1}(r_1)\cap\dots\cap f_n^{-1}(r_n)$; note that $\gamma$ is a smooth $n$-dimensional cycle. Let $\omega = \frac{g\,df_1\wedge\dots\wedge df_n}{f_1\cdots f_n}$. Then we have the Residue Theorem
$$\frac1{(2\pi i)^n}\int_\gamma \omega  = \text{Res}(\omega,a)\,,$$
provided we orient $\gamma$ correctly. 
Corresponding to the meromorphic $n$-form $\omega$, the Dolbeault isomorphism now gives a smooth $(n,n-1)$-form $\psi$ defined on a punctured neighborhood of $a$, and 
$$\text{Res}(\omega,a) = \int_{S^{2n-1}(\epsilon)} \psi$$
for small $\epsilon>0)$.
A very readable source, if you want to follow up on this, is the beginning of Chapter 5 of Griffiths and Harris.