In one complex dimension, the residue theorem states that the integral of a meromorphic function over the boundary of some domain $R$ is equal to the sum of residues of all poles inside that domain. What is the higher dimensional version of this statement?
In particular, given a region $R$ in $\mathbb{C}^n$, does there exist a natural contour such that the integral over this contour gives the sum of all residues inside the region? For $n>1$ it cannot be the boundary of $R$, since this has real dimension $2n-1$, while the contour should have dimension $n$. When $R$ is a product of the form $R_1 \times ... \times R_n$, where each $R_i$ is contained in a $\mathbb{C}$ factor, the answer seems clear, but is there a more general prescription?
As a side question, are there any useful techniques for visualizing higher dimensional complex contours? Even for $n=2$, there are already four real dimensions, and it seems difficult to draw any useful pictures.
