Suppose I have a rational function of $k$ complex variables:
$$ R(x_1,...,x_k) = P(x_1,...,x_k)/Q(x_1,...,x_k) $$
where $P$ and $Q$ are polynomials. Now I'd like to compute the integral of this function over the product of disks, $D^k$, inside $\mathbb{C}^k$, ie:
$$ I = \frac{1}{(2 \pi i)^k} \int_{D^k} dx_1...dx_k R(x_i) $$ I'd like to use a multi-variable version of Cauchy's theorem to express this as a sum over residues.
To take a trivial example, if $R(x,y,z) = 1/(xyz)$, then there is a single contribution from $x=y=z=0$, and $I=1$. More generally, I suggest the following possible strategy:
- Find irreducible polynomial factors of $Q(x_i)$, ie:
$$ Q(x_i) = f_1(x_i) ... f_n(x_i) $$ Each of these contributes a component on which $Q(x_i)$ vanishes, leading to a codimension one subspace on which $R(x_i)$ has a pole.
- Generically I expect contributions from the intersection of $k$ components, which generically intersect at a discrete points. In a neighborhood of a point, $p$, where, say, $f_1$, ..., $f_k$ vanish, I believe we can make a local linear change of variables, $x_i \rightarrow t_i$, such that we may write:
$$ R = \hat{R}(t)/(t_1^{n_1} ... {t_k}^{n_k}) $$
where $\hat{R}(t_i)$ is holomorphic near $t_i=0$.
- Then the residue is given by taking suitable derivatives of $\hat{R}(t)$, and summing this over all such points, $p$.
My question is, is this procedure correct? I worry what happens if $R(x_i)$ is not sufficiently generic, eg, more than $k$ components intersect at a point. And finally, is there a more elegant way to extract this integral directly from $R(x_i)$, without having to go through this algorithm?
