Definition of higher complex integral over polydisc

Question

I've come across the following generalization of Cauchy integral formula in higher dimensions.

Let $h \in \mathcal{O}(U)$, with $U \subset_{\text{open}}$ Let $(p_1, p_2) \in U$. There exists a bidisk $D(p_1, r_1) \times D(p_2, r_2) \subset U$, and we have\begin{align}{ f(p_1,p_2) = \frac {1}{(2\pi i)^2}}\oint\limits_{\partial D(p_1, r_1) \times \partial D(p_2, r_2)} {\frac{f(z,\xi)}{(z - p_1)(\xi - p_2)}} d\xi\,dz \end{align}

I understand that by applying Cauchy formula in one dimension, one has \begin{align*} f(p_1,p_2)={\frac {1}{(2\pi i)^2}}\oint \limits_{\partial D(p_1, r_1)_{\circlearrowleft}}{\frac {1}{z-p_1}} \Big(\oint_{\partial D(p_2, r_2)_{\circlearrowleft}} \frac{f(z,\xi)}{\xi - p_2} d\xi \Big)\,dz\\ \end{align*}

Nevertheless, I don't know how should one understand a double complex integral (i.e complex surface integral), as presented in the theorem. Most books I've encountered don't seem to go much over the details other that invoking compactedness and continuity with Fubini's theorem to go from the iterated line-integral to the formula. While I would sense this indeed is the right thing to do, it seems irrelevant as I don't know how to define such higher complex integral.

Answer 1

Let's stay in dimension $n=2$ as you are and let me rename the variables to $z_1$ and $z_2$. The so-called "distinguished boundary" of the disk is the 2 real dimensional submanifold $S^1 \times S^1$ (a torus) (product of the two boundaries of the discs). When integrating over this, you have to use something like differential forms, or you just think of it as an iterated integral as it is a product manifold. The differential form you integrate over it is $dz_1 \wedge dz_2$, where $z_j = x_j + i \, y_j$, so $dz_j = dx_j +i \, dy_j$. These are one forms, so $dz_1 \wedge dz_2$ is a 2 form so integrates over a 2 real dimensional surface. I would suggest a book on differential forms, such as Spivak's calculus on manifolds or similar.

Now the tricky bit for complex analysis is the orientation in higher dimension. There is a standard orientation of $\mathbb C$: if $z= x+i \,y$, then the orientation is the ordering $(x,y)$. But in $\mathbb C^2$, we have two possibilities $(x_1,y_1,x_2,y_2)$ or $(x_1,x_2,y_1,y_2)$. Neither is totally standard, the trick is to just pick one and stay consistent in your calculations.

The distinguished boundary is then simply oriented in such a way so that the Cauchy formula comes with a plus sign, not a minus sign.

However, if the worst thing you will integrate is the distinguished boundary of a polydisc, such as in Cauchy's theorem it is entirely fine to just use the "iterated integral" to think of what it is. I mean underneath, whenever you integrate over a submanifold, you in the end write the overall integral as a bunch of iterated integrals using something like a partition of unity. For the torus we are lucky that we can just write it as a single rather simple iterated integral. If you don't like the path integral there, just think of the arguments $\theta_1$ and $\theta_2$, and write the integral with respect to $d\theta_1$ and $d\theta_2$, which are then just two simple calculus integrals from $0$ to $2\pi$. In the end, all these integrals are computed by writing a bunch of Calculus I integrals.