---
tag: FG5448NZ18
kind: statement
role: theorem
name: Cauchy integral formula on a polydisc
msc: [32A26, 32A10]
uses: [507C34W0ZT, DJBWP7D3M8, A6HN207Y2T, R49Q9M519Z, 0DG10TVYPN]
about: [507C34W0ZT, DJBWP7D3M8]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula
---

## Statement

Let $a \in \C^n$ and $r = (r_1, \dots, r_n)$ with each $r_j > 0$.
Let $P = P(a, r)$ be the [[polydisc|DJBWP7D3M8]] with centre $a$ and polyradius $r$, and let $T = T(a, r)$ be its [[distinguished boundary|A6HN207Y2T]].
Let $f$ be a [[holomorphic function|507C34W0ZT]] on an open set of $\C^n$ that contains the closed polydisc $\overline{P}$.
Then for every $z \in P$,
$$f(z) = \frac{1}{(2\pi i)^n} \int_{T} \frac{f(\zeta)}{(\zeta_1 - z_1) \cdots (\zeta_n - z_n)} \, d\zeta_1 \cdots d\zeta_n,$$
where the integral is the [[torus integral|R49Q9M519Z]] over $T$.

## Remarks

The proof applies the one-variable [[Cauchy integral formula|0DG10TVYPN]] in $z_1$, then in $z_2$ inside the integral, and so on, $n$ times.
Continuity of $f$ on $T$ allows the iterated integrals to be combined into one torus integral.
The formula holds under the weaker hypothesis that $f$ is continuous on $\overline{P}$ and holomorphic on $P$.
Expanding $1/(\zeta_j - z_j)$ in a geometric series in $(z_j - a_j)/(\zeta_j - a_j)$ gives the power series of $f$ at $a$, with coefficients
$$c_\alpha = \frac{1}{(2\pi i)^n} \int_T \frac{f(\zeta)}{(\zeta - a)^{\alpha + \mathbf{1}}} \, d\zeta, \qquad (\zeta - a)^{\alpha + \mathbf{1}} = \prod_{j} (\zeta_j - a_j)^{\alpha_j + 1}.$$
Example: for $f(z) = z_1 z_2$ and $P$ the unit polydisc of $\C^2$, at $z = 0$ the right side is $(2\pi i)^{-2} \int_T d\zeta_1 \, d\zeta_2 = 0 = f(0)$.
Only the values of $f$ on the $n$-dimensional torus $T$ are used, not on the whole $(2n-1)$-dimensional boundary of $P$.

## Explanations

### L1 human

The formula says that a holomorphic function on a polydisc is determined by its values on a thin torus.
Each value inside is an average of boundary values, weighted by a simple kernel.
It is the one-variable Cauchy formula used once in each coordinate.

### L2 human

This is the basic tool of local theory in several variables.
From it one gets that holomorphic functions are locally sums of convergent power series, the Cauchy estimates $\abs{c_\alpha} \le \max_T \abs{f} / r^\alpha$, and the infinite differentiability of holomorphic functions.
Since the kernel is a product of one-variable kernels, it is tied to the product shape of the polydisc.
On other domains one uses integral formulas with other kernels, such as the [[Bochner–Martinelli formula|YMHYAEJQ80]].