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  "body": "\n## Statement\n\nLet $a \\in \\C^n$ and $r = (r_1, \\dots, r_n)$ with each $r_j \u003e 0$.\nLet $P = P(a, r)$ be the [[polydisc|DJBWP7D3M8]] with centre $a$ and polyradius $r$, and let $T = T(a, r)$ be its [[distinguished boundary|A6HN207Y2T]].\nLet $f$ be a [[holomorphic function|507C34W0ZT]] on an open set of $\\C^n$ that contains the closed polydisc $\\overline{P}$.\nThen for every $z \\in P$,\n$$f(z) = \\frac{1}{(2\\pi i)^n} \\int_{T} \\frac{f(\\zeta)}{(\\zeta_1 - z_1) \\cdots (\\zeta_n - z_n)} \\, d\\zeta_1 \\cdots d\\zeta_n,$$\nwhere the integral is the [[torus integral|R49Q9M519Z]] over $T$.\n\n## Remarks\n\nThe proof applies the one-variable [[Cauchy integral formula|0DG10TVYPN]] in $z_1$, then in $z_2$ inside the integral, and so on, $n$ times.\nContinuity of $f$ on $T$ allows the iterated integrals to be combined into one torus integral.\nThe formula holds under the weaker hypothesis that $f$ is continuous on $\\overline{P}$ and holomorphic on $P$.\nExpanding $1/(\\zeta_j - z_j)$ in a geometric series in $(z_j - a_j)/(\\zeta_j - a_j)$ gives the power series of $f$ at $a$, with coefficients\n$$c_\\alpha = \\frac{1}{(2\\pi i)^n} \\int_T \\frac{f(\\zeta)}{(\\zeta - a)^{\\alpha + \\mathbf{1}}} \\, d\\zeta, \\qquad (\\zeta - a)^{\\alpha + \\mathbf{1}} = \\prod_{j} (\\zeta_j - a_j)^{\\alpha_j + 1}.$$\nExample: for $f(z) = z_1 z_2$ and $P$ the unit polydisc of $\\C^2$, at $z = 0$ the right side is $(2\\pi i)^{-2} \\int_T d\\zeta_1 \\, d\\zeta_2 = 0 = f(0)$.\nOnly the values of $f$ on the $n$-dimensional torus $T$ are used, not on the whole $(2n-1)$-dimensional boundary of $P$.\n\n## Explanations\n\n### L1 human\n\nThe formula says that a holomorphic function on a polydisc is determined by its values on a thin torus.\nEach value inside is an average of boundary values, weighted by a simple kernel.\nIt is the one-variable Cauchy formula used once in each coordinate.\n\n### L2 human\n\nThis is the basic tool of local theory in several variables.\nFrom 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.\nSince the kernel is a product of one-variable kernels, it is tied to the product shape of the polydisc.\nOn other domains one uses integral formulas with other kernels, such as the [[Bochner–Martinelli formula|YMHYAEJQ80]].\n",
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