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  "body": "\n## Definition\n\nLet $c = (c_1, \\dots, c_n) \\in \\C^n$ and $r = (r_1, \\dots, r_n)$ with each $r_j \u003e 0$.\nLet $T(c, r)$ be the [[distinguished boundary|A6HN207Y2T]] $\\set{\\zeta : \\abs{\\zeta_j - c_j} = r_j \\text{ for all } j}$.\nFor $\\theta = (\\theta_1, \\dots, \\theta_n) \\in [0, 2\\pi]^n$, put $\\gamma(\\theta) = (c_1 + r_1 e^{i\\theta_1}, \\dots, c_n + r_n e^{i\\theta_n})$.\nLet $f \\colon T(c, r) \\to \\C$ be a function such that $\\theta \\mapsto f(\\gamma(\\theta))$ is [[Lebesgue integrable|DWJF7ABPR1]] on $[0, 2\\pi]^n$, for example a continuous function.\nThe *torus integral* of $f$ over $T(c, r)$ is\n$$\\int_{T(c, r)} f(\\zeta) \\, d\\zeta_1 \\cdots d\\zeta_n = \\int_{[0, 2\\pi]^n} f(\\gamma(\\theta)) \\prod_{j=1}^n \\left( i r_j e^{i\\theta_j} \\right) d\\theta_1 \\cdots d\\theta_n.$$\n\n## Remarks\n\nThe factor $i r_j e^{i\\theta_j} \\, d\\theta_j$ is $d\\zeta_j$ along the circle $\\zeta_j = c_j + r_j e^{i\\theta_j}$, so for $n = 1$ this is the [[contour integral|06ZGPYTWK8]] over the positively oriented circle.\nFor continuous $f$, Fubini's theorem makes it equal to the iterated integral of $n$ one-variable contour integrals, in any order.\nExample: for $f(\\zeta) = 1/(\\zeta_1 \\zeta_2)$ on $T(0, (1, 1))$, the integrand is $e^{-i\\theta_1} e^{-i\\theta_2} \\cdot i e^{i\\theta_1} \\cdot i e^{i\\theta_2} = -1$, so the integral is $-(2\\pi)^2 = (2\\pi i)^2$.\nNon-example: $f(\\zeta) = 1/(\\zeta_1 - 1)$ is not defined at the points of $T(0, (1, 1))$ with $\\zeta_1 = 1$.\nNear $\\theta_1 = 0$, $\\abs{f(\\gamma(\\theta))}$ behaves like $1/\\abs{\\theta_1}$, so it is not integrable and the torus integral is not defined.\nOther notation: $\\oint_{T(c, r)} f(\\zeta) \\, d\\zeta$.\n",
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