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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer.\nThe space $\\C^n$ is the set of $n$-tuples $z = (z_1, \\dots, z_n)$ of [[complex numbers|09FTAET5Z4]], with coordinatewise addition and scalar multiplication.\nIts norm is $\\abs{z} = \\left( \\abs{z_1}^2 + \\cdots + \\abs{z_n}^2 \\right)^{1/2}$, which makes it the Euclidean space $\\R^{2n}$ through $z_j = x_j + i y_j$.\nA *function of several complex variables* is a [[function|0XHDTE5YMZ]] $f \\colon D \\to \\C$, where $D$ is a subset of $\\C^n$.\n\n## Remarks\n\nExample: $f(z_1, z_2) = e^{z_1} \\sin z_2$ is a function on $D = \\C^2$.\nNon-example: $g(x, y) = x^2 + y^2$ on $\\R^2$ is a function of two real variables, not of complex variables.\nIt becomes a function of the complex variable $z = x + iy$ only when it is rewritten as $\\abs{z}^2$ on $\\C$, with $n = 1$.\nA map $F \\colon D \\to \\C^m$ is the same as $m$ such functions $F_1, \\dots, F_m$, its components.\nThe case $n = 1$ is the function theory of one complex variable.\n",
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