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  "body": "\n## Definition\n\nLet $D \\subseteq \\C^n$ be an open set, write $z_j = x_j + i y_j$ with $x_j, y_j$ real, and let $f = u + iv \\colon D \\to \\C$ with $u, v$ real-valued.\nSuppose that all first [[partial derivatives|09EGC6F8PT]] of $u$ and $v$ exist on $D$.\nThe function $f$ satisfies the *Cauchy–Riemann equations* on $D$ if for every $j = 1, \\dots, n$, at every point of $D$,\n$$\\frac{\\partial u}{\\partial x_j} = \\frac{\\partial v}{\\partial y_j}, \\qquad \\frac{\\partial u}{\\partial y_j} = -\\frac{\\partial v}{\\partial x_j}.$$\n\n## Remarks\n\nWith the [[Wirtinger derivatives|WMXDVF36ZT]] $\\frac{\\partial}{\\partial \\bar z_j} = \\frac12 \\left( \\frac{\\partial}{\\partial x_j} + i \\frac{\\partial}{\\partial y_j} \\right)$, the $2n$ real equations are the $n$ complex equations $\\partial f / \\partial \\bar z_j = 0$, $j = 1, \\dots, n$.\nFor $n = 1$ they are the classical [[Cauchy–Riemann equations|0T71SSYZ39]].\nA function $f$ of class $C^1$ on $D$ is [[holomorphic|507C34W0ZT]] if and only if it satisfies the Cauchy–Riemann equations on $D$.\nExample: $f(z_1, z_2) = z_1 z_2$ satisfies them, since $\\partial f / \\partial \\bar z_1 = 0$ and $\\partial f / \\partial \\bar z_2 = 0$.\nNon-example: $f(z_1, z_2) = z_1 \\bar z_2$ satisfies the pair for $j = 1$ but not the pair for $j = 2$, since $\\partial f / \\partial \\bar z_2 = z_1$, which is not zero when $z_1 \\ne 0$.\n",
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