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  "body": "\n## Definition\n\nLet $U \\subseteq \\C^n$ be an [[open set|02BH3MCBQB]] and let $F = (F_1, \\dots, F_m) \\colon U \\to \\C^m$ be a map with components $F_k \\colon U \\to \\C$.\nThe map $F$ is *holomorphic* if each component $F_k$, $k = 1, \\dots, m$, is a [[holomorphic function|507C34W0ZT]] on $U$.\nIf $V \\subseteq \\C^m$, a holomorphic map $U \\to V$ is a holomorphic map $F \\colon U \\to \\C^m$ with $F(U) \\subseteq V$.\n\n## Remarks\n\nEquivalently, $F$ is differentiable at each point $a \\in U$ in the real sense, and its derivative $DF(a) \\colon \\C^n \\to \\C^m$ is $\\C$-linear.\nThis derivative is given by the complex [[Jacobian matrix|047NDTEPCV]] $\\left( \\frac{\\partial F_k}{\\partial z_j}(a) \\right)_{k, j}$, an $m \\times n$ matrix.\nExample: $F(z_1, z_2) = (z_1 z_2, \\ z_1 - z_2)$ is holomorphic from $\\C^2$ to $\\C^2$, since both components are polynomials.\nNon-example: $G(z_1, z_2) = (z_1, \\ \\bar z_2)$ is not holomorphic, since its second component fails the Cauchy–Riemann equation $\\partial G_2 / \\partial \\bar z_2 = 0$.\nCompositions of holomorphic maps are holomorphic, by the chain rule.\n",
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