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  "body": "\n## Definition\n\nLet $D$ be an [[open set|02BH3MCBQB]] in $\\C^n$ and let $f \\colon D \\to \\C$.\nThe function $f$ is *complex differentiable* at $a \\in D$ if there is a $\\C$-[[linear map|0NRYHJPQG2]] $L \\colon \\C^n \\to \\C$ such that\n$$\\lim_{h \\to 0} \\frac{\\abs{f(a + h) - f(a) - L(h)}}{\\abs{h}} = 0.$$\nThe function $f$ is *holomorphic* on $D$ if it is complex differentiable at every point of $D$.\n\n## Remarks\n\nThe map $L$ is the [[Fréchet derivative|06EH97V2A7]] of $f$ at $a$, and it has the form $L(h) = \\sum_{j=1}^n \\frac{\\partial f}{\\partial z_j}(a) \\, h_j$.\nExample: $f(z_1, z_2) = z_1 z_2$ is holomorphic on $\\C^2$, since $f(a + h) - f(a) = a_2 h_1 + a_1 h_2 + h_1 h_2$ and $\\abs{h_1 h_2} \\le \\abs{h}^2$.\nNon-example: $g(z_1, z_2) = \\bar z_1$ is real differentiable with derivative $h \\mapsto \\bar h_1$, but this map is not $\\C$-linear, since it sends $i h$ to $-i \\bar h_1$.\nEquivalent forms: $f$ is holomorphic on $D$ if and only if $f$ is continuous and holomorphic in each variable separately.\nIt is also equivalent that $f$ is, near each point $a$, the sum of a convergent [[power series in several variables|R3QTCPTJK7]] centred at $a$.\nBy [[Hartogs's theorem on separate holomorphicity|ECJG25SAJR]], the continuity condition in the first equivalent form may even be dropped.\nThe set of holomorphic functions on $D$ is written $\\mathcal{O}(D)$.\n",
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