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  "body": "\n## Summary\n\nA holomorphic function of several variables is a function on an open subset of $\\C^n$ that is complex differentiable at every point.\nIt extends the notion of a [[holomorphic function|0NAHXR1FRH]] of one variable.\nSuch a function is holomorphic in each variable separately and infinitely differentiable.\nIt is locally the sum of a convergent [[power series in several variables|R3QTCPTJK7]].\nThese functions are the main objects of the theory of [[several complex variables|BH57D2FBNR]].\n\n## Explanations\n\n### L1 human\n\nA function of one complex variable is holomorphic when it has a complex derivative.\nFor several variables, one asks that near each point the function is well approximated by a complex linear function.\nPolynomials in $z_1, \\dots, z_n$, such as $z_1^3 - 2 z_1 z_2$, are holomorphic on all of $\\C^n$.\nThe function $1/(1 - z_1 - z_2)$ is holomorphic where $z_1 + z_2 \\ne 1$.\nA function that involves $\\bar z_1$, such as $\\abs{z_1}^2$, is not holomorphic.\n",
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