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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{C}$ be an open set and let $f : U \\to \\mathbb{C}$ be a function.\nThe function $f$ is holomorphic on $U$ if $f$ is [[complex differentiable|0D1C9Y1EXA]] at each point of $U$.\nLet $z_0 \\in \\mathbb{C}$. A function $f$ is holomorphic at $z_0$ if $f$ is holomorphic on some open disc with center $z_0$.\n\n## Remarks\n\nPolynomials in $z$ are holomorphic on $\\mathbb{C}$, and $z \\mapsto 1/z$ is holomorphic on $\\mathbb{C} \\setminus \\{0\\}$.\nSums, products, quotients with a nonzero denominator and compositions of holomorphic functions are holomorphic.\nEach holomorphic function is analytic, see [[holomorphic functions are analytic|0826SGV3R0]]. For this reason, many authors use the words holomorphic and analytic for the same functions.\nA function that is holomorphic on all of $\\mathbb{C}$ is an [[entire function|0419794R0T]].\n\n## Explanations\n\n### L1 human\n\nA holomorphic function is a function of a complex variable that has a complex derivative at each point of an open set.\nThis one condition has strong consequences: the function then has derivatives of all orders and is equal to its Taylor series near each point.\n",
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