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  "body": "\n## Statement\n\nLet $c \\in \\mathbb{C}$, let $R \\in (0, \\infty]$, and let $D = \\{z : |z - c| \u003c R\\}$ be the open disc with center $c$ and radius $R$.\nLet $f : D \\to \\mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.\nThen there is a holomorphic function $F : D \\to \\mathbb{C}$ with $F'(z) = f(z)$ for each $z \\in D$.\n\n## Remarks\n\nOne primitive is $F(z) = \\int_{[c, z]} f(w) \\, dw$, the [[contour integral|00FWNP9TXY]] along the segment from $c$ to $z$. [[Goursat's theorem|0BRG85WXYC]] for triangles shows that $F' = f$.\nTwo primitives of $f$ on $D$ differ by a constant, because $D$ is connected.\nThe disc can be replaced by a convex open set, or more generally by a simply connected open set.\nThe result fails on $\\mathbb{C} \\setminus \\{0\\}$: the function $1/z$ has no primitive there, by the [[integral of 1/z|0SXZ68HV6F]].\n\n## Explanations\n\n### L1 human\n\nOn a disc, each holomorphic function is the derivative of another holomorphic function.\nYou can build it by integrating along straight lines from the center, as in the real case $F(x) = \\int_c^x f(t) \\, dt$.\n",
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  "name": "Holomorphic functions on a disc have primitives",
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