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  "body": "\n## Problem\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] and let $(U_i)_{i \\in I}$ be an open cover of $X$.\nFor each $i$, let $f_i$ be a [[meromorphic function|W6FTSF57T1]] on $U_i$, such that $f_i - f_j$ is [[holomorphic|507C34W0ZT]] on $U_i \\cap U_j$ for all $i, j$.\nFind a meromorphic function $f$ on $X$ such that $f - f_i$ is holomorphic on $U_i$ for every $i$.\n\n## Remarks\n\nThe data $(U_i, f_i)$ are called Cousin I data; they prescribe the principal parts of $f$.\nThe differences $g_{ij} = f_i - f_j$ form a Čech $1$-cocycle with values in the sheaf $\\mathcal{O}$ of holomorphic functions.\nThe problem is solvable for these data if and only if this cocycle is a coboundary, that is, $g_{ij} = h_j - h_i$ for holomorphic $h_i$ on $U_i$.\nThen $f = f_i + h_i$ on $U_i$ is a solution.\nSo if $H^1(X, \\mathcal{O}) = 0$, every first Cousin problem on $X$ is solvable.\nBy Cartan's theorem B this holds on every Stein manifold, in particular on every domain of holomorphy in $\\C^n$.\nFor $n = 1$ and $X$ a domain in $\\C$, the problem is solved by the Mittag-Leffler theorem.\nIt can fail on other domains: on $X = \\C^2 \\setminus \\set{0}$, with $U_1 = \\set{z_1 \\ne 0}$, $U_2 = \\set{z_2 \\ne 0}$, $f_1 = 1/(z_1 z_2)$ and $f_2 = 0$, there is no solution.\nPierre Cousin studied the problem in 1895 for product domains in $\\C^n$.\n\n## Explanations\n\n### L1 human\n\nIn one variable, the Mittag-Leffler theorem builds a meromorphic function with prescribed poles and principal parts.\nThe first Cousin problem asks the same in several variables.\nThe local singular behaviour is given on pieces of a manifold, with the pieces agreeing up to holomorphic terms.\nOne wants one global meromorphic function with exactly this behaviour.\nWhether this is possible depends on the shape of the manifold.\nOn Stein manifolds, for example on $\\C^n$ and on convex domains, it always is.\n",
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      "first_stated": "1895",
      "posed_by": "Pierre Cousin"
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    "msc": [
      "32A20",
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    "name": "First Cousin problem",
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        "url": "https://en.wikipedia.org/wiki/Cousin_problems"
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  "msc": [
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  "name": "First Cousin problem",
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