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  "body": "\n## Definition\n\nLet $D \\subseteq \\C^n$ be an [[open set|02BH3MCBQB]].\nA *meromorphic function* on $D$ is given by an open cover $(U_i)_{i \\in I}$ of $D$ by connected open sets.\nFor each $i$ it gives two [[holomorphic functions|507C34W0ZT]] $g_i, h_i$ on $U_i$ such that:\n\n1. $h_i$ is not identically zero on $U_i$;\n2. $g_i h_j = g_j h_i$ on $U_i \\cap U_j$ for all $i, j \\in I$.\n\nTwo such data $(U_i, g_i, h_i)_{i \\in I}$ and $(V_k, p_k, q_k)_{k \\in K}$ define the same meromorphic function when $g_i q_k = p_k h_i$ on $U_i \\cap V_k$ for all $i$ and $k$.\n\n## Remarks\n\nOn the dense open set where some $h_i$ is nonzero, a meromorphic function $f$ is the holomorphic function $g_i / h_i$.\nIts polar set and its set of indeterminacy points are analytic subsets; for $n \\ge 2$ the indeterminacy set has codimension at least $2$.\n\nExample: on $D = \\C^2$ take the single set $U_1 = \\C^2$ with $g_1(z) = z_1$ and $h_1(z) = z_2$.\nClause 1 holds since $z_2$ is not the zero function, and clause 2 is empty, so $z_1 / z_2$ is meromorphic on $\\C^2$.\n\nNon-example: $e^{1/z_1}$, defined where $z_1 \\ne 0$, is not the restriction of a meromorphic function on $\\C^2$.\nSuppose it were $g/h$ near a point $(0, c)$.\nThen, for a suitable fixed $z_2$ close to $c$, the one-variable function $z_1 \\mapsto e^{1/z_1}$ would be meromorphic at $0$.\nBut it has an essential singularity there.\n\nFor $n = 1$ this definition gives the usual [[meromorphic functions|0ZZ7ZYDR8W]] on an open set of $\\C$.\nIn the language of sheaves, a meromorphic function is a global section of the sheaf of quotient fields of the sheaf of holomorphic functions.\n",
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