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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{C}$ be an open set.\nA function $f$ is meromorphic on $U$ if there is a set $P \\subseteq U$ with these properties.\n\n1. $P$ has no accumulation point in $U$.\n2. $f$ is [[holomorphic|0DG0Z72CQD]] on $U \\setminus P$.\n3. Each point of $P$ is a [[pole|0WPQYW8MTE]] of $f$.\n\n## Remarks\n\nRational functions $p/q$, with polynomials $p$ and $q \\neq 0$, are meromorphic on $\\mathbb{C}$. So are $\\tan z$, $\\frac{1}{\\sin z}$ and the Gamma function.\nIf $U$ is a [[domain|097KNJFJ7V]], $g$ and $h$ are holomorphic on $U$, and $h$ is not identically zero, then $g/h$ is meromorphic on $U$. The meromorphic functions on a domain form a field.\nThe function $e^{1/z}$ is not meromorphic on $\\mathbb{C}$, because its singularity at $0$ is [[essential|0YKDJBV26S]].\nThe set $P$ can be infinite. It is countable, and each compact subset of $U$ contains only finitely many points of $P$.\n\n## Explanations\n\n### L1 human\n\nA meromorphic function is holomorphic except at some isolated points, where it has poles.\nFor example, $\\frac{1}{z^2 + 1}$ is holomorphic on $\\mathbb{C}$ except at $i$ and $-i$, where it goes to infinity.\n",
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