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  "body": "\n## Definition\n\nLet $U \\subseteq \\C$ be an open set and let $f$ be a [[meromorphic function|0NSD1TPHR0]] on $U$.\nSo there is a closed discrete set $P \\subset U$ such that $f$ is [[holomorphic|0DG0Z72CQD]] on $U \\setminus P$ and each point of $P$ is a pole of $f$.\n\n1. A *zero* of $f$ is a point $a \\in U \\setminus P$ with $f(a) = 0$, as in the definition of a [[zero of a holomorphic function|0CEGEASY49]].\n2. A *pole* of $f$ is a point $a \\in P$, that is, an isolated singularity with $\\abs{f(z)} \\to \\infty$ as $z \\to a$, as in the definition of a [[pole|0WPQYW8MTE]].\n\n## Remarks\n\nExample: $f(z) = \\dfrac{z - 1}{z^2 + 1}$ on $\\C$ has one zero, at $1$, and two poles, at $i$ and $-i$, where the denominator vanishes and the numerator does not.\nNon-example: $e^{1/z}$ is not meromorphic on $\\C$, and $0$ is neither a zero nor a pole of it.\nThe point $0$ is an essential singularity: near $0$, $e^{1/z}$ is neither bounded nor tends to $\\infty$.\nIf $U$ is connected and $f$ is not identically $0$, the zeros of $f$ are isolated and the poles of $f$ are the zeros of $1/f$.\nThe definition extends to meromorphic functions on the Riemann sphere or on a Riemann surface, using local coordinates.\nA nonzero rational function, seen on the Riemann sphere, has as many zeros as poles when both are counted with their [[orders|K2S5TAW2BF]].\n",
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