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  "body": "\n## Definition\n\nLet $U \\subseteq \\C$ be an open set, let $f$ be a [[meromorphic function|0NSD1TPHR0]] on $U$, and let $a \\in U$.\nAssume that $f$ is not identically $0$ on any punctured disc around $a$.\nThe *order* of $f$ at $a$, written $\\operatorname{ord}_a f$, is the unique integer $n$ with the following property.\nThere are $r \u003e 0$ and a [[holomorphic|0DG0Z72CQD]] function $g$ on $\\set{z : \\abs{z - a} \u003c r}$ with $g(a) \\neq 0$ and $f(z) = (z - a)^n g(z)$ for $0 \u003c \\abs{z - a} \u003c r$.\nIf $f$ is identically $0$ near $a$, one sets $\\operatorname{ord}_a f = \\infty$.\n\n## Remarks\n\nIf $n \u003e 0$, then $a$ is a zero of [[order|064N002TX4]] $n$; if $n \u003c 0$, then $a$ is a pole of [[order|0YK27TJ4WF]] $-n$; if $n = 0$, then $f(a)$ is defined and nonzero.\nEquivalently, $n$ is the least index with a nonzero coefficient in the [[Laurent series|0X3XD2ATB0]] of $f$ at $a$.\nExample: $f(z) = \\dfrac{\\sin z}{z^3}$ has $\\operatorname{ord}_0 f = -2$, since $\\sin z = z \\, h(z)$ with $h$ holomorphic and $h(0) = 1$.\nNon-example: $e^{1/z}$ is not meromorphic near $0$ and has no order there.\nIts Laurent series at $0$ has infinitely many nonzero terms of negative index.\nThe order satisfies $\\operatorname{ord}_a(fg) = \\operatorname{ord}_a f + \\operatorname{ord}_a g$ and $\\operatorname{ord}_a(f + g) \\ge \\min(\\operatorname{ord}_a f, \\operatorname{ord}_a g)$.\nSome authors call it the valuation of $f$ at $a$.\n",
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