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  "body": "\n## Definition\n\nLet $D$ be a subset of $\\C$ that is unbounded, for example a sector or a ray, and let $f \\colon D \\to \\C$.\nA formal series $\\sum_{n \\ge 0} a_n z^{-n}$ with complex coefficients is a *Poincaré asymptotic series* of $f$ as $z \\to \\infty$ in $D$ if for every $N \\ge 0$\n$$z^N \\Big( f(z) - \\sum_{n=0}^{N} a_n z^{-n} \\Big) \\to 0 \\quad \\text{as } z \\to \\infty \\text{ in } D.$$\nThis is written $f(z) \\sim \\sum_{n \\ge 0} a_n z^{-n}$.\nIt is the [[asymptotic expansion|BJQTEPEZ9G]] of $f$ in the [[asymptotic scale|XEGB8Y44GG]] $\\varphi_n(z) = z^{-n}$.\n\n## Remarks\n\nAn equivalent condition is that $f(z) - \\sum_{n=0}^{N} a_n z^{-n} = O(z^{-N-1})$ for every $N$.\nThe coefficients are determined by $f$: $a_0 = \\lim f(z)$ and $a_N = \\lim z^N \\big( f(z) - \\sum_{n\u003cN} a_n z^{-n} \\big)$.\nThe same definition with $(z - z_0)^n$ in place of $z^{-n}$ gives Poincaré series at a finite point $z_0$.\n\nExample: $f(z) = \\frac{1}{z - 1}$ on $D = (2, \\infty)$ has $f(z) \\sim \\sum_{n \\ge 1} z^{-n}$.\nCheck: $f(z) - \\sum_{n=1}^{N} z^{-n} = \\frac{z^{-N}}{z - 1}$, and $z^N$ times this is $\\frac{1}{z - 1} \\to 0$.\n\nNon-example: $f(z) = \\sin z$ on $D = (0, \\infty)$ has no Poincaré asymptotic series, because $a_0 = \\lim_{z \\to \\infty} \\sin z$ does not exist.\n\nThe function $e^{-z}$ on $(0, \\infty)$ has the zero series, like the function $0$, so the series does not determine the function.\n",
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