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  "body": "\n## Definition\n\nLet $c \\in \\mathbb{C}$ and let $(a_n)_{n \\in \\mathbb{Z}}$ be complex numbers.\nThe Laurent series with center $c$ and coefficients $a_n$ is the series\n$$\\sum_{n = -\\infty}^{\\infty} a_n (z - c)^n$$\nin the complex variable $z$.\nIt converges at a point $z \\neq c$ if both series $\\sum_{n \\ge 0} a_n (z - c)^n$ and $\\sum_{n \\ge 1} a_{-n} (z - c)^{-n}$ converge. Its sum at $z$ is then the sum of the two limits.\n\n## Remarks\n\nA [[power series|0VARVQ9ZW1]] is a Laurent series with $a_n = 0$ for each $n \u003c 0$.\nIf the series converges on a nonempty open set, then it converges absolutely on an [[annulus|0AJ0YVK743]] $r \u003c |z - c| \u003c R$, and uniformly on each smaller closed annulus.\nFor example, $e^{1/z} = \\sum_{n \\ge 0} \\frac{1}{n!} z^{-n}$ for each $z \\neq 0$.\nThe part with negative powers is the [[principal part|0NDV5R6Z6X]].\n\n## Explanations\n\n### L1 human\n\nA Laurent series is like a power series, but it can also have negative powers of $z - c$.\nThe negative powers allow it to describe functions that blow up at $c$, such as $1/z$ or $e^{1/z}$.\n",
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