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  "body": "\n## Definition\n\nLet $c \\in \\mathbb{C}$, let $R \\in (0, \\infty]$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 \u003c |z - c| \u003c R$.\nLet $f(z) = \\sum_{n \\in \\mathbb{Z}} a_n (z - c)^n$ be the [[Laurent expansion|06WHRARZKY]] of $f$ on this punctured disc.\nThe principal part of $f$ at $c$ is the series\n$$\\sum_{n = 1}^{\\infty} a_{-n} (z - c)^{-n},$$\nthe part of the [[Laurent series|0X3XD2ATB0]] with the negative powers of $z - c$.\n\n## Remarks\n\nThe principal part converges for each $z \\neq c$. The difference $f(z) - \\sum_{n \\ge 1} a_{-n} (z - c)^{-n}$ extends to a holomorphic function on $|z - c| \u003c R$.\nThe principal part is $0$ at a [[removable singularity|0P18JC72RE]], it has finitely many nonzero terms at a [[pole|0WPQYW8MTE]], and it has infinitely many at an [[essential singularity|0YKDJBV26S]].\nFor example, the principal part of $\\frac{e^z}{z^2}$ at $0$ is $\\frac{1}{z^2} + \\frac{1}{z}$.\n\n## Explanations\n\n### L1 human\n\nThe principal part is the piece of the function that is responsible for its bad behavior at $c$.\nIf you subtract it, what is left is holomorphic at $c$.\n",
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    "name": "Principal part",
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        "locator": "Laurent series definition",
        "url": "https://en.wikipedia.org/wiki/Principal_part"
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        "locator": "Principal part",
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  "markdown": "https://math2.tamnd.com/node/0NDV5R6Z6X.md",
  "msc": [
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  "name": "Principal part",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/0ND/0NDV5R6Z6X.md",
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