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  "body": "\n## Statement\n\nLet $U \\subseteq \\C$ be an [[open set|01ZQQDCERP]], and let $A \\subseteq U$ be a set with no accumulation point in $U$.\nFor each $a \\in A$, let\n$$p_a(z) = \\sum_{k=1}^{N_a} \\frac{c_{a,k}}{(z - a)^k}$$\nbe a nonzero finite sum, with $N_a \\ge 1$ and complex coefficients $c_{a,k}$.\nThen there is a [[meromorphic function|0NSD1TPHR0]] $f$ on $U$ such that:\n\n1. $f$ is [[holomorphic|0DG0Z72CQD]] on $U \\setminus A$;\n2. for each $a \\in A$, the function $f - p_a$ is holomorphic near $a$.\n\nIn particular $f$ has a [[pole|0WPQYW8MTE]] at each $a \\in A$ with principal part $p_a$.\n\n## Remarks\n\nFor $U = \\C$ the proof is explicit.\nList $A$ as $a_1, a_2, \\dots$ with $\\abs{a_n}$ nondecreasing, and subtract from $p_{a_n}$ a Taylor polynomial $q_n$ at $0$ (empty when $a_n = 0$) so that $\\abs{p_{a_n} - q_n} \\le 2^{-n}$ on the disc $\\abs{z} \\le \\abs{a_n}/2$.\nThen $\\sum_n (p_{a_n} - q_n)$ converges uniformly on compact sets away from $A$ and gives $f$.\nFor a general open set, Runge's theorem on rational approximation replaces the Taylor polynomials.\nA classical example is\n$$\\frac{\\pi^2}{\\sin^2(\\pi z)} = \\sum_{n \\in \\Z} \\frac{1}{(z - n)^2},$$\nwhere here the plain sum already converges.\n\n## Explanations\n\n### L1 human\n\nThe theorem lets one place poles at any points that do not pile up inside the region, with any chosen principal parts.\nSumming the principal parts directly may diverge, so each term gets a correction that is holomorphic and does not change the pole.\nIt is the counterpart for poles of the [[Weierstrass factorization theorem|11VC51YHQJ]], which places zeros.\n\n### L2 human\n\nThe function $f$ is not unique: one can add any holomorphic function on $U$.\nThe theorem gives partial fraction expansions of meromorphic functions, such as $\\pi \\cot(\\pi z) = 1/z + \\sum_{n \\ne 0} \\left(1/(z-n) + 1/n\\right)$.\nCombined with Weierstrass products, it shows that one can prescribe both the principal parts at $A$ and finitely many Taylor coefficients at other points.\nIn the language of sheaf cohomology, it says that the first Cousin problem is solvable on every open subset of $\\C$; see also [[Runge's theorem|6DG4CTGXJD]].\n",
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