{
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{C}$ be a simply connected open set. This means that $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path.\nLet $a_1, \\dots, a_n$ be distinct points of $U$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on $U \\setminus \\{a_1, \\dots, a_n\\}$.\nLet $\\gamma$ be a closed, continuous and piecewise continuously differentiable path in $U \\setminus \\{a_1, \\dots, a_n\\}$.\nThen\n$$\\oint_\\gamma f(z) \\, dz = 2 \\pi i \\sum_{k=1}^{n} n(\\gamma, a_k) \\operatorname{Res}(f, a_k),$$\nwhere $n(\\gamma, a_k)$ is the [[winding number|0X7SBS8E0F]] of $\\gamma$ around $a_k$ and $\\operatorname{Res}(f, a_k)$ is the [[residue|0JFGKKWWFK]] of $f$ at $a_k$.\n\n## Remarks\n\nIf $\\gamma$ is a simple closed curve traveled once counterclockwise, then $\\oint_\\gamma f(z) \\, dz$ is $2 \\pi i$ times the sum of the residues at the points $a_k$ inside $\\gamma$.\nWith no points $a_k$, the theorem is [[Cauchy's integral theorem|01MPZXYVYF]].\nFor example, $\\oint_{|z| = 2} \\frac{dz}{z^2 + 1} = 2 \\pi i \\left( \\frac{1}{2i} - \\frac{1}{2i} \\right) = 0$, by the [[residues at the simple poles|01991NCP7H]] $i$ and $-i$.\nThe theorem is the main tool for the [[evaluation of real integrals|0476QH9846]].\n\n## Explanations\n\n### L1 human\n\nTo integrate around a closed curve, you only need to know the residues at the bad points inside the curve.\nEach bad point contributes $2 \\pi i$ times its residue, once for each turn of the curve around it.\n",
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    "name": "Residue theorem",
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        "locator": "Statement of Cauchy's residue theorem",
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