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  "body": "\n## Example\n\nThe [[circle integral|0T7ABX4CWE]] of $f(z) = 1/z$ over the unit circle $|z| = 1$ is\n$$\\oint_{|z| = 1} \\frac{dz}{z} = 2 \\pi i.$$\nComputation: the path is $\\gamma(\\theta) = e^{i\\theta}$ with $\\theta \\in [0, 2\\pi]$, so $\\gamma'(\\theta) = i e^{i\\theta}$. Then\n$$\\oint_{|z| = 1} \\frac{dz}{z} = \\int_0^{2\\pi} \\frac{i e^{i\\theta}}{e^{i\\theta}} \\, d\\theta = \\int_0^{2\\pi} i \\, d\\theta = 2 \\pi i.$$\nMore generally, let $c, w \\in \\mathbb{C}$ and $R \u003e 0$. If $|w - c| \u003c R$, then $\\oint_{|z - c| = R} \\frac{dz}{z - w} = 2 \\pi i$. If $|w - c| \u003e R$, the integral is $0$.\n\n## Remarks\n\nFor each integer $n \\neq -1$, $\\oint_{|z| = 1} z^n \\, dz = 0$, because $z^n$ has a primitive on $\\mathbb{C} \\setminus \\{0\\}$.\nThe value is not $0$, so $1/z$ has no primitive on $\\mathbb{C} \\setminus \\{0\\}$, and [[Cauchy's integral theorem|01MPZXYVYF]] does not apply on $\\mathbb{C} \\setminus \\{0\\}$.\nThis integral is the base of [[Cauchy's integral formula|0DG10TVYPN]], of the [[residue|0JFGKKWWFK]] and of the [[winding number|0X7SBS8E0F]].\n\n## Explanations\n\n### L1 human\n\nOn the unit circle, $dz$ is $i z \\, d\\theta$, so $dz / z$ is just $i \\, d\\theta$.\nGoing once around adds up $i \\, d\\theta$ over a full turn of $2\\pi$, which gives $2 \\pi i$.\n",
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