{
  "aliases": [
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{C}$ be an open set and let $f : U \\to \\mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.\nLet $c \\in \\mathbb{C}$ and $r \u003e 0$ be such that the closed disc $\\{z : |z - c| \\le r\\}$ is contained in $U$.\nThen for each $a \\in \\mathbb{C}$ with $|a - c| \u003c r$,\n$$f(a) = \\frac{1}{2 \\pi i} \\oint_{|z - c| = r} \\frac{f(z)}{z - a} \\, dz,$$\nwhere the integral is the [[circle integral|0T7ABX4CWE]] over the circle $|z - c| = r$.\n\n## Remarks\n\nThe values of $f$ on the circle determine the values of $f$ at each point inside the circle.\nWith $a = c$ and $z = c + r e^{i\\theta}$, the formula gives $f(c) = \\frac{1}{2\\pi} \\int_0^{2\\pi} f(c + r e^{i\\theta}) \\, d\\theta$. So $f(c)$ is the average of $f$ on the circle.\nFor $f = 1$, the formula is the [[integral of 1/(z - a)|0SXZ68HV6F]].\nThe formula gives [[the formula for the derivatives|0F0H5P5DED]], and it shows that [[holomorphic functions are analytic|0826SGV3R0]].\n\n## Explanations\n\n### L1 human\n\nIf you know a holomorphic function on a circle, you know it everywhere inside the circle.\nThe formula computes the value at an inside point $a$ as a weighted average of the values on the circle, with the weight $1/(z - a)$.\n",
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    "aliases": [
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    "kind": "statement",
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    "name": "Cauchy's integral formula",
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    "same_as": {
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      "wikipedia": "Cauchy's integral formula"
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    "sources": [
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        "locator": "Theorem",
        "url": "https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula"
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  "markdown": "https://math2.tamnd.com/node/0DG10TVYPN.md",
  "msc": [
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  "name": "Cauchy's integral formula",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0DG/0DG10TVYPN.md",
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  "tag": "0DG10TVYPN",
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  "url": "https://math2.tamnd.com/wiki/cauchys-integral-formula/",
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    "tag": "0DG10TVYPN",
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