{
  "aliases": [
    "fundamental-theorem-for-complex-line-integrals"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{C}$ be an open set and let $f : U \\to \\mathbb{C}$ be continuous.\nLet $F : U \\to \\mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] with $F'(z) = f(z)$ for each $z \\in U$.\nLet $\\gamma : [a, b] \\to U$ be a continuous and piecewise continuously differentiable path.\nThen the [[contour integral|00FWNP9TXY]] of $f$ along $\\gamma$ is\n$$\\int_\\gamma f(z) \\, dz = F(\\gamma(b)) - F(\\gamma(a)).$$\nIn particular, $\\int_\\gamma f(z) \\, dz = 0$ if $\\gamma$ is closed.\n\n## Remarks\n\nThe function $F$ is a primitive of $f$ on $U$. The integral depends only on the end points of $\\gamma$.\nFor each integer $n \\neq -1$, $z^n$ has the primitive $z^{n+1}/(n+1)$ on $\\mathbb{C} \\setminus \\{0\\}$. So $\\oint_\\gamma z^n \\, dz = 0$ for each closed path $\\gamma$ that does not pass through $0$.\nThe function $1/z$ has no primitive on $\\mathbb{C} \\setminus \\{0\\}$, because its [[integral over the unit circle|0SXZ68HV6F]] is $2 \\pi i \\neq 0$.\n\n## Explanations\n\n### L1 human\n\nThis is the complex version of $\\int_a^b F'(t) \\, dt = F(b) - F(a)$.\nIf you know a function whose derivative is $f$, you get the integral along any path from its values at the two ends.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "fundamental-theorem-for-complex-line-integrals"
    ],
    "kind": "statement",
    "msc": [
      "30E20"
    ],
    "name": "Fundamental theorem of contour integration",
    "role": "theorem",
    "sources": [
      {
        "locator": "Fundamental theorem for complex line integrals",
        "url": "https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem"
      },
      {
        "locator": "Contour integrals",
        "url": "https://en.wikipedia.org/wiki/Contour_integration"
      }
    ],
    "tag": "0YQ458D7PF",
    "uses": [
      "00FWNP9TXY",
      "0DG0Z72CQD"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0YQ458D7PF/",
  "links": [
    {
      "field": "uses",
      "target": "00FWNP9TXY"
    },
    {
      "field": "uses",
      "target": "0DG0Z72CQD"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0YQ458D7PF.md",
  "msc": [
    "30E20"
  ],
  "name": "Fundamental theorem of contour integration",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0YQ/0YQ458D7PF.md",
  "status": null,
  "tag": "0YQ458D7PF",
  "uri": "m2://0YQ458D7PF",
  "url": "https://math2.tamnd.com/wiki/fundamental-theorem-of-contour-integration/",
  "writing": {
    "tag": "0YQ458D7PF",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}