{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe show that $t \\mapsto F(\\gamma(t))$ has derivative $f(\\gamma(t)) \\gamma'(t)$ and apply the real fundamental theorem of calculus on each piece.\n\n1. Choose $a = t_0 \u003c t_1 \u003c \\dots \u003c t_m = b$ such that $\\gamma$ is continuously differentiable on each $[t_{j-1}, t_j]$, with one-sided derivatives at the end points.\n2. Let $p \\in U$. Define $E(w) = \\frac{F(w) - F(p)}{w - p} - F'(p)$ for $w \\in U$, $w \\ne p$, and $E(p) = 0$. Since $F$ is [[complex differentiable|0D1C9Y1EXA]] at $p$, $E$ is continuous at $p$.\n3. For every $w \\in U$, $F(w) - F(p) = (F'(p) + E(w)) (w - p)$.\n4. Fix $j$, put $\\varphi(t) = F(\\gamma(t))$ for $t \\in [t_{j-1}, t_j]$, and fix $t$ there. Put $p = \\gamma(t)$. By step 3, for $s \\ne 0$ with $t + s \\in [t_{j-1}, t_j]$, $\\frac{\\varphi(t+s) - \\varphi(t)}{s} = (F'(p) + E(\\gamma(t+s))) \\frac{\\gamma(t+s) - \\gamma(t)}{s}$.\n5. As $s \\to 0$, $\\gamma(t+s) \\to p$ because $\\gamma$ is continuous. So $E(\\gamma(t+s)) \\to 0$ by step 2. Also the last factor tends to $\\gamma'(t)$.\n6. Hence $\\varphi'(t) = F'(\\gamma(t)) \\gamma'(t) = f(\\gamma(t)) \\gamma'(t)$, one-sided at the end points. This is continuous in $t$, since $f$, $\\gamma$ and $\\gamma'$ are continuous on the piece.\n7. Apply the [[fundamental theorem of calculus|0NHGY0N0JB]] to $\\operatorname{Re} \\varphi$ and $\\operatorname{Im} \\varphi$ on $[t_{j-1}, t_j]$. It gives $\\int_{t_{j-1}}^{t_j} f(\\gamma(t)) \\gamma'(t) \\, dt = F(\\gamma(t_j)) - F(\\gamma(t_{j-1}))$.\n8. By the [[definition of the contour integral|00FWNP9TXY]] and [[additivity over intervals|05RCDKHZM6]], $\\int_\\gamma f(z) \\, dz$ is the sum over $j$ of the integrals in step 7.\n9. The sum telescopes: $\\int_\\gamma f(z) \\, dz = F(\\gamma(t_m)) - F(\\gamma(t_0)) = F(\\gamma(b)) - F(\\gamma(a))$.\n10. If $\\gamma$ is closed, then $\\gamma(b) = \\gamma(a)$, so $\\int_\\gamma f(z) \\, dz = 0$.\n\n## Idea\n\nAlong the path, $F(\\gamma(t))$ is an ordinary function of the real variable $t$. The chain rule says its derivative is the integrand $f(\\gamma(t)) \\gamma'(t)$, so the real fundamental theorem of calculus applies.\n",
  "commit": "68a8c3dc73fd55004b2796431c607c30960d1a85",
  "front": {
    "kind": "proof",
    "msc": [
      "30E20"
    ],
    "name": "Proof of the fundamental theorem of contour integration",
    "proves": "0YQ458D7PF",
    "sources": [
      {
        "locator": "Fundamental theorem for complex line integrals",
        "url": "https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem"
      }
    ],
    "tag": "R4HZTBV9VJ",
    "uses": [
      "00FWNP9TXY",
      "0DG0Z72CQD",
      "0D1C9Y1EXA",
      "0NHGY0N0JB",
      "05RCDKHZM6"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/R4HZTBV9VJ/",
  "links": [
    {
      "field": "proves",
      "target": "0YQ458D7PF"
    },
    {
      "field": "uses",
      "target": "00FWNP9TXY"
    },
    {
      "field": "uses",
      "target": "0DG0Z72CQD"
    },
    {
      "field": "uses",
      "target": "0D1C9Y1EXA"
    },
    {
      "field": "uses",
      "target": "0NHGY0N0JB"
    },
    {
      "field": "uses",
      "target": "05RCDKHZM6"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/R4HZTBV9VJ.md",
  "msc": [
    "30E20"
  ],
  "name": "Proof of the fundamental theorem of contour integration",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/R4H/R4HZTBV9VJ.md",
  "status": null,
  "tag": "R4HZTBV9VJ",
  "uri": "m2://R4HZTBV9VJ",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-fundamental-theorem-of-contour-integration/",
  "writing": {
    "tag": "R4HZTBV9VJ",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}