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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 $Q = \\{x + iy : a_1 \\le x \\le b_1, \\ a_2 \\le y \\le b_2\\}$ be a closed rectangle with $Q \\subseteq U$, where $a_1 \u003c b_1$ and $a_2 \u003c b_2$ are real numbers.\nLet $\\partial Q$ be the boundary of $Q$, traveled once counterclockwise. Then the [[contour integral|00FWNP9TXY]] of $f$ along $\\partial Q$ is\n$$\\oint_{\\partial Q} f(z) \\, dz = 0.$$\n\n## Remarks\n\nThe theorem does not assume that $f'$ is continuous. This is the difference with the older proof by Green's theorem.\nThe same statement holds for each closed triangle $T \\subseteq U$ and its boundary.\nProof idea: divide $Q$ into four equal rectangles and keep one where the integral is largest. Repeat, and use the complex derivative at the limit point.\nGoursat's theorem is the first step to [[primitives on a disc|02AC7JR6G6]] and to [[Cauchy's integral theorem|01MPZXYVYF]].\n\n## Explanations\n\n### L1 human\n\nIf $f$ is holomorphic on a rectangle and its inside, then the integral of $f$ around the edge of the rectangle is zero.\nThe surprise is that this needs only the existence of the complex derivative, and nothing about its continuity.\n",
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        "locator": "Goursat's theorem and its generalizations",
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