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  "body": "\n## Definition\n\nLet $\\Gamma$ be an oriented [[rectifiable curve|0HATW2VWAZ]] in $\\C$, closed or not, and let $\\varphi \\colon \\Gamma \\to \\C$ be continuous.\nThe *Cauchy-type integral* of $\\varphi$ over $\\Gamma$ is the function\n$$\\Phi(z) = \\frac{1}{2\\pi i} \\int_\\Gamma \\frac{\\varphi(\\tau)}{\\tau - z} \\, d\\tau, \\quad z \\in \\C \\setminus \\Gamma,$$\nwhere the integral is a [[contour integral|06ZGPYTWK8]].\nThe function $\\varphi$ is its *density*.\n\n## Remarks\n\nThe function $\\Phi$ is [[holomorphic|0NAHXR1FRH]] on $\\C \\setminus \\Gamma$, and $\\Phi(z) \\to 0$ as $z \\to \\infty$.\nExample: let $\\Gamma$ be the unit circle, oriented counterclockwise, and $\\varphi(\\tau) = \\tau^{-1}$.\nFor $\\abs{z} \u003c 1$, the integrand $\\frac{1}{\\tau(\\tau - z)} = \\frac{1}{z} \\left( \\frac{1}{\\tau - z} - \\frac{1}{\\tau} \\right)$ gives $\\Phi(z) = \\frac{1}{z}(1 - 1) = 0$; for $\\abs{z} \u003e 1$ it gives $\\Phi(z) = \\frac{1}{z}(0 - 1) = -\\frac{1}{z}$.\nSo the boundary values from inside and outside are $0$ and $-\\tau^{-1}$, and their difference is $\\varphi(\\tau)$.\nNon-example: for $z \\in \\Gamma$ the formula is not a convergent integral in general.\nThe reason is that $\\frac{1}{\\tau - z}$ is not integrable along $\\Gamma$ near $\\tau = z$.\nOn $\\Gamma$ one uses instead the [[Cauchy principal value|0SYR10HD4R]], as in [[singular integral equations|E30EA6ETZ4]].\nLet $\\Gamma$ be a closed curve bounding a domain $D$, and let $\\varphi$ be the boundary value of a function $F$ holomorphic on $D$ and continuous on its closure.\nThen $\\Phi = F$ on $D$ and $\\Phi = 0$ outside $\\overline{D}$, by [[Cauchy's integral formula|0DG10TVYPN]] and Cauchy's theorem.\nThe boundary values of $\\Phi$ for Hölder continuous $\\varphi$ are given by the [[Sokhotski–Plemelj formulae|DRT1E6WY6K]].\n",
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