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  "body": "\n## Summary\n\nA Cauchy-type integral is a function $\\Phi(z) = \\frac{1}{2\\pi i} \\int_\\Gamma \\frac{\\varphi(\\tau)}{\\tau - z} \\, d\\tau$ built from a curve $\\Gamma$ and a function $\\varphi$ on it.\nIt has the form of [[Cauchy's integral formula|0DG10TVYPN]], but $\\varphi$ is arbitrary and need not be the boundary value of a holomorphic function.\nIt is a [[holomorphic function|0NAHXR1FRH]] off $\\Gamma$, and in general it jumps across $\\Gamma$ by exactly $\\varphi$.\nThis makes it the basic tool for boundary value problems of Riemann–Hilbert type and for singular integral equations.\n\n## Explanations\n\n### L1 human\n\nCauchy's formula recovers a holomorphic function inside a curve from its values on the curve.\nA Cauchy-type integral uses the same formula with any function $\\varphi$ on the curve.\nThe result is still holomorphic on both sides of the curve, but the inside and outside functions need not fit together.\nThe amount by which they differ across the curve is $\\varphi$ itself.\nFor example, on the unit circle with $\\varphi(\\tau) = \\tau + \\tau^{-1}$, the integral equals $z$ inside and $-1/z$ outside, and the difference at $\\tau$ on the circle is $\\tau + \\tau^{-1}$.\n",
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