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  "body": "\n## Statement\n\nLet $\\Gamma$ be a smooth simple closed curve in $\\C$, oriented counterclockwise, with interior domain $D^+$ and exterior domain $D^-$.\nLet $\\varphi \\colon \\Gamma \\to \\C$ be [[Hölder continuous|012PBA0FMD]], and let $\\Phi(z) = \\frac{1}{2\\pi i} \\int_\\Gamma \\frac{\\varphi(\\tau)}{\\tau - z} \\, d\\tau$ for $z \\notin \\Gamma$ be its [[Cauchy-type integral|MGBKF52TQE]].\nLet $t \\in \\Gamma$.\nThen the limit $\\Phi^+(t)$ of $\\Phi(z)$ as $z \\to t$ with $z \\in D^+$ exists, and so does the limit $\\Phi^-(t)$ as $z \\to t$ with $z \\in D^-$.\nThey are\n$$\\Phi^{\\pm}(t) = \\pm \\frac12 \\varphi(t) + \\frac{1}{2\\pi i} \\, \\mathrm{p.v.} \\int_\\Gamma \\frac{\\varphi(\\tau)}{\\tau - t} \\, d\\tau,$$\nwhere $\\mathrm{p.v.}$ is the [[Cauchy principal value|0SYR10HD4R]].\nEquivalently,\n$$\\Phi^+(t) - \\Phi^-(t) = \\varphi(t), \\qquad \\Phi^+(t) + \\Phi^-(t) = \\frac{1}{\\pi i} \\, \\mathrm{p.v.} \\int_\\Gamma \\frac{\\varphi(\\tau)}{\\tau - t} \\, d\\tau.$$\n\n## Remarks\n\nProof idea: write $\\int_\\Gamma \\frac{\\varphi(\\tau)}{\\tau - z} \\, d\\tau = \\int_\\Gamma \\frac{\\varphi(\\tau) - \\varphi(t)}{\\tau - z} \\, d\\tau + \\varphi(t) \\int_\\Gamma \\frac{d\\tau}{\\tau - z}$.\nThe last integral is $2\\pi i$ for $z \\in D^+$ and $0$ for $z \\in D^-$.\nThe Hölder condition makes the first integral continuous up to $\\Gamma$ from both sides, with value $\\mathrm{p.v.} \\int_\\Gamma \\frac{\\varphi(\\tau) - \\varphi(t)}{\\tau - t} \\, d\\tau$ at $t$.\nFinally, $\\mathrm{p.v.} \\int_\\Gamma \\frac{d\\tau}{\\tau - t} = \\pi i$, half of $2\\pi i$, because the excised small arc around $t$ leaves half a turn.\n\nExample: on the unit circle with $\\varphi(\\tau) = \\tau^{-1}$, one has $\\Phi = 0$ on $D^+$ and $\\Phi(z) = -1/z$ on $D^-$.\nSo $\\Phi^+(t) - \\Phi^-(t) = t^{-1} = \\varphi(t)$, and the principal value integral equals $\\pi i (\\Phi^+(t) + \\Phi^-(t)) = -\\pi i \\, t^{-1}$.\n\nContinuity of $\\varphi$ alone is not enough: there are continuous densities whose principal value integral fails to exist at some points.\nThe formulae also hold on a smooth open arc, at every point other than the endpoints.\nThere $\\Phi^+$ is the limit from the left of the arc and $\\Phi^-$ the limit from the right.\nFor the real line they reduce to the [[Sokhotski–Plemelj theorem|RNB6QX0GCA]].\n\n## Explanations\n\n### L1 human\n\nA Cauchy-type integral defines two holomorphic functions, one inside the curve and one outside.\nAt the curve, the inside one and the outside one have limits.\nTheir difference is the density $\\varphi$, and their average is a principal value integral over the curve.\nSo the integral jumps by exactly $\\varphi$ when one crosses the curve.\n\n### L2 human\n\nThe formulae are the bridge between boundary value problems for holomorphic functions and singular integral equations.\nThe Riemann–Hilbert problem $\\Phi^+ = G \\Phi^- + g$ on $\\Gamma$ is solved by writing the unknowns as Cauchy-type integrals.\nConversely, substituting $\\Phi^\\pm$ into the formulae turns the [[singular integral equation|E30EA6ETZ4]] $a \\varphi + b S \\varphi = f$, with $S \\varphi = \\frac{1}{\\pi i} \\mathrm{p.v.} \\int_\\Gamma \\frac{\\varphi(\\tau)}{\\tau - t} d\\tau$, into the boundary condition $(a + b) \\Phi^+ - (a - b) \\Phi^- = f$.\nThis reduction is the basis of the theory of the [[index of a singular integral operator|GW3B4K1WA5]].\n",
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