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  "body": "\n## Problem\n\nLet $\\Gamma \\subset \\C$ be a smooth simple closed curve, oriented counterclockwise.\nLet $D^+$ be its interior domain and $D^-$ its exterior domain.\nLet $G \\colon \\Gamma \\to \\C \\setminus \\set{0}$ and $g \\colon \\Gamma \\to \\C$ be [[Hölder continuous|0WZTM0EFR6]] functions.\nFind a function $\\Phi$ that is [[holomorphic|0NAHXR1FRH]] on $D^+$ and on $D^-$ and vanishes at $\\infty$.\nIt must have continuous boundary values $\\Phi^+$ from $D^+$ and $\\Phi^-$ from $D^-$ on $\\Gamma$ with\n$$\\Phi^+(t) = G(t)\\, \\Phi^-(t) + g(t) \\quad \\text{for } t \\in \\Gamma.$$\nIn the matrix form of the problem, $\\Phi$ takes values in $\\C^n$.\nThen $G$ is a Hölder continuous map from $\\Gamma$ to the invertible $n \\times n$ complex matrices.\n\n## Remarks\n\nThe scalar problem is solved explicitly.\nLet $\\kappa$ be the [[winding number|0X7SBS8E0F]] of the curve $G(\\Gamma)$ about $0$, called the index of the problem.\nAssume $0 \\in D^+$; taking a branch of $\\log$ of $t^{-\\kappa} G(t)$, which has winding number $0$, turns the problem into a jump problem $\\Psi^+ - \\Psi^- = h$.\nThe Cauchy integral $\\frac{1}{2\\pi i} \\int_\\Gamma \\frac{h(\\tau)}{\\tau - z}\\, d\\tau$ solves it by the [[Sokhotski–Plemelj theorem|RNB6QX0GCA]].\nIf $\\kappa \\ge 0$, the problem is solvable for every $g$, and the solutions form an affine space of dimension $\\kappa$.\nIf $\\kappa \u003c 0$, it is solvable if and only if $g$ satisfies $-\\kappa$ linear conditions, and then the solution is unique.\n\nExample: for $G = 1$ the index is $0$ and the unique solution is the Cauchy integral of $g$.\n\nThe matrix problem cannot be solved by logarithms, since matrices do not commute.\nIt is closely tied to [[Hilbert's twenty-first problem|7RHW0KA5RY]] on linear differential equations with prescribed monodromy, and it underlies the Riemann–Hilbert method for integrable systems and random matrices.\n\n## Explanations\n\n### L1 human\n\nOne looks for a function that is holomorphic on both sides of a curve and jumps across the curve in a prescribed way.\nThe jump says that the value from inside equals a given function times the value from outside, plus another given function.\nFor scalar functions this is solved completely, and the answer depends on how many times the given function $G$ winds around $0$.\nFor matrix functions the problem is much harder and connects to differential equations and mathematical physics.\n",
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