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  "body": "\n## Statement\n\nLet $\\Gamma$ be a smooth simple closed curve in $\\C$, oriented counterclockwise.\nLet $a$ and $b$ be [[Hölder continuous|012PBA0FMD]] functions on $\\Gamma$ with $a(t)^2 - b(t)^2 \\ne 0$ for all $t \\in \\Gamma$, and let $k$ be Hölder continuous on $\\Gamma \\times \\Gamma$.\nLet $K$ be the operator of the [[singular integral equation|E30EA6ETZ4]]\n$$K \\varphi(t) = a(t) \\, \\varphi(t) + \\frac{b(t)}{\\pi i} \\, \\mathrm{p.v.} \\int_\\Gamma \\frac{\\varphi(\\tau)}{\\tau - t} \\, d\\tau + \\int_\\Gamma k(t, \\tau) \\, \\varphi(\\tau) \\, d\\tau,$$\nand let $K'$ be its adjoint operator\n$$K' \\psi(t) = a(t) \\, \\psi(t) - \\frac{1}{\\pi i} \\, \\mathrm{p.v.} \\int_\\Gamma \\frac{b(\\tau) \\, \\psi(\\tau)}{\\tau - t} \\, d\\tau + \\int_\\Gamma k(\\tau, t) \\, \\psi(\\tau) \\, d\\tau.$$\nAll functions below are Hölder continuous on $\\Gamma$.\nLet $\\kappa$ be the [[index|GW3B4K1WA5]] of $K$.\nThen:\n\n1. The homogeneous equations $K \\varphi = 0$ and $K' \\psi = 0$ each have a finite number of linearly independent solutions; call these numbers $\\ell$ and $\\ell'$.\n2. For Hölder continuous $f$, the equation $K \\varphi = f$ has a solution if and only if $\\int_\\Gamma f(t) \\, \\psi(t) \\, dt = 0$ for every solution $\\psi$ of $K' \\psi = 0$.\n3. $\\ell - \\ell' = \\kappa$.\n\n## Remarks\n\nThe adjoint is taken for the bilinear pairing $\\int_\\Gamma u(t) \\, v(t) \\, dt$; exchanging the order of integration in $\\int_\\Gamma (K \\varphi) \\psi \\, dt$ gives $\\int_\\Gamma \\varphi \\, (K' \\psi) \\, dt$.\nProof idea: the dominant operator $K^0 = a + b S$, with $S \\varphi = \\frac{1}{\\pi i} \\mathrm{p.v.} \\int_\\Gamma \\frac{\\varphi(\\tau)}{\\tau - t} \\, d\\tau$, is solved explicitly through the [[Sokhotski–Plemelj formulae|DRT1E6WY6K]] and a Riemann–Hilbert problem, which shows that $\\ell - \\ell' = \\kappa$ for $K^0$.\nOne then builds a regularizer $M$ with $M K = I + T$ for a compact operator $T$, and the Fredholm theory of $I + T$ transfers the results to $K$.\nThe term with $k$ is compact, so it changes neither the finiteness nor the index.\n\nExample: on the unit circle, take $a(t) = (t + 1)/2$, $b(t) = (t - 1)/2$ and $k = 0$, so $\\kappa = -1$.\nWith $P = \\frac12 (I + S)$ and $Q = \\frac12 (I - S)$, one has $K = a + bS = t P + Q$.\nOn $\\varphi = \\sum_n c_n t^n$, $P$ keeps the terms with $n \\ge 0$ and $Q$ the terms with $n \u003c 0$, so $K \\varphi = \\sum_{n \\ge 0} c_n t^{n+1} + \\sum_{n \u003c 0} c_n t^n$.\nSo $K \\varphi = 0$ forces $\\varphi = 0$, and $K \\varphi = f$ is solvable only if the constant Fourier coefficient of $f$ vanishes: $\\ell = 0$, $\\ell' = 1$, and $\\ell - \\ell' = -1 = \\kappa$.\n\nThe normal type condition $a^2 - b^2 \\ne 0$ is needed: without it, the homogeneous equation may have infinitely many independent solutions or the range may fail to be closed.\nWhen $\\kappa = 0$, the three statements give the classical Fredholm alternative, but in general $\\ell \\ne \\ell'$, unlike for Fredholm equations of the second kind.\n\n## Explanations\n\n### L1 human\n\nA singular integral equation of normal type behaves almost like a system of linear equations.\nThere are finitely many independent solutions of the homogeneous equation and finitely many conditions for the general equation to be solvable.\nThe difference of these two numbers is the index, and it can be computed by counting how many times a function made from the coefficients winds around $0$.\n\n### L2 human\n\nThese are the singular analogues of Fredholm's theorems.\nThey were proved by Fritz Noether, and the singular integral operators of normal type are classical examples of Fredholm operators with nonzero [[Fredholm index|2V9Q4KHY1T]].\nThe index formula $\\ell - \\ell' = \\kappa$ is an early case of an index theorem, which expresses an analytic quantity through a topological one, here a winding number.\nIn the Hölder setting the theorems are presented in Muskhelishvili's book on singular integral equations; the same results hold for operators on $L^p(\\Gamma)$, $1 \u003c p \u003c \\infty$, with continuous coefficients.\n",
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