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  "body": "\n## Definition\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 a Hölder continuous function on $\\Gamma \\times \\Gamma$.\nConsider the singular integral operator of a [[singular integral equation|E30EA6ETZ4]] of normal type\n$$K \\varphi(t) = a(t) \\, \\varphi(t) + \\frac{b(t)}{\\pi i} \\, \\mathrm{p.v.} \\int_\\Gamma \\frac{\\varphi(\\tau)}{\\tau - t} \\, d\\tau + \\int_\\Gamma k(t, \\tau) \\, \\varphi(\\tau) \\, d\\tau, \\quad t \\in \\Gamma.$$\nThe *index* of $K$ is the integer\n$$\\kappa = \\frac{1}{2\\pi} \\Big[ \\arg \\frac{a(t) - b(t)}{a(t) + b(t)} \\Big]_\\Gamma.$$\nHere the bracket is the change of a continuous branch of the argument of $(a - b)/(a + b)$ after one run of $t$ around $\\Gamma$.\nIn other words, $\\kappa$ is the [[winding number|0NQSV2EWDR]] around $0$ of the closed curve $t \\mapsto (a(t) - b(t))/(a(t) + b(t))$.\n\n## Remarks\n\nThe condition $a^2 - b^2 \\ne 0$ says that $a - b$ and $a + b$ do not vanish, so the quotient is a nonvanishing continuous function on $\\Gamma$ and its winding number is defined.\nExample: on the unit circle, $a(t) = (t + 1)/2$ and $b(t) = (t - 1)/2$ give $a + b = t$ and $a - b = 1$, so $a^2 - b^2 = t \\ne 0$.\nThe quotient is $1/t = e^{-i\\theta}$ for $t = e^{i\\theta}$, whose argument decreases by $2\\pi$, so $\\kappa = -1$.\nExample: for $a = 0$ and $b = 1$ the quotient is the constant $-1$, so $\\kappa = 0$.\nNon-example: $a = b = 1$ violates $a^2 - b^2 \\ne 0$; the quotient is $0$ everywhere, and no index is defined.\nThe index does not depend on $k$, and it is additive: the index of a product of two such operators is the sum of their indices.\n",
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