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  "body": "\n## Definition\n\nLet $\\operatorname{erf}$ be the [[error function|7Y56QCZSZJ]].\nThe *complementary error function* is the function $\\operatorname{erfc} \\colon \\R \\to \\R$ given by\n$$\\operatorname{erfc} x = 1 - \\operatorname{erf} x.$$\n\n## Remarks\n\nSince $\\frac{2}{\\sqrt{\\pi}} \\int_0^\\infty e^{-t^2}\\, dt = 1$, the definition is equivalent to\n$$\\operatorname{erfc} x = \\frac{2}{\\sqrt{\\pi}} \\int_x^\\infty e^{-t^2}\\, dt.$$\n\nExample: $\\operatorname{erfc} 0 = 1$, and $\\operatorname{erfc}(-x) = 2 - \\operatorname{erfc} x$ because $\\operatorname{erf}$ is odd.\n\nNon-example: $x \\mapsto 1 + \\operatorname{erf} x$ is not $\\operatorname{erfc}$; it tends to $2$ rather than $0$ as $x \\to \\infty$.\n\nFor large $x \u003e 0$, $\\operatorname{erfc} x \\sim \\frac{e^{-x^2}}{x\\sqrt{\\pi}}$.\nThe definition extends to complex $z$ by $\\operatorname{erfc} z = 1 - \\operatorname{erf} z$, which is an entire function.\nThe scaled function $e^{x^2} \\operatorname{erfc} x$ is often used in numerical work.\n",
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