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  "body": "\n## Definition\n\nThe *error function* is the function $\\operatorname{erf} \\colon \\R \\to \\R$ given by the [[definite integral|04TMP0XGMJ]]\n$$\\operatorname{erf} x = \\frac{2}{\\sqrt{\\pi}} \\int_0^x e^{-t^2}\\, dt,$$\nwhere $e^{-t^2}$ uses the [[exponential function|0J0FCWV32D]].\n\n## Remarks\n\nExample: $\\operatorname{erf} 0 = 0$, and the [[Gaussian integral|0DFT1GP1JP]] $\\int_0^\\infty e^{-t^2}\\, dt = \\frac{\\sqrt{\\pi}}{2}$ gives $\\lim_{x \\to \\infty} \\operatorname{erf} x = 1$.\nThe integrand is even, so $\\operatorname{erf}(-x) = -\\operatorname{erf} x$.\nIts derivative is $\\operatorname{erf}' x = \\frac{2}{\\sqrt{\\pi}} e^{-x^2} \u003e 0$, so $\\operatorname{erf}$ increases strictly from $-1$ to $1$.\n\nNon-example: $x \\mapsto \\int_0^x e^{-t^2}\\, dt$ is not the error function; without the factor $\\frac{2}{\\sqrt{\\pi}}$ its limit at $\\infty$ is $\\frac{\\sqrt{\\pi}}{2}$, not $1$.\n\nThe distribution function of the standard [[normal distribution|05AQE0QR0D]] is $\\Phi(x) = \\frac12\\left(1 + \\operatorname{erf}\\frac{x}{\\sqrt{2}}\\right)$.\nThe same integral, along any path from $0$ to $z$, defines $\\operatorname{erf} z$ as an entire function of $z \\in \\C$, with power series $\\operatorname{erf} z = \\frac{2}{\\sqrt{\\pi}} \\sum_{n \\ge 0} \\frac{(-1)^n z^{2n+1}}{n!\\,(2n+1)}$.\n",
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