Gaussian Error Function

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A Gaussian Error Function is an error function that is from a Gaussian function family.



  • (Wikipedia, 2013) ⇒
    • In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics and partial differential equations. It is defined as:[1][2] :[math]\displaystyle{ \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}\,\mathrm dt. }[/math]

      The complementary error function, denoted erfc, is defined as :[math]\displaystyle{ \begin{align} \operatorname{erfc}(x) & = 1-\operatorname{erf}(x) \\ & = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,\mathrm dt. \end{align} }[/math] The imaginary error function, denoted erfi, is defined as :[math]\displaystyle{ \operatorname{erfi}(z) = -i\,\,\operatorname{erf}(i\,z). }[/math] When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: :[math]\displaystyle{ w(z) = e^{-z^2}\operatorname{erfc}(-iz) . }[/math]

  1. Andrews, Larry C.; Special functions of mathematics for engineers
  2. Greene, William H.; Econometric Analysis (fifth edition), Prentice-Hall, 1993, p. 926, fn. 11