# Gaussian Error Function

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

## References

### 2013

• (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Error_function
• 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] :$\displaystyle{ \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}\,\mathrm dt. }$

The complementary error function, denoted erfc, is defined as :\displaystyle{ \begin{align} \operatorname{erfc}(x) & = 1-\operatorname{erf}(x) \\ & = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,\mathrm dt. \end{align} } The imaginary error function, denoted erfi, is defined as :$\displaystyle{ \operatorname{erfi}(z) = -i\,\,\operatorname{erf}(i\,z). }$ 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: :$\displaystyle{ w(z) = e^{-z^2}\operatorname{erfc}(-iz) . }$

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