Gaussian Error Function

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

• AKA: ERF, Gaussian Error Integral.
• Context:
  • It can be:
    • a Complementary Error Function.
    • a Complex Error Function.
• See: Gaussian Function, Gaussian Discrete Kernel, Forecast Error Function.

---

---

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: :[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]

---