# error function

The *error function ^{}* $\mathrm{erf}:\u2102\to \u2102$ is defined as follows:

$$\mathrm{erf}(z)=\frac{2}{\sqrt{\pi}}{\int}_{0}^{z}{e}^{-{t}^{2}}\mathit{d}t$$ |

The *complementary error function ^{}* $\mathrm{erfc}:\u2102\to \u2102$ is defined as

$$\mathrm{erfc}(z)=\frac{2}{\sqrt{\pi}}{\int}_{z}^{\mathrm{\infty}}{e}^{-{t}^{2}}\mathit{d}t$$ |

The name “error function” comes from the role that these functions^{} play in the theory of the normal random variable. It is also worth noting that the error function is a special case of the confluent hypergeometric functions and of the Mittag-Leffler function^{}.

Note. By Cauchy integral theorem (http://planetmath.org/SecondFormOfCauchyIntegralTheorem), the choice path of integration in the definition of $\mathrm{erf}$ is irrelevant since the integrand is an entire function^{}. In the definition of $\mathrm{erfc}$, the path may be taken to be a half-line parallel to the positive real axis with endpoint $z$.

Title | error function |
---|---|

Canonical name | ErrorFunction |

Date of creation | 2013-03-22 14:46:51 |

Last modified on | 2013-03-22 14:46:51 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 10 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 33B20 |

Related topic | AreaUnderGaussianCurve |

Related topic | ListOfImproperIntegrals |

Related topic | UsingConvolutionToFindLaplaceTransform |

Defines | complementary error function |