The ``error function'' encountered in integrating the Gaussian Distribution.

(1) | |||

(2) | |||

(3) |

where Erfc is the complementary error function and is the incomplete Gamma Function. It can also be defined as a Maclaurin Series

(4) |

(5) | |||

(6) |

It is an Odd Function

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

For , erf may be computed from

(14) | |||

(15) | |||

(16) | |||

(17) |

(Acton 1990). For ,

(18) |

Using Integration by Parts gives

(19) |

so

(20) |

(21) |

A Complex generalization of
is defined as

(22) | |||

(23) | |||

(24) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Error Function'' and ``Repeated Integrals of the Error Function.''
§7.1-7.2 in *Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables, 9th printing.* New York: Dover, pp. 297-300, 1972.

Acton, F. S. *Numerical Methods That Work, 2nd printing.* Washington, DC: Math. Assoc. Amer., p. 16, 1990.

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 568-569, 1985.

Spanier, J. and Oldham, K. B. ``The Error Function
and Its Complement
.''
Ch. 40 in *An Atlas of Functions.* Washington, DC: Hemisphere, pp. 385-393, 1987.

© 1996-9

1999-05-25