The Gaussian integral, also called the Probability Integral, is the integral of the 1-D Gaussian over
. It can be computed using the trick of combining two 1-D Gaussians

(1) |

and switching to Polar Coordinates,

(2) |

However, a simple proof can also be given which does not require transformation to Polar Coordinates (Nicholas and Yates 1950).

The integral from 0 to a finite upper limit can be given by the
Continued Fraction

(3) |

The general class of integrals of the form

(4) |

(5) | |||

(6) | |||

(7) |

Then

(8) |

For , this is just the usual Gaussian integral, so

(9) |

(10) |

(11) |

For Even,

(12) |

so

(13) |

(14) |

so

(15) |

(16) |

(17) | |||

(18) | |||

(19) | |||

(20) | |||

(21) | |||

(22) | |||

(23) |

A related, often useful integral is

(24) |

(25) |

**References**

Nicholas, C. B. and Yates, R. C. ``The Probability Integral.'' *Amer. Math. Monthly* **57**, 412-413, 1950.

© 1996-9

1999-05-25