There are several types of integrals which go under the name of a ``Dirichlet integral.'' The integral

(1) |

The integral

(2) |

Another integral is denoted

(3) |

There are two types of Dirichlet integrals which are denoted using the letters , , , and . The type 1 Dirichlet integrals are denoted , , and , and the type 2 Dirichlet integrals are denoted , , and .

The type 1 integrals are given by

(4) |

where is the Gamma Function. In the case ,

(5) |

The type 2 integrals are given for -D vectors and , and ,

(6) |

(7) |

(8) |

(9) | |||

(10) |

and are the cell probabilities. For equal probabilities, . The Dirichlet integral can be expanded as a Multinomial Series as

(11) |

(12) | |||

(13) |

where

(14) |

(15) | |||

(16) |

where

(17) |

**References**

Sobel, M.; Uppuluri, R. R.; and Frankowski, K.
*Selected Tables in Mathematical Statistics, Vol. 4: Dirichlet Distribution--Type 1.* Providence, RI: Amer. Math. Soc., 1977.

Sobel, M.; Uppuluri, R. R.; and Frankowski, K.
*Selected Tables in Mathematical Statistics, Vol. 9: Dirichlet Integrals of Type 2 and Their Applications.*
Providence, RI: Amer. Math. Soc., 1985.

Weisstein, E. W. ``Dirichlet Integrals.'' Mathematica notebook DirichletIntegrals.m.

© 1996-9

1999-05-24