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) |
(5) |
The type 2 integrals are given for -D vectors and , and ,
(6) |
(7) |
(8) |
(9) | |||
(10) |
(11) |
(12) | |||
(13) |
(14) |
(15) | |||
(16) |
(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 Eric W. Weisstein