Dirichlet Integrals

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

(1)

appears in Dirichlet's principle.

The integral

(2)

where the kernel is the Dirichlet kernel, gives the th partial sum of the Fourier series.

Another integral is denoted

(3)

for , ..., .

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)

where  is the gamma function. In the case ,

(6)

where the integration is over the triangle  bounded by the x-axis, y-axis, and line  and  is the beta function.

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

(7)

(8)

(9)

where

			(10)
			(11)

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

(12)

For small ,  and  can be expressed analytically either partially or fully for general arguments and .

			(13)
			(14)

where

(15)

is a hypergeometric function.

			(16)
			(17)

where

(18)

REFERENCES:

Jeffreys, H. and Jeffreys, B. S. "Dirichlet Integrals." §15.08 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 468-470, 1988.

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.