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

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

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
.

where

(15)

is a hypergeometric function .

where

(18)

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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. Referenced
on Wolfram|Alpha Dirichlet Integrals
Cite this as:
Weisstein, Eric W. "Dirichlet Integrals."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletIntegrals.html

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