Dirichlet Integrals
المؤلف:
Jeffreys, H. and Jeffreys, B. S.
المصدر:
"Dirichlet Integrals." §15.08 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press
الجزء والصفحة:
...
30-7-2019
1743
Dirichlet Integrals
There are several types of integrals which go under the name of a "Dirichlet integral." The integral
![D[u]=int_Omega|del u|^2dV](http://mathworld.wolfram.com/images/equations/DirichletIntegrals/NumberedEquation1.gif) |
(1)
|
appears in Dirichlet's principle.
The integral
![1/(2pi)int_(-pi)^pif(x)(sin[(n+1/2)x])/(sin(1/2x))dx,](http://mathworld.wolfram.com/images/equations/DirichletIntegrals/NumberedEquation2.gif) |
(2)
|
where the kernel is the Dirichlet kernel, gives the 
th partial sum of the Fourier series.
Another integral is denoted
![delta_k=1/piint_(-infty)^infty(sinalpha_krho_k)/(rho_k)e^(irho_kgamma_k)drho_k=<span style=]() {0 for |gamma_k|>alpha_k; 1 for |gamma_k|<alpha_k " src="http://mathworld.wolfram.com/images/equations/DirichletIntegrals/NumberedEquation3.gif" style="height:41px; width:304px" /> |
(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)
|
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.
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