There are several types of integrals which go under the name of a "Dirichlet integral." The integral
![D[u]=int_Omega|del u|^2dV](/images/equations/DirichletIntegrals/NumberedEquation1.svg) |
(1)
|
appears in Dirichlet's principle.
The integral
![1/(2pi)int_(-pi)^pif(x)(sin[(n+1/2)x])/(sin(1/2x))dx,](/images/equations/DirichletIntegrals/NumberedEquation2.svg) |
(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)
|
