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.svg) | (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.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) |
