The name "Dirichlet integral" is used for several types of integrals. The integral
![D[u]=int_Omega|del u|^2dV](/images/equations/DirichletIntegrals/NumberedEquation1.svg) |
(1)
|
appears in Dirichlet's principle.
The classical improper Dirichlet integral is
 |
(2)
|
For an integrable 
-periodic
function 
, the 
th partial sum of its Fourier
series is
![s_n(t)=1/(2pi)int_(-pi)^pif(x)(sin[(n+1/2)(t-x)])/(sin[1/2(t-x)])dx,](/images/equations/DirichletIntegrals/NumberedEquation3.svg) |
(3)
|
where 
is a nonnegative integer and the quotient
is the Dirichlet kernel, with its removable value
at 
modulo 
.
For 
and real 
,
another Dirichlet integral is the symmetric Cauchy
principal value
 |
(4)
|
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 
.
For the type 1 integrals, let 
, 
, and 
. For a continuous
function 
on ![[0,1]](/images/equations/DirichletIntegrals/Inline24.svg)
,
 |
(5)
|
where 
is the gamma function. In the case 
,
 |
(6)
|
for nonnegative integers 
and 
, 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 use a positive integer 
, real numbers 
and 
, and finite thresholds 
for 
. Write 
and ![K=Gamma(m+R)/[Gamma(m)product_(i=1)^(b)Gamma(r_i)]](/images/equations/DirichletIntegrals/Inline38.svg)
. The integrals
are
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
where ![Q_C=product_(i=1)^(b)[0,a_i]](/images/equations/DirichletIntegrals/Inline48.svg)
,
,
and ![Q_c=product_(i=1)^(c)[0,a_i]×product_(i=c+1)^(b)[a_i,infty)](/images/equations/DirichletIntegrals/Inline50.svg)
for integers 
. Here the products of intervals
denote Cartesian products. For 
the mixed integral is 
, and for 
it is 
. The factor 
normalizes the integral over
to 1.
When the thresholds arise from cell probabilities, write 
and 
.
If all 
cell probabilities, including 
, are equal, then 
.
If the 
are positive integers, the upper-tail integral
has the finite sum
 |
(10)
|
where 
and 
.
For a positive integer 
, the coefficient in this finite sum
is the multinomial coefficient 
.
For one coordinate with unit threshold, write 
and 
. Then
 |  |  |
(11)
|
 |  |  |
(12)
|
where 
and 
is the hypergeometric function. For two
dimensions, write 
, 
, 
, and 
. Then
 |  |  |
(13)
|
 |  |  |
(14)
|
