Dirichlet Integrals

There are several types of integrals which go under the name of a ``Dirichlet integral.'' The integral

$$D[u] =\int_\Omega \vert\nabla u\vert^2\,dV$$ (1)

appears in Dirichlet's Principle.

The integral

$${1\over 2\pi}\int_{-\pi}^\pi f(x){\sin[(n+{\textstyle{1\over 2}})x]\over \sin({\textstyle{1\over 2}}x)}\,dx,$$ (2)

where the kernel is the Dirichlet Kernel, gives the $n$th partial sum of the Fourier Series.

Another integral is denoted

$$\delta_k\equiv {1\over \pi}\int_{-\infty}^\infty {\sin \alph... ...rt > \alpha_k$\cr 1 & for $\vert\gamma_k\vert < \alpha_k$\cr}$$ (3)

for $k=1$, ..., $n$.

There are two types of Dirichlet integrals which are denoted using the letters $C$, $D$, $I$, and $J$. The type 1 Dirichlet integrals are denoted $I$, $J$, and $IJ$, and the type 2 Dirichlet integrals are denoted $C$, $D$, and $CD$.

The type 1 integrals are given by

$$I \equiv \int\!\!\!\int \cdots\int f(t_1+t_2+\ldots+t_n){t_1}^{\alpha_1-1}{t_2}^{\alpha_2-1} \cdots {t_n}^{\alpha_n-1}\, dt_1\, dt_2\, dt_n$$

$$= {\Gamma(\alpha_1)\Gamma(\alpha_2)\cdots\Gamma(\alpha_n)\over \Gam... ...\alpha_n}\right)} \int_0^1 f(\tau)\tau^{\left({\sum_n \alpha}\right)-1}\,d\tau,$$ (4)

where $\Gamma(z)$ is the Gamma Function. In the case $n=2$,

$$I = \int\!\!\!\int _T x^py^q\,dx\,dy = {p!q!\over (p+q+2)!} = {B(p+1,q+1)\over p+q+2},$$ (5)

where the integration is over the Triangle $T$ bounded by the $x$-axis, $y$-axis, and line $x+y = 1$ and $B(x,y)$ is the Beta Function.

The type 2 integrals are given for $b$-D vectors ${\bf a}$ and ${\bf r}$, and $0\leq c\leq b$,

$$C_{\bf a}^{(b)}({\bf r}, m) = {\Gamma(m+R)\over\Gamma(m)\pro... ...i}^{r_i-1}\,dx_i\over \left({1+\sum_{i=1}^b x_i}\right)^{m+R}}$$ (6)

$$D_{\bf a}^{(b)}({\bf r}, m) = {\Gamma(m+R)\over\Gamma(m)\pro... ...i}^{r_i-1}\,dx_i\over \left({1+\sum_{i=1}^b x_i}\right)^{m+R}}$$ (7)

$$CD_{\bf a}^{(c,d-c)}({\bf r}, m)= {\Gamma(m+R)\over\Gamma(m)... ...i}^{r_i-1}\,dx_i\over\left({1+\sum_{i=1}^b x_i}\right)^{m+R}},$$ (8)

where

$$R \equiv \sum_{i=1}^k r_i$$ (9)

$$a_i \equiv {p_i\over 1-\sum_{i=1}^k p_i},$$ (10)

and $p_i$ are the cell probabilities. For equal probabilities, $a_i=1$. The Dirichlet $D$ integral can be expanded as a Multinomial Series as

$D_{\bf a}^{(b)}({\bf r}, m)={1\over\left({1+\sum_{i=1}^b}\right)^m}\sum_{x_1<r_1}\cdots \sum_{x_b<r_b}{m-1+\sum_{a=1}^b x_i\choose m-1, x_1, \ldots, x_b}$
$\prod_{i=1}b \left({a_i\over 1+\sum_{k=1}^b a_k}\right)^{x_i}.\quad$	(11)

For small $b$, $C$ and $D$ can be expressed analytically either partially or fully for general arguments and $a_i=1$.

$$C_1^{(1)}(r_2; r_1) = {\Gamma(r_1+r_2)\,{}_2F_1(r_2, r_1+r_2; 1+r_2; -1)\over r_2\Gamma(r_1)\Gamma(r_2)}$$ (12)

$$C_1^{(2)}(r_2, r_3; r_1) = {\Gamma(r_1 + r_2 + r_3)\over r_2 \Gamma(r_1) \Gamma(r_2) \Gamma(r_3)} \int_0^1 {}_2F_1\, y^{r_3-1}(1+y)^{-(r_1+r_2+r_3)}\,dy,$$ (13)

where

$${}_2F_1\equiv {}_2F_1(r_2, r_1+r_2+r_3; 1+r_2, -(1+y)^{-1})$$ (14)

is a Hypergeometric Function.

$$D_1^{(1)}(r_2; r_1) = {\Gamma(r_1+r_2)\,{}_2F_1(r_1, r_1+r_2; 1+r_1; -1)\over r_1\Gamma(r_1)\Gamma(r_2)}$$ (15)

$$D_1^{(2)}(r_2, r_3; r_1) = {\Gamma(r_1 + r_2 + r_3)\over (r_1 + r_3) \Gamma(r_1) \Gamma(r_2) \Gamma(r_3)}\int_1^\infty {}_2F_1\, y^{r_3-1}\,dy,$$ (16)

where

$${}_2F_1\equiv {}_2F_1(r_1 + r_3, r_1 + r_2 + r_3; 1 + r_1 + r_3; -1 - y).$$ (17)

References

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.

Weisstein, E. W. ``Dirichlet Integrals.'' Mathematica notebook DirichletIntegrals.m.