# Help with multidimensional integral (Dirichlet type)

1. Jun 30, 2008

### DarkEternal

In a research problem (related to statistics), I've encountered a k-dimensional Dirichlet integral of the form:

$$f_k(x) = \int_{x_c}^\infty dx_1 \ldots \int_{x_c}^\infty dx_k\, \delta\left(x-\left(x_1+\ldots+x_k\right)\right)\ x_1^{-s} \ldots x_k^{-s}$$

where $$s>0$$. Essentially I am integrating the function $$x_1^{-s} \ldots x_k^{-s}$$ over the plane defined by $$x_1+\ldots+x_k=x$$, with the criterion $$x_k \geq x_c$$ providing the necessary cutoff so that the integral converges.

While I have been able to write a program to numerically integrate this, I am wondering if there is a closed form solution. It seems that this integral is related in some way to a k-dimensional generalization of the incomplete beta function or an integral of a Dirichlet probability distribution. However, when trying to generalize some of the derivations I've found in the literature, I find I run into trouble because of the cutoff (the integral is usually done for $$s<0$$). I've been able to derive

$$f_2(x) = x^{(-2(s-1)-1)}\, \theta\left(x-2x_c\right)\, \beta\left(\frac{x_c}{x},1-\frac{x_c}{x},1-s,1-s\right)$$ (where $$\theta(x)$$ is the Heaviside step function and $$\beta(x)$$ is the generalized incomplete beta function)

but have been unable to generalize the canonical derivations to higher k. It seems like the answer should look something like:

$$f_k(x)=x^{(-k(s-1)-1)}\, \theta\left(x-kx_c\right)\, g_k\left(\frac{x_c}{x},s\right)$$

where $$g_k\left(\frac{x_c}{x},s\right)$$ is the desired generalization.

Does anybody here have experience with these type of statistical integrals, or could point me to some literature? It seems like this problem must be solved, but I don't have much experience with these sorts of problems. I am also wondering if the approach taken in this paper, which evaluates the integral by a sum of the integrand evaluated at the vertices of the plane, can be applied to this problem (I suspect that it cannot?) Thanks.