Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What is the multi-dimensional analogue of the Beta-function called? The Beta-function being $$B(x,y) = \int_0^1 t^x (1-t)^y dt$$

I have a function $$F(x_1, x_2,\ldots, x_n) = \int_0^1\cdots\int_0^1t_1^{x_1}t_2^{x_2}\cdots(1 - t_1 - \cdots-t_{n-1})^{x_n}dx_1\ldots dx_n$$ and I don't know what it is called or how to integrate it. I have an idea that according to the Beta-function: $$F(x_1, \ldots,x_n) = \dfrac{\Gamma(x_1)\cdots\Gamma(x_n)}{\Gamma(x_1 + \cdots + x_n)}$$

Is there any analogue for this integral such as Gamma-function form for Beta-function?

share|improve this question
    
For the Beta function, the variable of integration should be $t$ not $x$. The definition is in fact $$B(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1} dt$$ How would you generalize then? –  Spenser Dec 3 '12 at 13:53
    
yes, of course. I've already edit it. –  stepkamipt Dec 3 '12 at 15:02
    
You need to change it in your definition of $F$ also. You can't integrate with the arguments of the function. For example, what happens when you evaluate it? Say $F(1,1,...,1)$? Then you integrate with respect to $d1d1...d1$ which makes no sense. Note also that the integral of the Beta function has powers $x-1$ and $y-1$, not $x$ and $y$. –  Spenser Dec 3 '12 at 22:24
add comment

2 Answers

What you can look at is the Selberg integral. It is a generalization of the Beta function and is defined by

\begin{eqnarray} S_n(\alpha,\beta,\gamma) &=& \int_0^1\cdots\int_0^1\prod_{i=1}^n t_i^{\alpha-1}(1-t_i)^{\beta-1}\prod_{1\leq i<j\leq n}|t_i-t_j|^{2\gamma}dt_1\cdots dt_n \\ &=& \prod_{j=0}^{n-1}\frac{\Gamma(\alpha+j\gamma)\Gamma(\beta+j\gamma)\Gamma(1+(j+1)\gamma)}{\Gamma(\alpha+\beta+(n+j-1)\gamma)\Gamma(1+\gamma)} \end{eqnarray}

share|improve this answer
add comment

Maybe the integrand has the form

$$ t_1^{x_1-1}(1-t_1)^{x_2-1}(1-t_1-t_2)^{x_3-1}\cdots(1 - t_1 - \cdots-t_{n-1})^{x_n-1} .$$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.