Analog of Beta-function 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?  

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}
