Multivariate gamma function

In mathematics, the multivariate Gamma distribution, $\Gamma_p(\cdot)$ , is a generalization of the Gamma function. It is useful in multivariate statistics.

It has two equivalent definitions:

$\Gamma_p(a)= \int_{S\in {\mathbf S}} \exp\left( -{\rm trace}(S)\right) \left|S\right|^{a-(p+1)/2} dS$

where ${\mathbf S}$ is the set of all positive-definite matrices. The other is more useful in practice:

$\Gamma_p(a)= \pi^{p(p-1)/4}\Pi_{j=1}^p \Gamma\left[ a+(1-j)/2\right].$

Thus

and so on.

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