Multivariate gamma function

In mathematics, the multivariate Gamma distribution, Γ_p(·), is a generalization of the Gamma function. It is useful in multivariate statistics.

It has two equivalent definitions. One is

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

where S is the set of all positive-definite matrices. The other one, more useful in practice, is

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

Thus

and so on.

See also: Multivariate gamma function, Gamma function, Mathematics, Multivariate statistics, Positive-definite matrix