Multivariate gamma function

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In mathematics, the multivariate Gamma function, Γp(·), 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 of size p\times p. 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

  • Γ1(a) = Γ(a)
  • Γ2(a) = π1 / 2Γ(a)Γ(a − 1 / 2)
  • Γ3(a) = π3 / 2Γ(a)Γ(a − 1 / 2)Γ(a − 1)

and so on.

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