# multivariate gamma function (real-valued)

The real-valued multivariate gamma function is defined by

 $\Gamma_{m}(a)=\int_{\mathfrak{S}}e^{-\operatorname{Tr}S}\left|S\right|^{a-{1% \over 2}(m+1)}\,{\rm d}S,$ (1)

where $\mathfrak{S}$ is the set of all $m\times m$ real, positive definite symmetric matrices, i.e.

 $\mathfrak{S}=\left\{S\in\mathbb{R}^{m\times m}\mid S>0,x^{\rm T}Sx>0\,\forall% \,x\in\mathbb{R}^{m\times 1}\setminus\{0\}\right\}.$ (2)

The real-valued multivariate gamma function can also be expressed in terms of the gamma function as follows

 $\Gamma_{m}(a)=\pi^{{1\over 4}m(m-1)}\prod\limits_{i=1}^{m}\Gamma\left(a-{1% \over 2}(i-1)\right).$ (3)

## Reference

A. T. James, “Distributions of matrix variates and latent roots derived from normal samples,” Ann. Math. Statist., vol. 35, pp. 475-501, 1964.

Title multivariate gamma function (real-valued) MultivariateGammaFunctionrealvalued 2013-03-22 14:22:06 2013-03-22 14:22:06 rspuzio (6075) rspuzio (6075) 15 rspuzio (6075) Definition msc 62H10 gamma function (multivariate real)