The real-valued multivariate gamma function is defined by \begin{equation} \Gamma_m(a) = \int_{\mathfrak{S}} e^{-\Tr S} \left|S\right|^{a-{1 \over 2}(m+1)}\, {\rm d} S, \end{equation} where $\mathfrak{S}$ is the set of all $m \times m$ real, positive definite symmetric matrices, i.e. \begin{equation} \mathfrak{S} = \left\{S \in \Bbb{R}^{m \times m} \mid S > 0, x^{\rm T}Sx > 0\, \forall\, x \in \mathbb{R}^{m \times 1}\setminus\{ 0\}\right\}. \end{equation}The real-valued multivariate gamma function can also be expressed in terms of the gamma function as follows

\begin{equation} \Gamma_m(a) = \pi^{{1 \over 4} m (m-1)} \prod\limits_{i=1}^{m}\Gamma\left(a-{1 \over 2}(i-1)\right). \end{equation}

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