The complex multivariate gamma function is defined as \begin{equation} \tilde{\Gamma}_m(a)=\int_{\mathfrak{A}} e^{-\Tr A}|A|^{a-m} {\rm d}A, \end{equation}where $ \mathfrak{A}$ is the set of all $ m \times m$ positive, complex-valued Hermitian matrices, i.e. \begin{equation} \mathfrak{A} = \left\{A \in \Bbb{C}^{m \times m} | A = A^H, A > 0\right\}. \end{equation}It can also be expressed in terms of the gamma function as follows \begin{equation} \tilde{\Gamma}_m(a)=\pi^{{1 \over 2}m(m-1)} \prod\limits_{i=1}^{m}\Gamma(a-i+1). \end{equation}

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