# multivariate gamma function (complex-valued)

The complex multivariate gamma function is defined as

 $\tilde{\Gamma}_{m}(a)=\int_{\mathfrak{A}}e^{-\operatorname{Tr}A}|A|^{a-m}{\rm d% }A,$ (1)

where $\mathfrak{A}$ is the set of all $m\times m$ positive, complex-valued Hermitian matrices, i.e.

 $\mathfrak{A}=\left\{A\in\mathbb{C}^{m\times m}|A=A^{H},A>0\right\}.$ (2)

It can also be expressed in terms of the gamma function as follows

 $\tilde{\Gamma}_{m}(a)=\pi^{{1\over 2}m(m-1)}\prod\limits_{i=1}^{m}\Gamma(a-i+1).$ (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 (complex-valued) MultivariateGammaFunctioncomplexvalued 2013-03-22 14:22:10 2013-03-22 14:22:10 mathpeter (5480) mathpeter (5480) 14 mathpeter (5480) Definition msc 62H10 gamma function (multivariate complex)