multivariate gamma function (complex-valued)

The complex multivariate gamma function is defined as

$${\tilde{\mathrm{\Gamma }}}_m(a)={\int }_{𝔄}{e}^{-\mathrm{Tr}A}{|A|}^{a-m}dA,$$(1)

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

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

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

$${\tilde{\mathrm{\Gamma }}}_m(a)={\pi }^{\frac{1}{2}m(m-1)}\prod _{i=1}^m\mathrm{\Gamma }(a-i+1).$$(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.