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.

Title	multivariate gamma function (complex-valued)
Canonical name	MultivariateGammaFunctioncomplexvalued
Date of creation	2013-03-22 14:22:10
Last modified on	2013-03-22 14:22:10
Owner	mathpeter (5480)
Last modified by	mathpeter (5480)
Numerical id	14
Author	mathpeter (5480)
Entry type	Definition
Classification	msc 62H10
Defines	gamma function (multivariate complex)