multivariate gamma function (complex-valued)

The complex multivariate gamma function is defined as

Γ~m(a)=𝔄e-TrA|A|a-mdA, (1)

where 𝔄 is the set of all m×m positive, complex-valued Hermitian matricesMathworldPlanetmath, i.e.

𝔄={Am×m|A=AH,A>0}. (2)

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

Γ~m(a)=π12m(m-1)i=1mΓ(a-i+1). (3)


A. T. James,“DistributionsDlmfPlanetmathPlanetmath 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)