MultivariateGammaFunctionComplexValued 2004-05-13 11:21:46 2006-10-17 07:16:12 Definition gamma function (multivariate complex) Gamma multivariate complex % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \DeclareMathOperator{\Tr}{Tr} 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} \subsection*{Reference} A. T. James,``Distributions of matrix variates and latent roots derived from normal samples,'' {\it Ann. Math. Statist.}, vol. 35, pp. 475-501, 1964.