MultivariateGammaFunctionRealValued 2004-05-13 11:09:38 2006-12-01 00:46:47 Definition gamma function (multivariate real) Gamma multivariate real % 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 real-valued multivariate gamma function is defined by \begin{equation} \Gamma_m(a) = \int_{\mathfrak{S}} e^{-\Tr S} \left|S\right|^{a-{1 \over 2}(m+1)}\, {\rm d} S, \end{equation} where $\mathfrak{S}$ is the set of all $m \times m$ real, positive definite symmetric matrices, i.e. \begin{equation} \mathfrak{S} = \left\{S \in \Bbb{R}^{m \times m} \mid S > 0, x^{\rm T}Sx > 0\, \forall\, x \in \mathbb{R}^{m \times 1}\setminus\{ 0\}\right\}. \end{equation} The real-valued multivariate gamma function can also be expressed in terms of the gamma function as follows \begin{equation} \Gamma_m(a) = \pi^{{1 \over 4} m (m-1)} \prod\limits_{i=1}^{m}\Gamma\left(a-{1 \over 2}(i-1)\right). \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.