Multivariate gamma function: Difference between revisions

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In [[mathematics]], the '''multivariate gamma function''', Γ_''p''(·), is a generalization of the [[gamma function]]. It is useful in [[multivariate statistics]], appearing in the [[probability density function]] of the [[Wishart distribution|Wishart]] and [[inverse Wishart distribution]]s, and the [[matrix variate beta distribution]].

It has two equivalent definitions. One is given as the following integral over the <math>p \times p</math> [[positive-definite matrix|positive-definite]] real matrices:

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:it follows that

::<math>

::<math>\frac{\partial \Gamma_p(a)}{\partial a} = \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma(a+(1-j)/2) \sum_{i=1}^p \psi(a+(1-i)/2) = \Gamma_p(a)\sum_{i=1}^p \psi(a+(1-i)/2).</math>

\begin{align}

::<math>\frac{\partial \Gamma_p(a)}{\partial a} & = \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma(a+(1-j)/2) \sum_{i=1}^p \psi(a+(1-i)/2) = \Gamma_p(a)\sum_{i=1}^p \psi(a+(1-i)/2).</math>[4pt]

& = \Gamma_p(a)\sum_{i=1}^p \psi(a+(1-i)/2).

\end{align}

</math>

{{no footnotes|date=May 2012}}