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multivariate gamma function (complex-valued)
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(Definition)
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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}
A. T. James,``Distributions of matrix variates and latent roots derived from normal samples,'' Ann. Math. Statist., vol. 35, pp. 475-501, 1964.
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"multivariate gamma function (complex-valued)" is owned by mathpeter.
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| Also defines: |
gamma function (multivariate complex) |
| Keywords: |
Gamma multivariate complex |
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Cross-references: terms, Hermitian matrices, positive, gamma function, complex
This is version 11 of multivariate gamma function (complex-valued), born on 2004-05-13, modified 2006-10-17.
Object id is 5854, canonical name is MultivariateGammaFunctionComplexValued.
Accessed 3390 times total.
Classification:
| AMS MSC: | 62H10 (Statistics :: Multivariate analysis :: Distribution of statistics) |
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Pending Errata and Addenda
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