# multivariate gamma function (complex-valued)

The complex multivariate gamma function is defined as

$${\stackrel{~}{\mathrm{\Gamma}}}_{m}(a)={\int}_{\U0001d504}{e}^{-\mathrm{Tr}A}{|A|}^{a-m}dA,$$ | (1) |

where $\U0001d504$ is the set of all $m\times m$ positive, complex-valued Hermitian matrices^{}, i.e.

$$\U0001d504=\{A\in {\u2102}^{m\times m}|A={A}^{H},A>0\}.$$ | (2) |

It can also be expressed in terms of the gamma function^{} as follows

$${\stackrel{~}{\mathrm{\Gamma}}}_{m}(a)={\pi}^{\frac{1}{2}m(m-1)}\prod _{i=1}^{m}\mathrm{\Gamma}(a-i+1).$$ | (3) |

## Reference

A. T. James,“Distributions^{} 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) |