In mathematics, the multivariate gamma function Γp is a generalization of the gamma function.
It is useful in multivariate statistics, appearing in the probability density function of the Wishart and inverse Wishart distributions, and the matrix variate beta distribution.
positive-definite real matrices: where
The other one, more useful to obtain a numerical result is: In both definitions,
is a complex number whose real part satisfies
reduces to the ordinary gamma function.
The second of the above definitions allows to directly obtain the recursive relationships for
Where G is the Barnes G-function, the indefinite product of the Gamma function.
The function is derived by Anderson[2] from first principles who also cites earlier work by Wishart, Mahalanobis and others.
There also exists a version of the multivariate gamma function which instead of a single complex number takes a
-dimensional vector of complex numbers as its argument.
It generalizes the above defined multivariate gamma function insofar as the latter is obtained by a particular choice of multivariate argument of the former.