Differential entropy of the multivariate student-t distribution The differential entropy of the multivariate student-t distribution when the covariance matrix is the identity matrix is given by 
$$
h = - \log \frac{ \Gamma \left( \frac{\nu+d}{2} \right) }
                      { \Gamma \left( \frac{\nu}{2} \right)
                        (\nu \pi)^{\frac{d}{2}}
                      }
+
\left( \frac{\nu+d}{2} \right)
\left(
\psi \left( \frac{\nu+d}{2} \right)
-
\psi \left( \frac{\nu}{2} \right)
\right)
$$ (source: Shannon Entropy and Mutual Information for Multivariate Skew-Elliptical Distribution by Arellano-Valle).
What about the general case ($\boldsymbol{\Sigma} \ne \boldsymbol{I}$)?
 A: To get the differential entropy in the general case, we draw on two properties:


*

*If $\boldsymbol{x}$ is a  standard Student-t random vector, then $\boldsymbol{y} = \boldsymbol{\mu} + \boldsymbol{L} \boldsymbol{x}$ is a Student-t random vector with mean $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma} = \boldsymbol{L}\boldsymbol{L}'$.

*It is known that for a vector valued random variable $\boldsymbol{x}$ and a matrix $\boldsymbol{A}$ we have $h(\boldsymbol{A} \boldsymbol{x}) = h(\boldsymbol{x}) + \log |\boldsymbol{A}|$ (see https://en.wikipedia.org/wiki/Differential_entropy#Properties_of_differential_entropy).


As a result, we can write for a Student-t random vector $\boldsymbol{x}$ with mean $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$
$$
h(\boldsymbol{x}) 
=
h_{\boldsymbol{\Sigma}=\boldsymbol{I}}
+
\frac{1}{2} \log | \boldsymbol{\Sigma} |
$$
where
$$
h_{\boldsymbol{\Sigma}=\boldsymbol{I}}
 = - \log \frac{ \Gamma \left( \frac{\nu+d}{2} \right) }
                      { \Gamma \left( \frac{\nu}{2} \right)
                        (\nu \pi)^{\frac{d}{2}}
                      }
+
\left( \frac{\nu+d}{2} \right)
\left(
\psi \left( \frac{\nu+d}{2} \right)
-
\psi \left( \frac{\nu}{2} \right)
\right)
$$
