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\begin{aligned} l & = \log (c) = - \frac{1}{2} \log (| Σ |) + \log Γ (\frac{ν + m}{2}) + (m - 1) \log Γ (\frac{ν}{2}) - m \log Γ (\frac{ν + 1}{2}) \\ - \frac{ν + m}{2} \log (1 + ζ^{T} Σ^{- 1} ζ / ν) + \frac{ν + 1}{2} \sum_{j = 1}^{m} \log (1 + \frac{ζ_{j}^{2}}{ν}) \\ \frac{\partial l}{\partial ν} & = \frac{1}{2} g (\frac{ν + m}{2}) + \frac{m - 1}{2} g (\frac{ν}{2}) - \frac{m}{2} g (\frac{ν + 1}{2}) \\ - \frac{1}{2} \log (1 + ζ^{T} Σ^{- 1} ζ / ν) + \frac{ν + m}{2 ν^{2}} \frac{ζ^{T} Σ^{- 1} ζ}{1 + ζ^{T} Σ^{- 1} ζ / ν} \\ + \frac{1}{2} \sum_{j = 1}^{m} \log (1 + ζ_{j}^{2} / ν) - \frac{ν + 1}{2 ν^{2}} \sum_{j = 1}^{m} \frac{ζ_{j}^{2}}{1 + ζ_{j}^{2} / ν} \\ - \frac{(ν + m)}{ν} \frac{ζ^{T} Σ^{- 1} (d ζ / d ν)}{1 + ζ^{T} Σ^{- 1} ζ / ν} + \frac{ν + 1}{ν} \sum \frac{ζ_{j} (d ζ_{j} / d ν)}{1 + ζ_{j}^{2} / ν} \end{aligned}

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\begin{align*} l & = \log (c) = -\frac12 \log ( |\Sigma |) +\log \Gamma \left(\frac{\nu +m}{2}\right) +(m-1) \log \Gamma \left(\frac\nu 2 \right) - m \log \Gamma \left(\frac{\nu +1}{2}\right)\\ & -\frac{\nu + m}{2} \log (1+ \zeta ^ T \Sigma ^{-1}\zeta /\nu ) +\frac{\nu +1}{2} \sum _{j=1}^ m \log \left(1+ \frac{\zeta _ j^2}{\nu }\right)\\ \frac{\partial l}{\partial \nu } & = \frac12 g\left(\frac{\nu +m}{2}\right) +\frac{m-1}{2} g\left(\frac\nu 2 \right)-\frac m2 g\left(\frac{\nu +1}{2}\right)\\ & -\frac12 \log (1+\zeta ^ T \Sigma ^{-1}\zeta /\nu ) + \frac{\nu +m}{2\nu ^2} \frac{\zeta ^ T \Sigma ^{-1}\zeta }{1+\zeta ^ T \Sigma ^{-1}\zeta /\nu }\\ & + \frac12 \sum _{j=1}^ m \log (1+\zeta _ j^2/ \nu ) - \frac{\nu +1}{2\nu ^2} \sum _{j=1}^ m \frac{\zeta _ j^2}{1+\zeta _ j^2/\nu }\\ & - \frac{(\nu + m)}{\nu }\frac{\zeta ^ T \Sigma ^{-1}(d\zeta /d \nu )}{1+\zeta ^ T \Sigma ^{-1}\zeta / \nu } +\frac{\nu +1}{\nu } \sum \frac{\zeta _ j (d\zeta _ j /d\nu )}{1+\zeta _ j^2/\nu } \end{align*}