Say I have to calculate the quantity, $Log Tr [ -\Delta - \frac{1}{4} + m^2]$ on $H^n$. Then looking up the spectral measure $\mu(\lambda)$ and the eigenvalues of the Laplacian ($= -\Delta = - \nabla^i \nabla_i $) in the equations 2.36 and 2.34 in this paper, I would have written down the answer as,

$\int d\lambda \left (\mu_n(\lambda) = \vert \frac { \Gamma (i\lambda + (n-1)/2 ) } { \Gamma( i \lambda) } \vert ^2 \frac{ \pi }{2^{2(n-2)}\Gamma^2(n/2) } \right ) log ( \lambda^2 + m^2) $

Is the above answer correct? Am I reading the normalizations correctly?

Now in the special case of $n=2$ I see a certain paper to be writing the expression as, $\frac{Vol(\mathbb{H}^2) }{4\pi } \int tanh(\pi \sqrt{\lambda} ) log(\lambda + m^2) $

Now its not clear to me that the two expressions are equal! Can someone help fix the conventions required in the general expression so as to be able to match the specific $n=2$ result.

One notes this identity useful for the $n=2$ case that, $\mu_2 (\lambda) = \pi \lambda tanh(\pi \lambda)$