Can I obtain the expression $$ \nu \log (1+\frac{\theta}{\alpha}) $$ as a limit from the expression $$ \delta \gamma- \delta (\gamma^{1/\kappa} + 2 \theta)^\kappa $$ with $\kappa \to 0$ where $\nu$ and $\alpha$ can be chosen dependent of $\delta, \gamma, \kappa$ but not $\theta$.
The motivation is this document where it is claimed that the $\Gamma$ distribution arises as a limit of the tempered stable distribution, the first is the cumulant transform of the $\Gamma$ distribution and the second of the tempered stable.
I'm guessing to somehow use the limit $$ \lim_{w\to \infty} w (z^{1/w}-1) = \log (z) $$ but I haven't made any mentionable progress.
