I have parameters of two Gumbel distributions ($\mu_1, \beta_1)$ and $(\mu_2, \beta_2)$. Since max of 2 Gumbels is a Gumbel, I'd like to compute $\mu_m, \beta_m$, so that:

$Gumbel(\mu_m,\beta_m)$ = $max(Gumbel(\mu_1, \beta_1), Gumbel(\mu_2, \beta_2))$

Any hints how to do this? I've found a solution for the case when $\beta_1 = \beta_2$ in here (point 6, page 105):

but I'm even not sure how this was computed.