I would like to know the general closed form of the expected maximum of i.i.d Gumbel variables. I found this onlie: Expectation of the Maximum of iid Gumbel Variables
In the linked page, it shows the log-sum form for $E[\max_{i}(v_i+\epsilon_i)]$ when each $\epsilon_i$ are i.i.d and follows $G(\mu,1)$ where the scale parameter $\sigma=1$.
But, I could not find the general form of the expected maximum when the scale parameter is not equal to 1. Especially, I am interested in the closed log-sum form when $\mu=0$.
Since this is in the line of random utility models, I found this: $$E[\max_{i}(v_i+\epsilon_i)]=ln(\sum_{i}exp(\frac{v_i}{\sigma}))+\gamma$$ where $\gamma$ is Euler constant and $\sigma$ as a dispersion parameter ($\mu$=0, $\sigma\neq 1$).
Is there any paper or proof that shows this solution with a dispersion parameter?