difficulty understanding branch of the logarithm Here is one past qual question,
Prove that the function $\log(z+ \sqrt{z^2-1})$ can be defined to be analytic on the domain $\mathbb{C} \setminus (-\infty,1]$
(Hint: start by defining an appropriate branch of $ \sqrt{z^2-1}$ on    $\mathbb{C}\setminus (-\infty,1]$ )
It just seems typical language problem. I do not see point of being rigorous here. But
I think I have difficulty understanding the branch cut. I don't know if someone will explain me in some easy way and try to explain the solution to the problem. Any help will be much appreciated.

Alternatively, you can just take the standard branch for $\sqrt{z}$ excluding $(-\infty,0]$ and then compute $\sqrt{z-1}\sqrt{z+1}$ which is defined for $z+1,z-1\notin(-\infty,0]$, that is, for $z\notin(-\infty,1]$
