difficulty understanding branch of the logarithm

Question

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.

Answer 1

We know that $\sqrt{f(z)}=\exp(\frac12\log f(z))$. You choose a value for $\log f(z_0)$. Here the path of integration does not matter because the region supplied is simply connected and $f(z)$ which here is $z^2-1$ is non zero in the region. So the integral $I(z)=\int_{z_0}^z\frac{f'(z)dz}{f(z)}$ is well-defined. Check that it is analytic. You can thereafter define $\log f(z)=\log f(z_0)+I(z)$. This gives an analytic branch of $\log f$ and the choice of $\log f(z_0)$ determines the branch.

Having gotten a well defined branch for the square root. You can get one for $\log(z+\sqrt{z^2-1})$

Answer 2

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 $z\notin(-\infty,1]$