I am trying to find out the maximum domain on $\mathbb{C}$, where the function $f=\sqrt{z^2 -1}$, $z\in \mathbb{C}$ becomes holomorphic.
I know that $\sqrt{z^2-1} = e^{\frac{1}{2}(log\left | z^2-1 \right | + iArg(z^2-1) +2\pi k)}$, $k \in \mathbb{Z}$
and that there are two discontinuous solutions given, but how do I manipulate the complex plane so that those solutions become continuous? An intuive as well as an mathematical explanation would be much appreciated.
Thank you.