I know we can rewrite $\sqrt{z^2 +1}$ as $z^2 (1+z^{-2})$ and use this by looking at the principal branch of the function $\exp{\left(\frac{1}{2} \log(1+z^{-2})\right)}$.
However I am struggling to find the domain of holomorphicity that includes the exterior of the unit disc.
I keep trying to find where $\exp(\frac{1}{2} \log(1+z^{-2})$ has cuts and see that it is when $1+z^{-2}$ is negative real but I don't see how would relate to the unit circle as the value for $z$ I find is the complement of the interval $[-i,i]$.
Any help would be appreciated