Analytic branch of a square root

If $\ell$ is an analytic branch of the logarithm on a connected domain avoiding zero, then $\exp (\ell /2)$ is an analytic branch of the square root. Any two such branches differ everywhere by one constant sign.

For $z=r{e}^{i\theta }$ with $-\pi <\theta \le \pi$, the principal square root is $\sqrt{r}{e}^{i\theta /2}$. Its standard analytic domain removes the nonpositive real axis.