I know that $\sqrt z$ has two distinct values, one corresponds to the principle analytic branch of logarithm on $D_{\pi} = \mathbb C \setminus \{Re^{\pi}:R \geq 0 \}$ and other corresponds to the analytic branch of logarithm on $D_{3 \pi} = \mathbb C \setminus \{Re^{3 \pi}:R \geq 0 \}$. But I observe that $D_{\pi} = D_{3 \pi}$. Then both the branches of logarithm have the same branch cut and so they have the same points of singularities. Hence any branch of $\sqrt z$ is analytic on the entire complex plane other than the non-positive real axis.
Is it the true fact or not? Please help me.
Thank you in advance.