Analyticity of the branches of $\sqrt z$.

Question

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.

Answer 1

No, it's not true. There are many other branches of $\sqrt{z}$. All you need to do is remove enough points from the plane so that the complement has a continuous branch of the argument $\theta$, then define $\sqrt{re^{i\theta}} = re^{i\theta/2}$. In your example, you are removing the nonpositive real axis. But you can choose any curve going from 0 to $\infty$ as the branch cut, and define an analytic branch of $\sqrt{z}$ on the complement.