Analyticity of √z+√z\sqrt{z+\sqrt{z}}

I am trying to find where f(z)=√z+√zf(z)=\sqrt{z+\sqrt{z}} is analytic when considering the branch (−π,π)(-\pi,\pi). My attempt was this: I know that z+√zz+\sqrt{z} is analytic on the set A=C−{x+iy | x≤0,y=0}A=\mathbb{C}-\lbrace x+iy~|~ x\leq 0, y=0\rbrace, hence if I show that z+√zz+\sqrt{z} has image contained in AA, then ff is analytic on AA. I am having trouble doing this, though. Any help?

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As long as z∈A=C∖(−∞,0]z \in A = \Bbb{C}\setminus(-\infty, 0], we can easily check that √z\sqrt{z} is contained in the right-half plane. Then writing z+√z=√z(1+√z)z + \sqrt{z} = \sqrt{z}(1 + \sqrt{z}), we find that this is again in AA.
– Sangchul Lee
Oct 21 at 0:43

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