How can we define a branch for $\sqrt{ 1 + \sqrt{z} } $? I know a branch for $\sqrt{z}$ is the negative real axis. But, how can we deal with the square root of $1+ \sqrt{z} $ ?
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You only need a branch when what's under the radical can be zero. If you take the negative real axis as the branch for $\sqrt z$, then $1+\sqrt z$ can't be zero, so no further branch-taking is needed. – Gerry Myerson Sep 29 '15 at 07:05
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So, we can take this branch for $\sqrt{1 + \sqrt{z} }$ and so it is analytic outside of it? – Sep 29 '15 at 07:14
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Sure looks like it. – Gerry Myerson Sep 29 '15 at 07:18
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the more interesting question would be how to handle a minus sign under the square root.. – tired Sep 29 '15 at 13:03