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I am trying to solve this problem:

Does there exist a function $f(z)$, that is analytic at $E=\{x+iy :x>y\}$ and provides $f^2(z)=z$ for every $z \in \mathbb C$.

I have seen a solution that assumes $f(z)=\sqrt z$, but i have a bad feeling about this way.

Do you have a good solution (or convince me that the square root solution is good).

Thanks.

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    First you say $f$ is analytic in $E$, but then you apply $f$ to every $z \in \mathbb C$ ... what did you really mean?2012-01-19

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