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Looking at a domain which is like an annulus (inner radius $r$ and outer radius $s>r$) can there be a (square)root function defined on it?

If it can't be single valued, is it possible to define one which is not single valued?

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Do you mean a single-valued square root? –  Giuseppe Negro Sep 11 '12 at 19:16
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the brief answer is no, not single valued if the origin is in the middle. In fact, Volume I of John B. Conway, page 202, Theorem VIII.2.2(h) a region $G$ is simply connected if and only if every analytic $f \in H(G)$ such that $f(z) \neq 0 \; \; \forall z \in G,$ there exists a $g \in H(G)$ such that $f(z) = \left( g(z) \right)^2.$ I believe $H(G)$ must mean holomorphic and single-valued, have not found the definition yet. Got it, page 151, section VII.2. –  Will Jagy Sep 11 '12 at 19:24
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