TITLE: Existence of square roots and logarithms
QUESTION [13 upvotes]: Does there exist an open connected set in the complex plane on which the identity function has an analytic square root but not an analytic logarithm?

REPLY [5 votes]: Suppose $U \subseteq \mathbb{C}$ is a connected open set on which an analytic square root can be defined.  Then it follows easily that $U$ cannot contain the origin, and every closed curve in $U$ must have even winding number around the origin.  But any closed curve in the plane with nonzero winding number around the origin contains in its image a simple closed curve with winding number one around the origin, so $U$ cannot have any closed curves with nonzero winding number around the origin, and hence an analytic logarithm exists on $U$.
See also this question.