 Holomorphic square root

A holomorphic square root of a nonvanishing holomorphic function $f$ is a holomorphic function $g$ satisfying $g^2=f$. It exists locally. On a connected open domain it exists globally exactly when the winding number of $f\circ \gamma$ around zero is even for every closed path $\gamma$ in the domain. Continuing a local square root once along such a path multiplies it by $(-1{)}^m$, where $m$ is that winding number. Thus the obstruction concerns parity, whereas a holomorphic logarithm needs every such winding number to be zero.

On $\mathbb{C}\setminus \{0\}$, the function $f(z)=z$ has no global holomorphic square root: its value winds once on the unit circle, and the lift shown under monodromy changes sign. In contrast, $f(z)=z^2$ winds twice there and has global square roots $g(z)=z$ and $g(z)=-z$. It still has no holomorphic logarithm on that domain. This distinguishes the two global continuation conditions.