= Holomorphic square root
{title2=$g^2=f$}
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.
Back to article page
