= No entire square root of the identity function
If an <entire function> $h$ satisfied $h(z)^2=z$, then $h(0)=0$. Differentiating would give $2h(0)h'(0)=1$, a contradiction. More generally, the order of every zero of a square of a <holomorphic function> is even, so a simple zero prevents a holomorphic square root on any neighborhood containing it. A square root can still exist on a slit domain avoiding the zero. The <identity theorem> can force this obstruction from values specified only on a sequence accumulating elsewhere in a connected domain.
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