= Solution
Suppose an analytic extension $F$ existed on the punctured plane. The <identity theorem> would extend $F^2=z$ from $D$ to that connected domain, so $F$ would never vanish and differentiation would give $2F'/F=1/z$. Integrate around the positively oriented unit circle. The <argument principle> says $\int F'/F$ is $2\pi i$ times the integer winding number of the closed curve $F(e^{it})$ about zero, whereas
$$
2\oint\frac{F'}Fdz=\oint\frac{dz}{z}=2\pi i.
$$
This would make that winding number $1/2$, a contradiction. Hence \b[no such analytic extension exists]. Equivalently, continuation of a square root once around zero reverses its sign, preventing a single-valued holomorphic function on the punctured plane.
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