The literature seems rather coy on this point.
While $\sqrt{z}$ is not meromorphic on the complex plane $\mathbb{C}$, can it be regarded as globally meromorphic on the appropriate Riemann surface (two branched copies of $\mathbb{C}$), or (equivalently?) locally meromorphic at $z=0$? Moreover, can the root of the function at $z=0$ be regarded as a zero of order $1/2$?
And moreover, is $1/\sqrt{z}$ also meromorphic on the surface, and can it be regarded as having a pole of order $1/2$?
EDIT: Clarified(?) that I was asking whether the function globally meromorphic on $2 \mathbb{C}$.