= Solution
Use the principal <complex square root>, $\sqrt z=\sqrt r\,e^{i\theta/2}$ with $-\pi<\theta<\pi$. For $F(z)=e^{z-\sqrt z}/z$, the local expansion on this <holomorphic branch> is
$$
F(z)=\frac1z-\frac1{\sqrt z}+\frac32+O(\sqrt z).
$$
The fractional power prevents a single-valued <Laurent series> in a punctured disc. In particular, zero is a <branch point>, not an <isolated singularity> classified as a <pole>. Merely observing that $zF(z)\to1$ on the slit plane would not establish a <simple pole>: $zF(z)$ does not extend as a <holomorphic function> through a whole disc around zero.
The negative real half-line is a chosen <branch cut>. Its upper and lower boundary values differ because $\sqrt{-x+i0}=i\sqrt x$ and $\sqrt{-x-i0}=-i\sqrt x$ for $x>0$. Continuation once around zero changes the sign of the <complex square root>, and hence changes $F$ in general. Other <branch cuts> are possible; this one defines the principal <holomorphic branch>.
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