= Solution
Choose a continuous <branch of a multivalued function> of the <complex logarithm> on a simply connected region avoiding zero. Then
$$
g=e^{-\theta}e^{i\ln r}=e^{i\operatorname{Log}z},\qquad
\boxed{g'(z)=\frac{i}{z}g(z).}
$$
Alternatively $u_r=-e^{-\theta}\sin(\ln r)/r=v_\theta/r$ and $v_r=e^{-\theta}\cos(\ln r)/r=-u_\theta/r$, verifying the <Cauchy-Riemann equations>. The <holomorphy> is local to a chosen <branch of a multivalued function>: increasing $\theta$ by $2\pi$ multiplies $g$ by $e^{-2\pi}$, so this is not a single-valued <holomorphic function> on the whole punctured plane.
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