= Solution
The logarithm is defined away from the $z$ axis, where $x^2+y^2>0$. For the <vector> potential $\psi\widehat{\boldsymbol e}_z=(0,0,\psi)$, its <curl> is $(\psi_y,-\psi_x,0)$. Since
$$
\psi_x=\frac{ax}{x^2+y^2},\qquad \psi_y=\frac{ay}{x^2+y^2},
$$
we obtain
$$
\boxed{\boldsymbol u=\left(\frac{ay}{x^2+y^2},-\frac{ax}{x^2+y^2},0\right),\qquad x^2+y^2>0.}
$$
For $a>0$ this is clockwise circulation about the axis, the velocity structure of a <point vortex> in each transverse plane.
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