= Solution
A <velocity potential> satisfies $\mathbf u=\nabla\phi$. Integration of $\phi_r=Q/(2\pi r)$ and $r^{-1}\phi_\theta=0$ gives
$$
\boxed{\phi=\frac Q{2\pi}\log r.}
$$
Choose the <streamfunction> convention $u=\psi_y$, $v=-\psi_x$. Its <streamfunction in polar coordinates> satisfies $u_r=r^{-1}\psi_\theta$ and $u_\theta=-\psi_r$, so
$$
\boxed{\psi=\frac Q{2\pi}\theta,\qquad w(z)=\phi+i\psi=\frac Q{2\pi}\log z.}
$$
Additive constants are immaterial. This <complex potential of a two-dimensional potential flow> is defined on a chosen <logarithm branch>. Its <complex velocity> $w'=Q/(2\pi z)=u-iv$ is single-valued away from the origin. Continuing once around the <line source> increases $\psi$ by $Q$, consistently with its nonzero <volume flux>.
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