= Solution
{c}
Take positive $\Gamma$ to mean counterclockwise <circulation>. A <line vortex> at $\zeta_0$ has <complex potential of a two-dimensional potential flow> $\Gamma\log(\zeta-\zeta_0)/(2\pi i)$. The <method of images> imposes the <no-penetration boundary condition> on the real axis by placing an image <line vortex> of strength $-\Gamma$ at $\bar\zeta_0$. Thus
$$
\boxed{W(\zeta)=\frac{\Gamma}{2\pi i}
\log\!\left(\frac{\zeta-\zeta_0}{\zeta-\bar\zeta_0}\right)+C}.
$$
For real $\zeta$, the numerator and denominator have equal modulus, so the <streamfunction> is
$$
\operatorname{Im}W=-\frac\Gamma{2\pi}
\log\left|\frac{\zeta-\zeta_0}{\zeta-\bar\zeta_0}\right|
=\text{constant}.
$$
The real axis is therefore a <streamline>. The image <line vortex> lies outside the mapped fluid region, while the <complex velocity>
$$
W'(\zeta)=\frac\Gamma{2\pi i}
\left(\frac1{\zeta-\zeta_0}-\frac1{\zeta-\bar\zeta_0}\right)
$$
decays as $|\zeta|\to\infty$. This selects the flow due to the <line vortex> and the boundary, without superposing an imposed background <velocity field>.
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