= Solution
{c}
The physical <complex potential of a two-dimensional potential flow> is $w(z)=W(g(z))$. Applying the <chain rule> gives the <complex velocity>
$$
\boxed{u-iv=w'(z)=\frac\Gamma{2\pi i}
\left(1-\frac1{z^2}\right)
\left[\frac1{z+z^{-1}-c-c^{-1}}
-\frac1{z+z^{-1}-\bar c-\bar c^{-1}}\right]}.
$$
The physical <velocity> is $(\operatorname{Re}w',-\operatorname{Im}w')$. A useful equivalent expression follows from
$$
g(z)-g(c)=\frac{(z-c)(z-c^{-1})}{z}.
$$
Factoring the image expression in the same way makes the $-1/z$ terms cancel, leaving
$$
\boxed{u-iv=\frac\Gamma{2\pi i}
\left[\frac1{z-c}+\frac1{z-c^{-1}}
-\frac1{z-\bar c}-\frac1{z-\bar c^{-1}}\right]}.
$$
This represents the physical <line vortex> at $c$, an image <line vortex> of the same strength at $c^{-1}$, and image <point vortices> of opposite strength at $\bar c$ and $\bar c^{-1}$. All three image locations are outside the fluid region. Both boundary pieces inherit the constant <streamfunction> of the real axis in the mapped <complex plane>.
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