= Solution
{c}
A free <point vortex> is advected by the regular part of the <velocity field> at its position. Its own singular <complex velocity> $\Gamma/[2\pi i(z-c)]$ must first be subtracted. The image expression therefore gives
$$
\boxed{\dot x_c-i\dot y_c
=\lim_{z\to c}\left[w'(z)-\frac\Gamma{2\pi i(z-c)}\right]
=\frac\Gamma{2\pi i}
\left[\frac1{c-c^{-1}}-\frac1{c-\bar c}
-\frac1{c-\bar c^{-1}}\right]}.
$$
Equivalently, $\dot c$ is the <complex conjugate> of the boxed expression. This is the instantaneous <velocity>, evaluated with the current value of $c$.
To see why mapping the image <velocity> alone would miss a term, expand the mapped physical singularity:
$$
\frac{g'(z)}{g(z)-g(c)}
=\frac1{z-c}+\frac{g''(c)}{2g'(c)}+O(z-c).
$$
Thus the same <point-vortex drift under a conformal map> can be written
$$
\dot x_c-i\dot y_c
=\frac\Gamma{2\pi i}
\left[\frac{g''(c)}{2g'(c)}
-\frac{g'(c)}{g(c)-\overline{g(c)}}\right].
$$
The finite term involving $g''$ is part of the physical regular <velocity field>.
If $c=id$ with $d>1$, direct substitution into the image expression gives
$$
\begin{aligned}
\dot x_c-i\dot y_c
&=\frac\Gamma{2\pi}
\left[-\frac d{d^2+1}+\frac1{2d}+\frac d{d^2-1}\right]\\
&=\boxed{\frac\Gamma{2\pi}
\left[\frac1{2d}+\frac{2d}{d^4-1}\right]}.
\end{aligned}
$$
The bracket is strictly positive, and the result is real. Hence a positive <line vortex> initially moves horizontally to the right, a negative <line vortex> moves to the left, and the initial vertical <velocity> is zero. As $d\to\infty$, the leading term $\Gamma/(4\pi d)$ is the familiar drift of a <point vortex> above a straight wall.
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