= Point-vortex drift under a conformal map
{title2=$\overline{\dot c}=\frac{\Gamma}{2\pi i}\frac{g''(c)}{2g'(c)}+g'(c)R'(g(c))$}
Let $\zeta=g(z)$ be a <conformal map>, and let a <point vortex> of <circulation> $\Gamma$ lie at $z=c$. Suppose the mapped <complex potential of a two-dimensional potential flow> is
$$
W(\zeta)=\frac\Gamma{2\pi i}\log(\zeta-g(c))+R(\zeta),
$$
where $R$ is a <holomorphic function> near $g(c)$ and represents the other <point vortices>, boundaries, and any imposed flow. The physical <complex velocity> is $w'=g'W'$. The <point vortex> follows its regular <velocity>, found by removing $\Gamma/[2\pi i(z-c)]$ before taking the limit. A <Taylor expansion> proves
$$
\frac{g'(z)}{g(z)-g(c)}=\frac1{z-c}+\frac{g''(c)}{2g'(c)}+O(z-c),
$$
which gives the title formula. The <complex conjugate> occurs because the <complex velocity> convention is $w'=u-iv$, whereas $\dot c=u+iv$.
For a mapped <upper half-plane> with no imposed flow, the <method of images> gives $R(\zeta)=-\Gamma\log(\zeta-\overline{g(c)})/(2\pi i)$, hence
$$
\overline{\dot c}=\frac\Gamma{2\pi i}
\left[\frac{g''(c)}{2g'(c)}-\frac{g'(c)}{g(c)-\overline{g(c)}}\right].
$$
The first term is essential: excluding a mapped singularity and then transforming its regular <velocity> does not in general equal subtracting the physical singularity after transformation.
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