= Solution
For each case omit the dipole's own singular field when evaluating the surrounding <complex velocity>. With the convention $W'=u-iv$ this gives $d\bar a/dt=W'_{\mathrm{image}}(a)$, which is the conjugated equation requested in the source. In case (a), differentiating $\bar m z/(1-\bar a z)$ gives
$$
\boxed{\frac{d\bar a}{dt}=\frac{\bar m}{(1-|a|^2)^2}}.
$$
For case (b), differentiate the three image terms:
$$
\boxed{\frac{d\bar a}{dt}=-\frac{\bar m}{(a-\bar a)^2}+\frac{\bar m}{(a+\bar a)^2}+\frac{m}{4a^2}
=\frac{\bar m}{4(\operatorname{Im}a)^2}+\frac{\bar m}{4(\operatorname{Re}a)^2}+\frac{m}{4a^2}}.
$$
For case (c), differentiation gives
$$
\boxed{\frac{d\bar a}{dt}=\frac{i\bar m}{(a+i\bar a)^2}=\frac{\bar m}{2(\operatorname{Re}a+\operatorname{Im}a)^2}}.
$$
The physical equation for $da/dt$ is the complex conjugate of the displayed equation. These expressions apply away from the walls, where the point-dipole model is nonsingular.
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