= Solution
For an analytic complex potential $f$ whose singularities lie outside $|z|=a$, the <circle> theorem adds the reflected potential
$$
F(z)=f(z)+\overline{f(a^2/\bar z)}.
$$
The added term is analytic outside the <circle> wherever the reflected original is regular; in particular no new exterior singularities occur when $f$ is regular inside the <circle>. On $|z|=a$, the relation $a^2/\bar z=z$ makes $F=f+\bar f$ real. Thus its imaginary part, the stream function, is constant on the <circle> and the normal velocity vanishes. This proves the theorem, with any harmless additive constant left free. Its usual decay condition at infinity selects the surrounding flow inherited from $f$.
Write $a_1=z_1$ for the <planar potential dipole>[dipole] position to distinguish it from the <circle> radius, now one. Substitution gives
$$
\boxed{F(z)=\frac m{z-a_1}+\frac{\bar m\,z}{1-\bar a_1z},\qquad|a_1|>1.}
$$
The image singularity is at $1/\bar a_1$, strictly inside the solid <circle>. This exterior potential has the original <planar potential dipole>[dipole] and an impermeable <cylinder> boundary.
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