= Solution
The <Milne-Thomson circle theorem> constructs a circular impermeable boundary in planar <potential flow>. Suppose $f(\zeta)$ is an incident <complex potential of a two-dimensional potential flow> with no singularities on or inside $|\zeta|=R$. Then
$$
\boxed{W(\zeta)=f(\zeta)+\overline{f\!\left(\frac{R^2}{\bar\zeta}\right)}}
$$
is a <complex potential of a two-dimensional potential flow> outside the <circle>, with the same exterior singularities and their strengths as the incident flow; the added image singularities lie inside the excluded disk. On $|\zeta|=R$, $R^2/\bar\zeta=\zeta$, so $W=f+\bar f$ is real. Its imaginary part, the <streamfunction>, is constant on the <circle>, proving the <no-penetration boundary condition>. The theorem gives the usual zero-<circulation> construction; a separately prescribed <circulation> can be added by its logarithmic vortex potential.
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