Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / K / 3 / a / Solution

For an analytic complex potential $f$ whose singularities lie outside $|z|=a$, the circle theorem adds the reflected potentialThe 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 dipole position to distinguish it from the circle radius, now one. Substitution givesThe image singularity is at $1/{\bar{a}}_1$, strictly inside the solid circle. This exterior potential has the original dipole and an impermeable cylinder boundary.