 Solution

The Milne-Thomson circle theorem states that if $f$ is an analytic complex potential with no singularity on or inside $|z|=a$, thenmakes $|z|=a$ a streamline while retaining the original exterior singularities and adding reflected singularities inside the circle. On the circle the two terms are conjugates, so the resulting streamfunction is zero there.

Put $A=m{e}^{i\alpha }$ and $c=a^2/b$. Applying the formula to the exterior planar potential doublet givesEquivalently, up to an irrelevant constant,The image planar potential doublet is inside the obstacle at $c<a$; the only physical fluid singularity is the prescribed one at $b$.