Milne-Thomson circle theorem

Remarks

On the circle, $a^2/z = \bar z$, so $\bar f(a^2/z) = \overline{f(z)}$ and $w(z) = 2\operatorname{Re} f(z)$ is real. Hence the stream function $\operatorname{Im} w$ vanishes on the circle. The function $\bar f$ is holomorphic wherever $f$ is holomorphic at the conjugate point. So the singularities of $\bar f(a^2/z)$ are the image points $a^2/\bar z_0$ of the singularities $z_0$ of $f$ outside the disc. These image points lie inside the disc. Example: the uniform stream $f(z) = Uz$ gives $w(z) = U(z + a^2/z)$, the potential flow around a circular cylinder. A point source at $z_0$ outside the circle produces an image source at $a^2/\bar z_0$ and a sink of the same strength at the centre.

Going deeper

For a student

The theorem is the circle version of the method of images. Combined with conformal maps such as the Joukowsky transform, it produces potential flows past airfoils and other shapes from flows past a circle. The forces on the cylinder can then be computed with the Blasius theorem.