Past exam of the complex analysis: conformal maps and geometry course of the University of Oxford / 2023 / 3 / a

The radial harmonic function with value zero at $|z|=r$ and one at $|z|=1$ isIndeed $\log |z|$ is harmonic away from zero, and the displayed expression has the required boundary values. Uniqueness of the Dirichlet problem identifies it as the harmonic measure of an annulus.

The concentric normalization of an eccentric annulus here usesThis Möbius transformation preserves the unit circle and sends the inner circle's real endpoints $0,4/5$ to $-1/2,1/2$. Its image circle is symmetric about the real axis and hence is $|w|=1/2$. Equivalently, expanding $|T(z){|}^2=1/4$ gives $|z{|}^2=(4/5)\mathrm{Re}z$, precisely the inner-circle equation. Thus $T$ sends the domain to $1/2<|w|<1$. Conformal invariance of harmonic measure and i giveThe zero of $T$ lies in the removed inner disk, so this logarithm is harmonic throughout $\Omega$.