Solution
= Solution
The radial <harmonic function> with value zero at $|z|=r$ and one at $|z|=1$ is
$$
\boxed{\omega_{A(r,1)}(z,\mathbb T)=\frac{\log(|z|/r)}{\log(1/r)}}.
$$
Indeed $\log|z|$ 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>.