Harmonic measure of an annulus (${\omega }_{A(r,1)}(z,\mathbb{T})=\log (|z|/r)/\log (1/r)$)

For $0<r<1$, the harmonic measure of the outer circle of $r<|z|<1$ is the title expression. The function $\log |z|$ is harmonic away from zero; the normalization gives boundary values zero on the inner circle and one on the outer circle. Uniqueness of the Dirichlet problem proves the formula. A conformal map to an annulus transports it to a doubly connected domain.