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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.