 Angular average of a logarithmic potential

For $r<1$, average the real part of the convergent power series for $\log (1+r{e}^{i\theta })$: every nonconstant Fourier mode vanishes. For $r>1$, factor out $r$ and apply the same argument to $r^{-1}$. The boundary singularity at $r=1$ is logarithmically integrable. This identity computes the expected logarithmic distance after an isotropic Gaussian displacement.