= Angular average of a logarithmic potential
{title2=$\frac1{2\pi}\int_0^{2\pi}\log|1+re^{i\theta}|\,d\theta=\log\max(1,r)$}
For $r<1$, average the real part of the convergent power series for $\log(1+re^{i\theta})$: 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.
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