= Circle integral of a logarithm with a radial branch cut
{title2=$\int_{|z|=R}\operatorname{Log}z\,dz=2\pi iR e^{i\beta}$}
Take a <complex logarithm> whose radial <branch cut> has angle $\beta$. On a positively oriented circle, use the one-sided branch values on either side of its crossing with the cut. Parametrize from $\theta=\beta$ to $\beta+2\pi$; adding an integer multiple of $2\pi i$ to the branch does not change the integral because the integral of $dz$ is zero. Along the open arc, a primitive is $z\operatorname{Log}z-z$. Its endpoint difference is $2\pi i$ times the cut-crossing point $Re^{i\beta}$, proving the displayed formula. The circle crosses the cut, so this is a piecewise contour integral of boundary values, not the integral of a single holomorphic logarithm on a neighborhood of a closed circle. The <Cauchy integral theorem> therefore does not make it zero.
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