Logarithmic potential

Summary

The logarithmic potential of a finite measure $\mu$ with compact support in the plane is $p_\mu(z) = \int \log(1/\abs{z - w}) \, d\mu(w)$. It is the potential of the measure $\mu$ with respect to the logarithmic kernel. It is harmonic off the support of $\mu$ and superharmonic on the whole plane. Logarithmic potentials connect potential theory with complex analysis. For a polynomial $p$ with zeros $z_1, \dots, z_d$, $\log(1/\abs{p})$ is, up to a constant, the logarithmic potential of the point masses at the zeros.