Logarithmic kernel

Summary

The logarithmic kernel is the function $k(z, w) = \log(1/\abs{z - w})$ of two points of the plane. It is the planar counterpart of the Newtonian kernel. It equals $2\pi$ times the fundamental solution of the Laplacian of the plane, translated to the pole $w$. Integrating it against a measure gives the logarithmic potential, which links potential theory with complex analysis, polynomial approximation and the distribution of zeros of polynomials. Unlike the Newtonian kernel, it takes negative values when the two points are more than one unit apart.