A comparison of scalar multipole expansions

Scalar multipole expansions have been widely used in many areas of physics as general solutions to Laplace’s equation. However, comparison of work by different authors is often complicated by the existence of several different forms of multipole expansions in different coordinate systems. In this paper we compare the spherical harmonic, Taylor’s series, direction cosine and traceless tensor expansions and present models of dipole configurations that in the proper limit give the individual multipoles. These pictures simplify the comparison of the series and aid in interpretation of the various multipole moments.