Scalar multipole expansions and their dipole equivalents

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Abstract
If the source of a field that satisfies Poisson’s equation can be written as the divergence of a vector s, then a scalar multipole expansion of the source can be evaluated in terms of s, which is a dipole density. A multipole expansion in terms of the dipole density can be computed about different origins. This allows us to evaluate the expansion of a dipole displaced from the origin and find a method of approximating some multipole expansions by displaced dipoles. In many physical applications it is known that the source is a displaced dipole, and we can find its location from a multipole expansion at some convenient location. It is possible to derive pictures in terms of dipole densities that in the proper limit become the individual multipoles. There are, however, ambiguities in that for some multipoles more than one picture gives the proper field.

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