Fast spheroidal multipole imaging of elementary magnetic sources on the axis

The multipole image (MI) is the set of coefficients in a harmonic expansion of the scalar magnetic potential of a magnetic source. Compared to the common spherical harmonics, spheroidal harmonics provide an improved description of the field near an elongated source. The total (spherical or spheroidal) MI can be retrieved using a superposition of elementary source images. This article presents fast two-term recursive formulae for multipole imaging of an elementary dipolar source on the axis. These results are compared to one-term recursions for the spherical MI. Another useful result is formulae linking the spherical and spheroidal MI by use of addition theorems and expansions of the Green function. Although it is possible to obtain the spheroidal MI from the spherical MI, perfect accuracy for a complex elongated source is not possible. In contrast, the spherical MI, which is appropriate for the magnetic field at the remote region, is accurately generated from the spheroidal MI, which provides the most precise description at the near zone.