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 com mon spherical harmonics, spheroidal harmonics provide an improved descripti on of the field near an elongated source. The total (spherical or spheroida l) MI can be retrieved using a superposition of elementary source images. T his 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 formula e linking the spherical and spheroidal MI by use of addition theorems and e xpansions of the Green function. Although it is possible to obtain the sphe roidal MI from the spherical MI, perfect accuracy for a complex elongated s ource is not possible. In contrast, the spherical MI, which is appropriate for the magnetic field at the remote region, is accurately generated from t he spheroidal MI, which provides the most precise description at the near z one. (C) 2001 American Institute of Physics.