We have already seen how if we let **E** or **B** be given by

or | (15.1) |

then

- Both the fields given above and their partner fields (given by the curl) have zero divergence.
- The fields given above are completely transverse, since (operator).
- The partner fields given by the curl are not purely transverse.
- In order to be consistent, the fields above are also the curls of the partner fields. In fact, this follows from vector identities for divergenceless fields.

It is therefore sensible to define, once and for all, a set of
multipoles that embody these properties. In addition, anticipating a
need to treat longitudinal fields as well as transverse fields, we will
define a third kind of multipoles with zero curl but non-zero
divergence. These will necessarily be ``connected'' to sources (why?).
We will call these ``pre-computed'' combinations of bessel functions,
vector spherical harmonics, and their curls the *Hansen* Multipoles
(following unpublished notes from L. C. Biedenharn as I have been
unable to determine his original reference):

- The Hansen Multipoles

- Green's Functions for the Vector Helmholtz Equation
- Multipolar Radiation, revisited
- A Linear Center-Fed Half-Wave Antenna
- Connection to Old (Approximate) Multipole Moments
- Angular Momentum Flux
- Concluding Remarks About Multipoles
- Table of Properties of Vector Harmonics