The virtue of the Hansen solutions is that they ``automatically'' work
to decompose field components into parts that are mutual curls (as
required by Faraday/Ampere's laws for the fields) or divergences (as
required by Gauss's laws for the fields):
_L & = & 0

_L & = & 0

_L & = & i k f_&ell#ell;(kr) Y_L()
Hence
and
are divergenceless, while the divergence of
is a scalar solution to the HHE!
is related to the scalar field
and the gauge invariance of the theory in an interesting way we will
develop. Also:
_L & = & -ik _L

_L & = & ik _L

_L & = & 0
which shows how
and
are now ideally suited to form the
components of electric and magnetic multipole fields mutually linked by
Ampere's and Faraday's law.

Robert G. Brown 2017-07-11