Now that we have that under our belts we can address the multipolar
expansion of the vector potential intelligently. To begin with, we
will write the **general solution** for the vector potential in
terms of the multipolar expansion for the outgoing wave Green's
function defined above:

(11.113) |

where, by convention, means that the is conjugated but the bessel/neumann/hankel function is

We can therefore simplify our notation by defining certain functions
of the radial variable:

(11.114) |

(11.115) | |||

(11.116) |

Clearly and for , . At the origin the
solution is completely regular and *stationary*. Outside the
bounding sphere of the source distribution the solution behaves like a
linear combination of outgoing spherical multipolar waves. From now
on we will concentrate on the latter case, since it is the one
relevant to the zones.