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:
() & = & ik &sum#sum;_L [ J_L()
&int#int;_r^&infin#infty;&mu#mu;_0 (') H_L(')^(&ast#ast;) d^3r' .

& & + . H^+_L() &int#int;_0^r &mu#mu;_0 (')
J_L(')^(&ast#ast;) d^3r' ]
where, by convention,
means that the
is
conjugated but the bessel/neumann/hankel function is *not*.
This is because the only point of the conjugation is to construct
from the
-sum for each
via the addition
theorem for spherical harmonics. We certainly *don't* want to
change
into
, which changes the time dependent behavior of
the solution^{13.9}. Note that the integral over all space is broken
up in such a way that the Green's function expansions above always
converge. This solution is exact everywhere in space *including
inside the source itself!*

We can therefore simplify our notation by defining certain functions
of the radial variable:
() = &sum#sum;_L i k[ C_L(r) J_() +
S_L(r) H^+_L() ].
In this equation,
C_L(r) & = & &int#int;_r^&infin#infty;&mu#mu;_0 (') H_L(')^(&ast#ast;) d^3r'

S_L(r) & = & &int#int;_0^r &mu#mu;_0 (') J_L(')^(&ast#ast;) d^3r'.

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.