Exactly the opposite is true in the **far zone**. Here
and the exponential oscillates *rapidly*. We can approximate the
argument of the exponential as follows:

(13.9) |

where we have assumed that and used a binomial expansion of the root sum. We neglect higher order terms. Note that this approximation is good

Then

(13.10) |

In the far zone, the solution behaves like an outgoing spherical wave times an amplitude that depends on integral over the source that depends on angles in an intricate fashion.

At this point I could continue and extract

(13.11) |

(if the source is actually small enough to allow expansion of the exponential in a series

Instead we are going to do it right. We will begin by reviewing the
solutions to the *homogeneous* Helmholtz equation (which should
really be discussed before we sweat solving the *inhomogeneous*
equation, don't you think?) and will construct the *multipolar
expansion* for the outgoing and incoming (and stationary) wave Green's
function. Using this, it will be a trivial matter to write down a
formally exact *and* convergent solution to the integral equation
*on all space* that we can chop up and approximate as we please.
This will provide a much more natural (and accurate) path to
multipolar radiation. So let's start.