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:

(11.76) |

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

(11.77) |

At this point I could continue and extract

(11.78) |

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.