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:
At this point I could continue and extract
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.