The next term in the multipolar expansion is the term:

(11.140) |

- -sum the product of the 's
- use the small expansion for in the integral and combine it with the explicit form for the resulting to form a dot product
- cancel the 's
- explicitly write out the hankel function in exponential form

(11.141) |

Of course, you can get it directly from J9.9 (to a lower approximation) as
well, but that does *not* show you what to do if the small
approximation is not valid (in step 2 above) and it neglects part of the
outgoing wave!

There are two important and independent pieces in this expression. One of the
two pieces is symmetric in
and
and the other is antisymmetric
(get a minus sign when the coordinate system is inverted). Any vector
quantity can be decomposed in this manner so this is a very general step:

(11.142) |