Multipole expansion of the electromagnetic field

You can try Modern Problems in Classical Electrodynamics by Charles Brau. I think it is relatively clear in this book.

At least I hope you can find the end-results in Zangwill or Brau. Then you should after trying be able to get there yourself.

Start from the expression for the electric potential. In the integral there is a $frac{1}{|bar{r}-bar{r}'|}$. You just need to make a Taylor expansion of that where $bar{r}'$ is small. Then you put that term for term into the integral and integrate. You will then get the multipole expansion for the electric potential. Taking the gradient times a minus sign will give you the electric field.

The Taylor series expansion along with Spherical wave expansion is presented in the book "Theory of Electromagnetic Wave Propagation" by Papas.

http://store.doverpublications.com/0486656780.html

The three dimensional Taylor expansion of the scalar and vector potentials due to a monochromatic current density is used to derive the Electric and magnetic fields. The fields of electric dipole and quadruple are derived in this manner. The expansion uses

$$ cfrac{e^{ikleft|mathbf{r}-mathbf{r}'right|}}{left|mathbf{r}-mathbf{r}'right|}=sum_{n=0}^{infty}cfrac{1}{n!}left(-mathbf{r}'cdotnablaright)^{n}cfrac{e^{ikr}}{r} $$

This method is more suited "if the source can be described as a superposition of an electric dipole, a magnetic dipole and an electric quadrupole"

Also, the spherical harmonic expansion given in Jackson was proposed by "Casimir and Bouwkamp" in their paper, which has elaborate presentation.