Multipole expansion of the electromagnetic field In Jackson's Classical Electrodynamics, section 9.7, he develops the multipole expansion of the electromagnetic fields in terms of the vector spherical harmonics and the spherical Bessel and Hankel functions. His expansion is somewhat confusing, and I was wondering any other reference doing the same expansion, in some other manner?
 A: 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.
A: 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
$$
\cfrac{e^{ik\left|\mathbf{r}-\mathbf{r}'\right|}}{\left|\mathbf{r}-\mathbf{r}'\right|}=\sum_{n=0}^{\infty}\cfrac{1}{n!}\left(-\mathbf{r}'\cdot\nabla\right)^{n}\cfrac{e^{ikr}}{r}
$$
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.
A: It is better to avoid vector spherical harmonics by expanding the electric and magnetic scalar potentials instead of the vector fields.  The electric potential
is expanded in every EM textbook.  The magnetic scalar potential is treated in Section 7.10 of "Classical Electromagnetism, 2nd Edition" by Jerrold Franklin
