Green's Function of a homogeneous cylinder

[PeteyCoco]

I've been reading this article for a prof this summer: http://arxiv.org/pdf/1302.0245v1.pdf
I'm having some trouble following the math in Appendix B: Green's Function Of A Homogeneous Cylinder (page 9). Can someone explain to me why there is a factor of
$\frac{1}{\rho\rho'}$
in front of the Green function? Can someone walk me through the representation of the function in eq (B3) Also, how did equation (B4) come about? It looks similar to eq (1) of the article, but something is happening that I can't follow.

 

[PeteyCoco]

Okay, I've been looking through Jackson and it seems that it might have what I need. Any help is still appreciated because I'm sure I'll hit a roadblock in Jackson (I've only were read griffiths)Sent from my iPhone using Physics Forums

 

[Born2bwire]

That factor comes about because they simply pulled it out of the radial part as explained in the sentence following Equation (B3). It's a simple definition. My guess is if you substitute that \sqrt{\rho \rho'} factor into Equation (B4) and work out the math then you will end up with the scalar wave equation in cylindrical coordinates for the radial component. This would be Equation (3.141) in my third edition of Jackson. So, once again, just substitute the \tilde{gm} into the gm in the Jackson equation and I bet you get Equation (B4).