Separation of Variables In Electrostatics

[M-Speezy]

I am curious how legitimate a solution Separation of Variables tends to give. I've been working problems out of Griffith's book on Electromagnetism, and am often uneasy as to the way things are done. I have two specific issues. The first, is that in spherical it is often necessary to remove entire portions of the solution, as you would divide by zero otherwise. This makes perfect sense at surface level, because otherwise the solution wouldn't make any sense. But how do you know that is then the solution, that you haven't left off important parts? The other issue I have is with finding the constants through a Fourier Series. Often, the potential would be given at a specific point, so everything is simplified to be analyzed at JUST that specific point, so that the constant can be found. My question is how do we know that the constant found at that point will apply to all space (that is being considered)? It all 'makes sense', but seems very dubious to me. If anyone would be able to explain some of these ideas, or maybe why I'm perhaps being silly, I would really appreciate it. Thanks!

 

[M-Speezy]

I think I may have answered one of my questions, the one about ignoring portions of solutions. Griffiths discusses the fact that sines or cosines can be used to construct any solution, so it is somehow justifiable. I think to understand it at a deeper level I'd be looking at a lot of math. My other question still stands, if anyone knows anything about it!