In other words, are the primitves (*) a complete set of solutions (in the mathematical sense of completeness)?. Can any solution be found as combination of primitives? If so, how is the proof?

Thank you alexb.

This is contradicted by what immediately follows

>study the
>primitives (*) solutions of waves(called "modes" which are
>"transversal magnetic TM, transversal electric TE and
>transversal electromagnetic TEM" and then we form
>combinations of these modes.

Right, this is exactly what I wrote.

>What if there are others
>"modes" that we don't know about but which can actully
>arise?

Sure, but this is less important than the question whether a combination (in some sense) of the modes at hand is generic enough to produce all solutions in some range.

>In other words, are the primitves (*) a complete set of
>solutions (in the mathematical sense of completeness)?.

First, this question differs from "whether there exist other modes?"
Second, the answer to your question depends on the set of primitives, on the equation, domain, and boundary conditions. As it is, it is too general to have a unique answer. Third, of course wherever the method of combining the primitives is employed in classical problems, it is shown that such combinations span a broad class of solutions.

>Can
>any solution be found as combination of primitives? If so,
>how is the proof?

Again, this is too general a question. But, for example, wherever the border consists of a straight line, a half-line, or a segment, coordinates are are selected such that on the border the primitives take a simpler form, by say losing one of the factors. The remaining functions might serve as the basis of Fourier transform which would demonstrate completeness for a particular kind of boundary problems.

Thanks again

I think this is a small question, completely of no consequence.