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I'm looking for a reference book that treats the theory behind the eigenfunctions solution of the so called

*vector*Helmholtz equation and its Neumann and Dirichlet problems.

I've already found a theory inside the last chapter of Morse & Feshbach's Methods of theoretical physics, vol.2, but that treatment I think is really general (and also quite complicated, for me, to follow) and it separates the solution into three eigenvector groups

**L**,

**N**and

**M**and the solutions will be a linear combinations of the three. For sure I wasn't able to fully understand this treatment, but I'm interested in this kind of theory applied to the electromagnetic context, in particular for resonant cavities.

In my (italian) waveguides book, I can find in an appendix where the solutions of that kind of problem can be divided in three kind: solenoidal, irrotational and harmonic (and this last one is generated when the surface is not simply-connected). Can anyone suggest me a reference for this kind of treatment, oriented to the electromagnetism? The next step is to express these solutions into TE, TM and TEM solutions: how can you connect the solenoidal/irrotational/harmonic siolutions with the TE/TM/TEM? It's just something like solenoidal -> TE, irrotational ->TM, harmonic ->TEM? I'm not sure about this.

Thank you!