Forget capacitors. Forget inductors. Forget resistors. Assume a generous background in elementary complex analysis and algebra. I'm trying to arrive at an elegant, minimalist, mathematically simplified, linear circuit theory. So let's assume all we know is that the voltage or current at any point on a geometrically lineal conductor (i.e., exclude current junctions) in the circuit is a mathematically linear function of the voltage or current at any (other or same) point on a geometrically lineal conductor in the circuit. (Formally, the linear function would be called a "linear operator"; and mathematically a "domain" and a "range" would need to be defined for the operator; the domain would consist of real-valued functions of time; same for the range; but we can skip further detail on this for now.) Assume an arbitrary reference point in the circuit for voltage ground.
There are two results that I suppose are true about this circuit. I would like to find a simple, mathematically elegant, rigorous proof of these results. Hopefully the proofs will be intuitively meaningful -- but this is not strictly necessary. (The intuitive aspects can be a subsequent development if necessary.)
First, a proof is needed that the voltage or current at any point in the circuit has the form Acos(wt+theta) if the voltage or current at any single (same or other) point has the form Bcos(wt+phi), for a fixed, given angular frequency w. (Some simplifying assumptions might be needed to, say, preclude additional inputs, say with different frequencies. But hopefully this can be done in a simple, elegant manner without the proof degenerating, for example, into many sub-cases.)
Second, a proof is needed that the ratio of any voltage in its complex form [Aexp(jwt+theta)] to any current also in its complex form [Bexp(jwt+phi)] in the circuit is a constant complex number (which we can call "impedance") independent of the amplitudes A, B and phases theta, phi. It seems such a result needs to be proved before we can talk meaningfully about impedance. (And by the way, does this accurately capture the usual meaning of "impedance" in electrical engineering?)
The second result may be a very simple consequence of the first. But also of interest here is how can we modify the assumptions to make them as simple and minimal as possible and yet still have a "powerful" set of theorems for the development of a mathematically simple, elegant (and useful) linear circuit theory.
To illustrate the relevance of this question and where it could lead, I'm thinking that a fairly simple, linear circuit theory should be possible for a very complicated, arbitrary interconnection of coaxial cable segments in a complicated topology involving arbitrary interconnections between shields and center conductors. And hopefully a useful linear circuit theory for such a circuit would not have to degenerate first to a discussion of capacitance, inductance and resistance, although a theory of capacitance, inductance and resistance (say based on empirical electrical sampling of the circuit) might be a nice offshoot derived from the main theory.
A subsequent development might be to ask similar questions about an electrically linear 3D continuum, say like cookie dough, with a few wires stuck into it. But we don't need to go there yet.
I have not been able to find anything that comes close to such a development. So I'm asking this question because I don't want to reinvent the wheel, so to speak. Does anyone know of any theoretical development of this nature? (And while I'm at it, do (rigorous) mathematicians and electrical engineers talk to each other?)
I'm a Amateur Extra ham radio operator, and commentary about "impedance" exudes all over the place in ham radio; I also have a strong background in the rigors of pure mathematics. So, in particular, I'd feel a whole lot more comfortable if I could find a rigorous, mathematically elegant (i.e., mathematically "simple") definition of impedance. Circuit theory textbooks, physics textbooks, and E&M textbooks I've looked at and comments I've found on the Internet just don't cut it for me. Invariably, any definition of impedance I've found first degenerates to a discussion of discrete capacitors and inductors and then impedance is defined only for very specific (and typically very simple) circuits involving resistors, capacitors, and inductors. That seems to create a huge limitation in the formal idea of impedance. But it seems to me that impedance is probably a much more general concept, especially, for example, when people start talking about impedance at various points on say a dipole antenna. In that case there are typically no discrete capacitors and inductors in the antenna itself aside from potential such elements of sorts in tuners, RF chokes, capacitance hats, and loading coils.