I don't know about any book along the lines of "Electronics for mathematicians," perhaps because while there are many beautiful and complex mathematical constructions in electronics, they are much more classical and treaded than the cutting edge in QFT. Electronic related maths are not so fertile in Fields medals, I guess.
The Art of Electronics is an engineering book, so they elegantly skip the deepest part of the fundamentals. I think writing the first chapter of the book was a daunting task, since the authors had to make sure the basic points were covered rigurously without getting swamped in details that don't add real value to the contents. An example is the treatment of Ohm's law: they say it works for metallic conductors, warn about strong nonlinearities in semiconductors and rarefied gasses, and let the reader interested in going down the rabbit hole consult Purcell and Morin's
Electricity and Magnetism. I don't have the book, but I imagine they describe the conduction electrons in a metal as a degenerate Fermi gas, and explain DC resistivity as scattering in the crystal lattice as the electrons drift in the E field. From there you can interpret the lattice excitations as a boson field (phonons) and entertain yourself for weeks on end with a mix of thermodynamics and electron-boson nonrelativistic QFT, all of that covered in many condensed matter physics textbooks. If the frequency of the electromagnetic field is increased, I guess the electron-boson-photon physics become so complicated that a semiclassical analysis, or directly Maxwell's equations, or some specific version of them (Pocklington equation), or even transmission line analysis, is enough to understand what's going on. There are many levels of interpretation, they are covered in many books of different branches of physics and engineering, and for the purposes of the book only Ohm's law is needed at first, with some additions about matching and termination of transmission lines much later on.
I think electronics is so high in the physics ladder that almost any specific topic chosen requires a full library as backup. Not long ago two port terminals were being discussed, which derived in a discussion of air core coils, from there into their lumped equivalent at higher frequencies, from there into the axial and helyx propagation modes, turning them into waveguides, which can be treated with frequency-dependent reflection coefficients measured experimentally, or mathematically with models derived from simplifications of Maxwell's equations treated in several research papers
.
I'm a theory oriented guy, and I'm dismayed by the fact that I can spend weeks thinking about more and more details of my projects, instead of building them. I find the Art of Electronics refreshing; the authors build things that work in radio telescopes and deep sea probes, manage all these levels of complexity, and explain the working circuits in a simple, logical and exhaustive way. It's admirable.
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