## 1 Introduction  
The family of gyro-devices includes both oscillators (gyrotron (called also gyromonotron), gyro backward wave oscillator (Gyro-BWO), cyclotron autoresonance maser (CARM)) and amplifiers (gyro traveling wave tube (Gyro-TWT), gyro-klystron, gyro-twistron). All of these tubes are also known as electron cyclotron resonance masers (CRMs) because their operation is based on a physical phenomenon called electron cyclotron maser instability. CRMs were independently proposed in 1959 by [PERSON] in the USSR, [PERSON] and [PERSON] in the USA, and [PERSON] in Australia [1]. Their development began with the discovery, construction, and successful demonstration of the first gyrotron nearly 60 years ago in Gorky, USSR (now Nizhny Novgorod, Russia). The remarkable history of this invention has been described in many articles and books (see, for example, [2; 3; 4; 5]). Nowadays, gyro-devices are recognized as powerful sources of coherent electromagnetic radiation in a wide range of frequencies, recently extended to sub-THz and THz bands. The current status of their development and achievements is presented in the annually updated prominent report [6] issued regularly by Professor [PERSON]. The numerous applications of gyro-devices in various fields of science and technology are discussed in plentiful reviews [7; 8; 9; 10; 11; 12; 13; 14]. The structure and main parts of a gyrotron are shown schematically in Figure 1. The tube contains an electron-optical system (EOS) in which the initial formation of the electron beam is performed by a magnetron injection gun. It is then compressed in an adiabatically increasing magnetic field, and the energy of the orbiting electrons associated with their rotational motion increases at the expense of their axial momentum. The interaction between beam electrons and an electromagnetic wave takes place in the cavity resonator, where the magnetic field intensity reaches its maximum. The generated radiation is transformed into a well-collimated (Gaussian-like) wave beam by a quasi-optical (QO) mode converter comprising a launcher (of the Vlasov or Denisov type) and a system of mirrors. The output wave beam is coupled to the outside world through a vacuum window. The energy of the spent electron beam is dissipated in a water-cooled collector, which is equipped with magnetic coils used to evenly distribute the deposited energy and reduce the thermal load. In some of the most advanced designs (e.g., in gyrotrons for fusion), various sophisticated depressed collectors are used to recover some of the electron beam energy and increase overall efficiency. The example in Figure 1 with an internal mode converter and radial output of the generated radiation is typical for powerful gyrotrons for fusion. For other applications, sometimes more simple designs (e.g., without an internal mode converter and with an axial output of the wave) are used. In this paper, we provide a comprehensive and systematic overview of some selected well-known fundamental analytical relations of gyrotron theory, which are usually treated separately and without sufficient details in otherwise excellent monographs on gyrotron physics (see, e.g., [15; 16; 17; 18; 19; 20; 21; 22]). The focus here is on the important constants of electron motion (invariants) that are insightful for understanding the underlying mechanisms of beam-wave interaction in gyro-devices. We believe that a systematic and consistent presentation of these fundamentals of gyrotron theory in one place (and in the proper order) would be helpful to a wide readership, not limited to experts in the development and application of gyrotrons. The paper is organized as follows: In Section 2, we consider the dispersion of electromagnetic waves in waveguides, transverse waves in electron beams, Brillouin dispersion diagrams, and cyclotron resonance conditions in beam-wave synchronization. Special attention is given to the derivation of the autoresonance integrals in the context of the CARM interaction.