Applied Physics and Mathematics Annotation << Back Mathematical Basis of Transverse Plane Waves Theory.Wave Particles of the Class ical Field Kukushkin A.V. The package of solutions of Maxwell’s equations in vacuum has been analysed to show that it exhaustively contains all possible solutions for transverse plane waves propagating in a preferential direction at the speed of light without attenuation. With the extended use of the complex variable theory, the general rule has been formulated which outlines the exhaustive set of amplitude distribution functions for any wave of the package in the equal phase plane. By this rule (Cauchy-Riemann conditions), a close link is established between the classical theory of functions of a complex variable and the mathematical description of a specific package of the Maxwell fields. The field expanded to wave functions of this package is being considered for applicability to diverse physical problems the exhaustive list of which is defined by the internal properties of these functions alone. Examples are given for field modelling in electrodynamics boundary problems, in particular, for the field with zero critical points. In this connection, the use of 2-sheeted (n-sheeted) functions of a complex variable for coordinate functions in constructing curvilinear coordinate systems in the plane is analysed. It is shown that among singular solutions of the package there are also some with power flow logarithmic singularity, and those can be found helpful for modelling massless particles possessing expressive wave properties and of the classical field with granular energy distribution in free space-time. It is also shown that a wave-particle in the field of the linearly polarized wave is a composite of two wave sub-particles that can freely pass through two holes in an impermeable screen interfering after passing the screen. Keywords: package of solutions of Maxwell’s equations, transverse plane waves, inhomogeneous (non-uniform) plane waves, self-conjugate fields, conformal mapping, coordinate function of a complex variable, amplitude function of a complex variable, Cauchy-Riemann conditions, zero critical points of the field, critical force lines of the field, singular solutions, massless waveparticles of the classical field, composite particles of the field, interference between wave sub-particles of the classical field. Contacts: E-mail: avkuku @gmail.com Pp. 98-119.