Solutions to Time-Harmonic Maxwell Equations via Transmutation Theory

We construct solutions of a time-harmonic Maxwell-type system within the framework of the algebra of complex quaternions. Using quaternionic analysis, we establish a connection between this system and certain first-order differential operators whose kernels consist of monogenic functions. Building on known representations of harmonic and monogenic functions, we develop a constructive procedure based on transmutation operators for generating explicit solutions of the equations (D±λ)u=0, and consequently of the corresponding Maxwell system. This approach provides a systematic method for reconstructing electromagnetic fields from harmonic and monogenic data, yielding an explicit link between quaternionic operator theory, transmutation methods, and the classical formulation of Maxwell equations.