Cauchy – Riemann Conditions for the Maxwell’s Equations of a Single-Frequency Quaternion

As is known, hypercomplex numbers have both a scalar part and imaginary parts. Unlike the well-known Maxwell equations, the equations written for a quaternion in 4D hypercomplex space also have a scalar part. Since these equations are obtained mathematically by multiplying a quaternion in vector representation by a differential operator of a quaternion in matrix representation, the quaternion is a solution to this equation. It is shown that by solving the presented equations, it is possible to obtain two types of waves: magnetic and electric, using, respectively, magnetic and electric intensities. Quaternion waves contain particles that are formed from magnetic or electric intensities through the operation of scalar multiplication by the Hamiltonian operator. Magnetic waves have charged electrons as particles, and electric waves have electron spins (rotors). The obtained equations satisfy the Cauchy-Riemann conditions and, consequently, the requirements of conservation of energy during transformations.