On the Hilbert operator and the Hilbert formulas on the unit sphere for the time-harmonic Maxwell equations

Abstract

In this work we establish some analogues of the Hilbert formulas on the unit sphere for the theory of time-harmonic (monochromatic) electromagnetic fields. Our formulas relate one of the components of the limit value of a time-harmonic electromagnetic field in the unit ball to the rest of components. The obtained results are based on the close relation between time-harmonic solutions of the Maxwell equations and the three-dimensional α-hyperholomorphic function theory. Hilbert formulas for α-hyperholomorphic function theory for α being a complex number are also obtained, such formulas relate a pair of components of the boundary value of an α-hyperholomorphic function in the unit ball to the other pair of components, in an analogy with what happens in the case of the theory of functions of one complex variable.