Quaternionic time-harmonic Maxwell operator

Abstract

The theory of quaternionic alpha -hyperholomorphic functions (synonyms: monogenic, regular, spatial holomorphic vectors) is currently developing rapidly. In particular, many integral formulae with explicit reproducing kernels have been obtained. In this work we establish a one-to-one correspondence between time-harmonic (=monochromatic) electromagnetic fields and pairs of 'mutually conjugate' hyperholomorphic functions. This leads to the Cauchy-type integral associated with Maxwell's equations. Some main integral formulae for Maxwell's equations involving this Cauchy-type integral are obtained. It should be mentioned that, in fact, we introduce and study a somewhat more general quaternionic object which has better algebraic and analytic properties than the 'physical' Maxwell operator and which contains the latter as a special case.