Helmholtz Operator with a Quaternionic Wave Number and Associated Function Theory

Abstract

A class of the hyperholomorphic quaternion-valued functions is introduced. It is related to the set of the metaharmonic functions (i.e., elements of the kernel of the Helmholtz operator with a quaternionic parameter (=quaternionic wave number)) just in the same way as the usual quaternionic analysis is related to the set of the harmonic functions. Theorems about the connection between these metaharmonic functions and hyperholomorphic functions were proved. There were constructed fundamental solutions of analogon of the Cauchy-Riemann operator and of the Helmholtz operator with a complex-quaternionic parameter. It is shown that there exists a bijection of a set of monochromatic electromagnetic fields in a homogenious medium, in absence of currents and charges, onto a set of pairs of functions belonging to the conjugate classes of a hyperholomorphy The last allowed to obtain some assertions for such electro-magnetic fields as the consequences of a theory of the hyperholomorphic functions. A part of these assertions is known, but a part is possibly new.