Explicit representations of the regular solutions to the time‐harmonic Maxwell equations combined with the radial symmetric Euler operator

In this paper we consider a generalization of the classical time‐harmonic Maxwell equations, which as an additional feature includes a radial symmetric perturbation in the form of the Euler operator $E:={\textstyle\sum\nolimits_{i}}\,x_i {\partial }/{\partial x_i}$ . We show how one can apply hypercomplex analysis methods to solve the partial differential equation (PDE) system [ D –λ–α E ] f =0, where D is the Euclidean Dirac operator and where λ and α are arbitrary non‐zero complex numbers. When α is a positive real number, the vector‐valued solutions to this PDE system provide us precisely with the solutions to the time‐harmonic Maxwell equations on the sphere of radius 1/α. We give a fully explicit description of the regular solutions around the origin for general complex α, λ ∈ ℂ\{0} in terms of hypergeometric functions and special homogeneous monogenic polynomials. We also discuss the limit cases λ → 0 and α → 0. In the latter one, we actually recognize the classical time‐harmonic solutions to the Maxwell equations in the Euclidean space. Copyright © 2008 John Wiley & Sons, Ltd.