Title: Explicit representations of the regular solutions to the time-harmonic Maxwell equations combined with the radial symmetric Euler operator Authors: Caçao, IsabelConstales, Denis ×Krausshar, Rolf Sören # Issue Date: Jan-2009 Publisher: John Wiley and Sons Ltd. Series Title: Mathematical Methods in the Applied Sciences vol:32 issue:1 pages:1-11 Abstract: In this paper we consider a generalization of the classical time-harmonic Maxwell equations which as additional feature includes a radial symmetric perturbation in form of the Euler operator $E:=\sum_{i} x_i \frac{\partial }{\partial x_i}$. We show how one can apply hypercomplex analysis methods to solve the PDE system $[D - \lambda - \alpha E]f = 0$, where $D$ is the Euclidean Dirac operator and where $\lambda$ and $\alpha$ are arbitrary non-zero complex numbers. When $\alpha$ 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 $1/\alpha$. We give a fully explicit description of the regular solutions around the origin for general complex $\alpha, \lambda \in \mathbb{C} \backslash \{0\}$ in terms of hypergeometric functions and special homogeneous monogenic polynomials. We also discuss the limit cases $\lambda \to 0$ and $\alpha \to 0$. In the atter one we actually recognize the classical time-harmonic solutions of the Maxwell equations in the Euclidean space. Description: IMPACTFACTOR 2007: 0.594 ISSN: 0170-4214 Publication status: published KU Leuven publication type: IT Appears in Collections: Analysis Section