Biquaternionic Treatment of Inhomogeneous Time-Harmonic Maxwell’s Equations Over Unbounded Domains

Abstract

We study the inhomogeneous equation \({\text {curl}}\vec {w}+\lambda \vec {w}=\vec {g},\,\lambda \in {\mathbb {C}},\,\lambda \ne 0\) over unbounded domains in \({\mathbb {R}}^{3}\), with \(\vec {g}\) being an integrable function whose divergence is also integrable. Most of the results rely heavily on the “good enough” behavior near infinity of the \(\lambda \) Teodorescu transform, which is a classical integral operator of Clifford analysis. Some applications to inhomogeneous time-harmonic Maxwell equations are developed. Moreover, we provide necessary and sufficient conditions to guarantee that the electromagnetic fields constructed in this work satisfy the usual Silver–Müller radiation conditions. We conclude our work by showing that a particular case of our general solution of the inhomogeneous time-harmonic Maxwell equations coincide with the integral representation generated by the dyadic Green’s function.