Does the Helmholtz theorem of vector decomposition apply to the wave fields of electromagnetic radiation?

Abstract

The derivation of the Helmholtz theorem of vector decomposition of a 3-vector field requires that the field satisfy certain convergence properties at spatial infinity. This paper investigates if time-dependent electromagnetic radiation wave fields, which are of long range, satisfy these requirements. It is found that the requirements are satisfied because the fields give rise to integrals over the radial distance r of integrands of the form sin(kr)/r and cos(kr)/r. These Dirichlet integrals converge at infinity as required.