Root nature of the transverse electric characteristic equation for a dissipative sphere

Natural frequencies of the transverse electric (TE) modes of a dissipative sphere with Debye-type permittivity are evaluated numerically for different relaxation times. Results provided strong evidence for the influence of dielectric dispersion and dissipation on the resonant behavior. It is show that the TE characteristic equations of the lossless spheres have only simple roots. Based on numerical studies on the dissipative sphere, it is deduced that if a higher-order root should occur, it is not an intrinsic feature of the problem, but merely a consequence of the choice of the numerical values for the physical parameters. From the continuity of the scattering coefficients on the branch cut associated with the square root of the complex permittivity, it is concluded that the branch cut does not contribute to the transient scattered field. It is shown that the branch points do not give rise to extra poles