In Section 4, we consider these conditions in the particular case of layered spheroids, compare our results with available numerical calculations, concern the question on singularities of the analytic continuations of wave fields in presence of a spheroidal particle that is far from being clear and that gave rise to controversial results of the earlier analysis, and finally discuss the applicability of the results obtained to the light scattering case.
The values of [[sigma].sub.1] = [square root of [a.sup.2] - [b.sup.2]] and [[sigma].sub.2] = 2ab/[square root of [a.sup.2] - [b.sup.2]], where a and b are the spheroid semiaxes, are determined by the same equations as the distances to singularities of the analytic continuations of the scattered and internal field analogs, respectively.
In  we have considered such particles in electrostatics and found singularities of the analytic continuations of the scattered and internal field analogs at the distances [[??].sub.1] = [square root of [a.sup.2] - [b.sup.2]]/2 and [[??].sub.2] = ab/[square root of [a.sup.2] - [b.sup.2]], respectively.
Despite these differences, the similarity of the electrostatic and light scattering cases is strong enough because of the resemblance of the wave functions, of the wave fields in the presence of a particle, of their expansions, and of singularities of their analytic continuations (see for more details ).
A comparison of the theoretical studies showed that their main difference is related to an unclear status of singularities of the analytic continuation of the internal field.
Analytic continuations The complex analyticity of the distribution [y.sup.z.sub.+][ln.sup.m] [absolute value of y] for -1 < Re (z) together with the principle of analytic continuation makes that (35) continues to hold, [for all]z [member of] C\[Z.sub.p],
We will now derive a more explicit expression in order to evaluate the right-hand side of (41) after analytic continuation. To this end, we first need the following n-dimensional projection operator [T.sup.n.sub.p,q] : D ([R.sup.n]) [right arrow] D ([R.sup.n]) such that [phi] [right arrow] [T.sup.n.sub.p,q][phi], defined by
The right-hand side of (46) shows that the analytic continuation of the regular distribution [[absolute value of x].sup.z] [ln.sup.m] [absolute value of x] is no longer a regular distribution.
By analytic continuation this product is uniquely extended to all z [member of] C\[Z.sub.p].
After analytic continuation we get the distributions [T.sup.*] ([y.sup.z.sub.+] [ln.sup.m] [absolute value of y]), [for all]z + n [member of] C\[Z.sub.e,-]], in spherical coordinates as
Asymptotic expansions and analytic continuations
of the H-function have been discussed by Braaksma ((1)).
Sondow, Double integrals and infinite products for some classical constants via analytic continuations
of Lerch's transcendent, to appear in the Ramanujan Journal.