Finally, for completeness, we invoke the exact expressions of gradient action upon interior and exterior solid spheroidal harmonics (41) via the surface ones for the shake of clarity.
Proceeding to the analysis, we introduce the interior [u.sup.m/q.sub.l,in] (regular as [tau] [right arrow] [1.sup.+]) and the exterior [u.sup.m/q.sub.l,ex] (regular as [tau] [right arrow] +[infinity]) prolate spheroidal harmonic eigenfunctions of degree l [greater than or equal to] 0 and of order m = 0, 1, 2, ..., l, in terms ofthe associated Legendre functions of the first [P.sup.m.sub.l] and of the second [Q.sup.m.sub.l] kind (for more information about these functions and their properties, refer to [7, 14, 15], where a detailed appendix contains such details) according to
Indeed, even though the scattered electromagnetic fields have been calculated for n = 0,1,2,3 (higher order terms are not of substantial interest) in a closed analytical form of infinite series in terms of the spheroidal harmonic eigenexpansions, they are not given in fully compact fashion.
Although in theory the case to be considered here is a special case of that studied in , the authors still believe it is much beneficial to study it separately because the Neumann functions and their image systems in the ellipsoidal geometry have to be constructed using ellipsoidal harmonics, while those in the prolate spheroidal geometry can be constructed using prolate spheroidal harmonics, but the ellipsoidal harmonics are much more complicated to handle than the spheroidal ones.
The interior prolate spheroidal harmonics are [P.sup.m.sub.n]([xi])[P.sup.m.sub.n]([eta]) cos m[phi] and [P.sup.m.sub.n]([xi])[P.sup.m.sub.n]([eta]) sin m[phi], and the exterior prolate spheroidal harmonics are [Q.sup.m.sub.n]([xi])[P.sup.m.sub.n]([eta]) cos m[theta]and [Q.sup.m.sub.n]([xi])[P.sup.m.sub.n]([eta]) sin m[phi], for n = 0,1,..
The surface prolate spheroidal harmonics form a complete set of eigenfunctions over a prolate spheroid [S.sub.[xi].
Due to the azimuthal symmetryofthe system, the interior Neumann function [N.sup.i.sub.a](r, [r.sub.s]) can be expressed in terms of the even interior prolate spheroidal harmonics as