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.
Dassios, Ellipsoidal Harmonics, Eencyclopedia of Mathematics and its Aapplications, Cambridge University Press, 2012.
in view of (42), comprise the exterior solid ellipsoidal harmonics. The complete orthogonal set
form the surface ellipsoidal harmonics on the surface of any prescribed ellipsoid [rho] = [[rho].sub.s], which, with respect to the weighting function factor [[([[rho].sup.2.sub.s] - [[mu].sup.2])([[rho].sup.2.sub.s] - [v.sup.2])].sup.-l/2] for every [mu] [member of] [[h.sub.2], [h.sub.2]] and v [member of] [0, [h.sub.3]], satisfy the orthogonality relation
Finally, in order to collect the basic tools for solving boundary value problems in fundamental domains with ellipsoidal boundaries, we introduce Heine's expansion formulae for any singular point [r.sub.0], which express the fundamental solution of the Laplacian in terms of ellipsoidal harmonics as
This contribution offers a generalization of the results obtained in  for the particular physical application, but using the theory of ellipsoidal harmonics until a certain order l = 0, 1, 2, 3 and with m = 1,2, 21+1 as the degree.
Thus, in terms of the primary field (13), in view of the unit dyadic, and taking the three projections of the magnetic dipole in Cartesian coordinates from (2), the condition (59), the gradient operator (36), and the unit normal vector (39) in ellipsoidal coordinates, we apply orthogonality of the surface ellipsoidal harmonics [S.sup.m.sub.l]([mu], v) = [E.sup.m.sub.l]([mu]) [E.sup.m.sub.l](v) for l [greater than or equal to] 0 and m = 1,2, ..., 2l + 1.
Combining now (65) and (69), in view of (63), we obtain the unknown constant coefficients, when orthogonality of the surface ellipsoidal harmonics [E.sup.m.sub.l]([mu])[E.sup.m.sub.l](v) for l [greater than or equal to] 0 and m = 1,2, ..., 2l + 1 is applied through (48).
for every value of [micro] [member of] [[h.sub.3], [h.sub.2]] and v [member of] [0, [h.sub.3]], concerning the [rho]-component of boundary condition (71), where we have implied the Cartesian representations of ellipsoidal harmonics of the first-order such as
Although the electromagnetic fields in the present investigation have been obtained for n = 0,2,3 (there is no first-order field and higher order terms are not of substantial interest) in a closed analytical form of infinite series in terms of the ellipsoidal harmonics, they are not given in fully compact formation.
The advantages of the formulation lie in the analytical expressions that yield closed-type compact forms, involving simple analytically known constant coefficients for any order of ellipsoidal harmonics introduced into the potentials.