Rotational-translational addition theorems for scalar spheroidal wave functions

Abstract

Abstract. Rotational-translational addition theorems for the scalar spheroidal wave function ^\(h; r],£, (f>), with / = 1,3,4, are deduced. This permits one to represent the mnth scalar spheroidal wave function, associated with one spheroidal coordinate system (hq\ rjq, $q,<j>q), centered at its local origin Oq, by an addition series of spheroidal wave functions associated with a second rotated and translated system (hr; Tjr, ^r,<j>r), centered at Or. Such theorems are necessary in the rigorous analysis of radiation and scattering by spheroids with arbitrary spacings and orientations. ','7, due to Dalmas and Deleuil [6, Sec. 2;7], These theorems have found application in the problem of electromagnetic scattering of a plane wave from a pair of perfectly conducting prolate spheroids whose major axes are in parallel alignment [6,8,9]. Moreover, very thin conducting spheroids can be used to model thin-wire dipole antennas, and translational addition

Rotational-translational addition theorems for spheroidal vector wave functions

“…[5,21] and [2], we get, in the case r' < d: OO OO ■Mjiftr, M) = E E {[i)A™WMyvr\r',d',<j>') + 0')} H=-oov=\n\ (24) and a similar expression for is deduced from Eq. (24) by substituting N for M. For calculations of the coefficients and ^B™" related to ^a™", the reader is referred to the works by Cruzan [5] and by Dalmas and Deleuil [16].…”

“…3 of Ref. [2], As can be seen, Eq. (22) is identical to the (B-l) form provided by Cruzan [5] but Eq.…”

“…'Nmn and Nm/(h) denote the normalization factors of the associated Legendre functions (Ref. [2], p. 748) and the spheroidal angle function respectively (Ref. [6], p. 22).…”

Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives

“…King and Van Buren [12] used the product expansion in 1973 in the derivation of a general addition theorem for the spheroidal functions that includes both translation and rotation. We note that the addition theorem was rediscovered and published by others in the 1980s, first for translation alone [13] and then with rotation included [14].…”

“…There is no subtraction error involved in evaluating the right-hand side of (13), but a modest subtraction error occurs in (14) for the limited region where £ is near unity, m is equal to zero, and I is odd and small. This error results from a subtraction of the two terms on the right-hand side of (14).…”