On the accurate computation of the prolate spheroidal radial functions of the second kind
The series expansion of the prolate radial functions of the second kind, expressed in terms of the spherical Neumann functions, converges very slowly when the spheroid's surface coordinate £ approaches 1 (thin spheroids). In this paper an analytical series expansion in powers of (£2 — 1) is obtained to facilitate the convergence. Then, by using the Wronskian test, it is shown that this newly developed expansion has been computed with a double precision accuracy. Introduction. The prolate spheroidal radial functions satisfy the following differential equation [1], [2]: i ty " !) (ATM h2¥ + ^rr^) Rmn(h,Z) = 0, m = 0,l,2,..., n = m, m + 1, m + 2,... where £ is the spheroid's radial coordinate (£ > 1), and Xmn is the spheroid's eigenvalue for the given h parameter, i.e., h = kF where k = 2-k/X is the operating wavenumber and F is the semi-interfocal distance of the spheroid. According to Flammer [2] the second solution of the above differential equation, which can be expressed in terms of the associated Legendre functions of the first and second kinds, is given by {oo OO \ dr(h)QE+r(o+ E' r=2m,2m+l r=2m+2,2m+l J (2) where denotes the summation over even or odd values of r if n — m is even or odd. Also, the expansion coefficients and follow the recursion relations given in [2] along with the spheroid's joining factor Kmh{h) [2, p. 33]. Another representation of the prolate spheroidal wave function of the second kind is given in terms of the following spherical Neumann expansion [2]: ' f^ar(h)nm+r(hO, (3) V ^ / r=0,l Received May 2, 1994. ©1996 Brown University 677 678 T. DO-NHAT and R. H. MACPHIE where aTMn are the normalized expansion coefficients, and nm+r(/i£) are the spherical Neumann functions. Equations (2) and (3) are the main interest. It is historically well known that the series in (3) converges very slowly when /i£ is small. According to Morse and Feshbach [3, p. 1506], "the series does not converge well for h£ small, in fact it is an asymptotic series not being absolutely convergent for any finite value of /i£." Recently, Sinha and MacPhie [4] summed this series up to 40 terms and replaced the residual series by an integral. However, the integrand of this integral is a curve-fitting function which may not be reliable for large to or n. In this paper we focus on the series given by (2) expressed in terms of the associated Legendre functions of the first and second kinds. Here, due to the lack of the development of the QTM+r(£) function, Flammer [2] expanded the prolate spheroidal function of the second kind in powers of (£2 — 1) by using that of the first kind and the Wronskian of Rmh(h,{;) and Rmh{h,£). However, Flammer's prolate spheroidal wave expression of the second kind is cumbersome and complicated, and is limited to some lower values of m. For the above purpose we first derive the representations of the associated Legendre function of the second kind QTM+r(£) for any integer m + r (r — —2m, —2m + 1,..., m — 0,1,2,...). By using the linear hypergeometric transformation, QTM+r{£) is given in closed form for —2m < r < — 1. However, for r > 0, QTM+r(£) is explicitly expressed in terms of the associated Legendre functions of the first kind. By using these representations it will be proved that the prolate spheroidal radial function of the second kind can be expressed in terms of its first kind. Nevertheless, when £ is near to 1 an analytical series expansion in powers of (£2 — 1) is obtained and all the expansion coefficients are expressed in closed forms in terms of the dTMn and coefficients. 2. Closed form expression of <517(0 {v > —to). First, for v = —to, —to + 1,..., -1, we start with the general definition of Barnes [5, Ch. XV, p. 326] for positive integers m as follows: = sin(/z + to)tt T{u + m + l)F(l/2) (£2 l)m/2 " sin utt 2"+1r(is + 3/2) 1 „ /v m v m 1 3 ,_9\ . .. ' F ( o + TT + 1) o + ~o~ + o' V + o ' £ )' l£l — (4) where F(a, 6; c; z) is the hypergeometric function with z = £~2, a — v/2 + to/2 +1,6 = v/2 + to/2 + 1/2, and c = a + b — m. By using the linear hypergeometric transformation for to = 1, 2,3,..., F(a, 6; c; z) is given [6, p. 560, Eq. 15.3.12] as follows: F(a b. a+b — mz) — r(m)r(a + 6-m)(1 z)— Y (a ~ rn)k(b m)k _ k r (a,o,a+o m,z) r(a)r(6) 1 ' k\ (1 m)k [ ' _(_pm r(a + 6 to) y, (g)k(b)k F(a — m)T(b — m) fc! (fc + to)! V(k + 1) V(k + to + 1) + V(k + a) + <S>(k + 6)] PROLATE SPHEROIDAL RADIAL FUNCTIONS OF THE SECOND KIND 679 for m = 1,2,, | arg(l — z)\ < n, |1 — z\ < 1, where ^(a:) = j-[InT(a;)] is the Digamma function. In our case F(a — m) = T{v/2 — to/2 + 1), T(6 — m) = T(v/2 — to/2 + 1/2) and, since u — —to, —to + 1,..., —1, the Gamma function T(a — to), or T(6 — to) with argument 0, —1, -2,..., tends to infinity. Hence, (5) is reduced to m—1 F(a b-a + b-m-z)= ^ [ ^ ~ ' (1 z)-m V — — )k[ ~ ,k fj _ z)k [ ' ' + ' ' r(a)r(6) 1 j ^ fc! (1 — m)fc ( j ' m = 1,2,... . (6) If we now substitute (6) into (4), then QTM(£) (i/ = m + r) is given in closed form as follows: oIC) I 1 ra-'fo, + V,„+rK)-( i) •! lTM 1). £,+1 Zj fe! (1 — m)t {" ' (7) r — —2m, —2m + 1,..., —1. If to = 1 and r = —2, by using (7) we obtain Q—i(0 — —£(£2 ~ l)"1^2) which agrees with that which is derived by using the recursion formula of the associated Legendre functions [6, p. 334, Eq. 8.5.3], For v = 0,1,2,..., from the definition of Hobson for the associated Legendre functions of argument greater than 1 [5, p. 325],