# Cấu trúc sóng chức năng trong điện lý thuyết P2

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## Cấu trúc sóng chức năng trong điện lý thuyết P2

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Spheroidal Coordinates and Wave Functions 2.1 SPHEROIDAL COORDINATE SYSTEMS Prolate and oblate spheroidal coordinate systems are formed by rotating the two-dimensional elliptic coordinate system, consisting of confocal ellipses and hyperbolas, about the major and minor axes of the ellipses, respectively [l] (shown in Figs. 2.1 and 2.2, where d is the interfocal distance).

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## Nội dung Text: Cấu trúc sóng chức năng trong điện lý thuyết P2

1. Spheroidal Wave Functions in Electromagnetic Theory Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong Copyright  2002 John Wiley & Sons, Inc. ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic) 2 Spheroidal Coordinates and Wave Functions 2.1 SPHEROIDAL COORDINATE SYSTEMS Prolate and oblate spheroidal coordinate systems are formed by rotating the two-dimensional elliptic coordinate system, consisting of confocal ellipses and hyperbolas, about the major and minor axes of the ellipses, respectively [l] (shown in Figs. 2.1 and 2.2, where d is the interfocal distance). The prolate spheroidal coordinates shown in Fig. 2.1 are related to the rectangular coordinates by the following transformations: xc- ; I/ K2 - 1) cos 4, (2.la) ?/=2 !Jcl- q2) (S2 - l)sinq$(2.lb) d z = --?I(, (2.lc) 2 where -l< - q - 1, l 1 forms an elongated ellipsoid of revolution with major axis of length e and minor axis of length d&?i. The degenerate surface [ = 1 is the straight line along the z-axis from z = --id to z = +id. The surface at 171= constant < 1 forms a hyperboloid of revolution of two sheets with an asymptotic cone whose generating line passesthrough the origin and is inclined at the angle e = cos-l 7 to the z-axis. The degenerate surface 171= 1 is a part of the 13 2. 14 SPHEROIDAL COORDINATES AND WAVE FUNCTIONS 4-0 \ Xclrr -“- --’ Fig. 2.1 Prolate spheroidal coordinate system. 3. SPHEROIDAL COORDINATE SYSTEMS 15 Fig. 2.2 Oblate spheroidal coordinate system. 4. 16 SPHEROIDAL COORDINATES AND WAVE FUNCTIONS z-axis for which 1x1>$. The surface at 4 = constant represents a plane through the z-axis forming the angle 4 with respect to the X, z-plane. The prolate spheroidal coordinate system is a curvilinear orthogonal sys- tem. Its metric coefficients are given by h, = {($)2+(g)2+(g)2=$/~7 (2.2a) (2.2b) (2.2c) The oblate spheroidal coordinates shown in Fig. 2.2 are related to the rectangular coordinates by the following transformations: Y=:! %A1 - q2) (r2 + l)sin#, (2.3b) d z = -TIC (2.3~) 2 where -l< - q - 1, 0 - < < 00, 0 < 4 - 27r. < < - < (2.3d) In an oblate spheroidal system, the surface at c = constant > 0 forms a flattened ellipsoid of revolution with a major axis of length d&GT and minor axis of length 4. The degenerate surface at c = 1 is a circular disk of radius a = id which lies in the x, y-plane and is centered at the origin. The surface at 1~1 constant < 1 is a hyperboloid of revolution of one sheet = with an asymptotic cone whose generating line passesthrough the origin and is inclined at the angle 8 = cos- ’ r) to the z-axis. The degenerate surface at 1~1 1 is the z-axis. The surface at 4 = constant is a plane through the x-axis = forming the angle q5with respect to the x, z-plane. The oblate spheroidal coordinate system is also a curvilinear orthogonal system. Its metric coefficients are given by h, = /($)2+($)2+($)2=$/~, (2.4a) (2.4b)
5. SPHEROIDAL SCALAR WAVE FUNCTIONS 17 h4 = &)2+(~)2+(~)2=~~(1-n2)(t2+1). (2.4~) The prolate and oblate spheroidal systems are two of the orthogonal curvi- linear coordinate systems. The coordinates 7,
6. 18 SPHEROIDAL COORDINATES AND WAVE FUNCTIONS for prolate* spheroidal coordinates and for oblate spheroidal coordinates. The functions Smn(c, q), I&&c, c), $&--ic, q), and &&ic, it> satisfy the following ordinary differential equations respectively: -f$- - q~~)--$i~~(C,$] - C2q2 [(I + [Amn - 5-j Smn(V7~ (2*8a) = O, $[((j2 l)$&,(c,
7. SPHEROlDAL ANGULAR HARMONICS 19 = g ‘C’YJQK+r (rl) r= -2m,-2m+l + 5 ‘qF(,,p,“_m- (2.11) r=2m+2,2m+l In Eqs. (2.10) and (2.11), l stands for either c or -ic, P:(z) is the associated Legendre function of the first kind (range within - 1 < x < l), and Qr (z) is the associated Legendre function of the second kind
8. 20 SPHEROIDAL COORDINATES AND WAVE FUNCTIONS , (n - m) odd; ,O(y (n- m) even, ( n - m) odd. (2.13b) Here l stands for either c or -ic. The spheroidal angular harmonics can also be obtained from solution of the auxiliary different ial equation and its boundary conditions in different forms, as stated in [69] and [70]. The only difference between this method and the method given by Abramov et al. is in the boundary conditions utilized. Abramov et al. [69,70] applied the boundary and normalized conditions S(c, 0) = 0, S(c, 1) is bounded/finite, s -1 1 g-&Y-l) = 1; S’(c, 0) = 0, S(c, 1) is bounded/finite, and s -1 1 to solve the differential equation numerically. s;,(cJ) = 1; However, the boundary conditions given here are numerically more straight- forward and efficient. Whatever the boundary conditions are, solution of the differential equation is computationally very expensive compared with the methods mentioned above. 2.3.2 Power Series Representation Expanding the associated Legendre functions as a power series of 1 - q2, we have the following expressions of the angular spheroidal harmonics for even
9. SPHEROIDAL ANGULAR HARMONICS 21 and odd (n - m>: I (1 - ~~)“/2~c;“k”(e)(l - ?j2)“, k=O (n- m) even, (2.14) q(l - $)m@c~n(.)(l - q2)‘“, k=O 4 ( n-m) odd, where l again stands for either c or -ic and the coefficients cykn(*)s are related to dyn(.)s by 1 cykn (l > = / 00 2mk!(m + k)! x r=k P 2rn (n- m) even, X (2.15) 00 (2m + 2r + l)! >: (2r + l)! r=k and .1) ( (- - Yrn (r-h ml! + (n- m) even, (2.16) (-1) V(n+m+ l)! 7 ( n - m) odd. Besides the power series of 1 - q2 above, an alternative representation of the angular spheroidal harmonics in terms of the following simple ascending power series of q is given by Smn(C,?j) = (1 - q2)m’2 ~'A~n(C)~4, (2.17a) e=o, 1 00 Smn(-iG rl) = (-I)“-“(1 - r12)m/2e-c” x B,mn(-ic)(l + r# e=o 10. 22 SPHEROIDAL COORDINATES AND WAVE FUNCTIONS = (1 - ?j2)m/2ec?7 2B,m”(-ic)(l - ?7)e, (2.17b) e=o where AT”(c) and B,“” ( AC) satisfy, respectively, the recurrence relations as follows: (l + l)(k’+-2)&!!(c) - [e(k’ + 2772 1) + m(m + 1) + - X,,(C)] A?“(C) - CHAFF = 0, (2.18a) 2(l+ l)(e + T7.J l)Bem+nl(c) [e(e + 2m + 1 + 4c) + - + (m + l)(m + 2~) - A,,(-ic) - c2] Brn(-ic) + 2(! + m)cBem_nl(--ic)= 0, (2.18b) with lim AE’(c) = 0 and (2.19) e-00 A;nn(c) The seriesforms in Eqs. (2.17a) and (2.17b) serve as alternative exact rep- resentations of the angular spheroidal harmonics where the associated Legen- dre functions are not readily available. The computational and convergence speedsfor evaluations of Eq. (2.14) is almost the same as that of Eq. (2.10). Eqs. (2.17a) and (2.17b) can also be used to obtain accurate results of the angular spheroidal functions, but the convergence of the series is not as rapid as that of the series in Eq. (2.10) or in (2.17). 2.4 EIGENVALUES A,, AND EXPANSION COEFFICIENTS dr” The variable-separation constants Am, in Eqs. (2.8a), (2.8b), (2.9a), and (2.9b) are the eigenvalues of the prolate spheroidal angular and radial func- tions. Basically, there are five methods available for evaluation of the eigen- values of spheroidal harmonics: (1) the exact evaluation by solving the tran- scendental equation in continued fractional form [1,7] or its equivalence [70], (2) accurate evaluation by the relaxation method [3,68,79], (3) approximate evaluation by power series expansion [18,74,77,78,80], (4) approximate es- timation by asymptotic expansions [18,4], and (5) systematic evaluation by casting the eigenvalue problem in a tridiagonal, symmetric matrix [80]. The first method serves as an exact technique whose results are very accurate and have been used as referenced data for comparison [72,73,81]. The second method, developed by Caldwell [79], has become very popular recently and has been incorporated into the programs of both [3] and [68]; it converges quickly and gives reasonably good agreement between the evaluated results [3,68,79] and the exact results of Flammer [l], even when the value of c2 be- comes quite large. The third method has the advantage of rapid convergence when the value of c2 is small; but its convergence becomes quite slow or the 11. EIGENVALUES A,, AND EXPANSION COEFFICIENTS qn 23 method may even fail when c2 is large (e.g., c > 10) [74,77,78,80,4]. The fourth method provides simple and easy-to-use formulas for evaluation of the eigenfunctions, but it is valid only for small c. The fifth method suggested by Hodge [80], reduces the eigenvalue problem to that of finding the eigen- values of a real (or imaginary) tridiagonal symmetric matrix. Thus, it allows for well-known procedures, which are rapid and accurate, to be used for the eigenvalue computation. It is more direct and systematic than other meth- ods. Therefore, it is considered to be a valuable tool when the computation of numerous eigenvalues is required. 2.4.1 Case I: 1~1~ < 1000 By substituting Eq. (2.10) into Eq. (2.8a), the following recurrence relation for the expansion coefficients can be established: 43447J#) + [B,“(c) - &nn(C)]~n(C) + C34(4Fl!$(C) = 0, r 2 2, (2.20) where (2m + r + 2)(2m + r + 1) c2 Ay(c) = (2.21a) (2m + 2r + 3)(2m + 2r + 5) ’ 2(m+r)(m+r+l)-2m2-lC2 B,“(c) = (2m + 2r - 1)(2m + 2r + 3) + (m+r)(m+r+l), (2.21b) (2.21c) c,m(c) = (2m + 2r “;,Z + 2r - 1)c2 and Abet + [By - a&n(C)]cn(C) = 0 (r = 0), (2.22a) Abet + [By(c) - &n(c)]cn(,> = 0 (r = 1). (2.22b) Note that the coefficients in the recursion relations do not depend on n. The n dependence is introduced solely through the eigenvalue X,,(c). The ex- pansion of Eq. (2.10) converges for all finite 7 only when the ratio qn/$!T2 converges. Therefore, the eigenvalues should satisfy the following transcen- dental equation [l] : ul(&nn) + U2(&nn) = 0, (2.23) where m m P n-m-2 P n-m-4 . . . , (2.24a) Yrn n-m-4 - x mn- n-m-6 Yrn - x ‘W&n- 12. 24 SPHEROIDAL COORDINATES AND WAVE FUNCTIONS P F-m+4 x mn- Tr-m+4 - P ?-m+6 . . . . (2.24b) ?z-m+fj - x mn- In Eqs. (2.24a) and (2.24b), the compact notation b d f h a-- - - -7- l e9 c- e- g- z- denotes the continued fraction b a- ; d C- f e- h s-i - ... and the intermediates p,” and or are defined according to Flammer [l] by r(r - 1)(2772 + r)(2m + r - 1) P’ = (2m-k 2r - 1)2(2m + 2r - 3)(2m + 2r + 1) c4 (r 2 2); (2.25a) “Irm = (m+r)(m+r+l)+G [l- 4m2 - 1 (2m + 2r - 1)(2m + 2r + 3) (r 2 0). 1 (2.25b) As can be seen from the transcendental equation, the solution can be found by expanding the eigenvalues Xmn into a Taylor series and then solving the polynomial equation, which was discussed in several published papers [l&74, 77,78,80]. Detailed expansion with a few coefficients was addressed by Flam- mer [11. However, the expansion is accurate only when the values of ]c12 are not very large. When ]c12 is very large, the Taylor series expansion technique fails because the series representation does not converge. In a similar fashion, the relaxation method [3,79] is also not accurate when ]c12 is very large. However, the solution for any ]c12 can be found by solving the transcenden- tal equation (2.23) directly. For (cl2 - 1000, Newton’s numerical technique < can be employed to solve for its roots efficiently, where the estimated value of x and the starting and end points of the iterative technique are found by LKt al. [19] as follows: xestimate = n(n+l)+Re El g , (2.26a) (2m 1 - 1)(2m + 1) xstart = n(n+l)-c2 l- n- 1)(2n + 3) ’ (2.26b) [ (2 (2 m - 1)(2m + 1) (2.26~) x end = n(n+l)+c2 (2 n- 1)(2n+3) I ’ 13. EIGENVAL UES A,, AND EXPANSION COEFFICIENTS qn 25 The coefficients qn (0) used in many places for positive index e 2 0 varying from 0 (or 1) to 00 for even (or odd) n - nz and c;(o) with negative index -k < 0 varying from -2m (or -2m + 1) to -2 (or -1) for even (or odd) n - m are determined from the following continued fraction formulas [11: Pt+n2 - = - a+2 7 e=, - (2.27a) P en Qe+4 Pe+a - pe+4 - qe+6 Pe+6 - ’ l P a-k - 2 = - y--ka-k-2 ’ (2.27b) P x+2 p-k - P-k-2 - l l l It is found from the definition that P2 n d - rnn = --PO, 5 = -PI, d0 1 Pe = pe/ae, 4e = Ye/se- The intermediates oe, &, and Ye used above are given by (2m + e + 2)(2m + e + 1) c2 CQ = (2.28a) (2m+2&+3)(2m+%-- 1) ’ Pe = (m+e)(m+e+l)-A,,(C) +2(m+e)(m+e+l)-2m2-lC2 (2.28b) (2m+2e- 1)(2m+2e+3) ’ e(e- 1) y4 = C2 (2.28~) (2m+2e-3)(2m+26- 1) ’ where Xmn(c)‘s denote the eigenvalues for a given c and assumed values m and n. To determine the unique solutions of the cn(*)‘s, the following two equa- tions are also used: for even n - m, and for odd n - m, (2.2913) With the equations above, we can determine the coefficients en(c), where most important, c can be a complex number and T = -2m, . . ., -2, 0, 2, . . . for even n - m or -2m + 1, . . .,-1, 1,3,. . . for odd n - m. 14. 26 SPHEROlDAL COORDINATES AND WAVE FUNCTIONS 2.4.2 Case II: 1~1~ > 1000 For ]c12 > 1000, further modifications must be made. This is because when the value of c is large, the required accuracy for &tart must be exceptionally high for the Newton’s method to evaluate the correct eigenvalues. A very accurate initial guess is necessary in such cases,but to generate such a guess is not systematic. Hence, it is found practical to solve for the eigenvalue by Newton’s method in caseswhere ]c12- 1000 [19]. In the case of ]c12> 1000, < the method proposed by Hodge [2,80] represents a more appropriate way of solving for the eigenvalues. A compact program capable of fast computation of eigenvalues for large c has been developed by Li et al. [19]. In this method, the eigenvalue problem is cast in matrix form, and a tridi- agonal symmetric matrix is obtained. Standard matrix manipulation proce- dures may then be used for determination of the eigenvalues. The procedure is discussedbriefly here. In Eq. (2.20) we let D9 = c 2q+s9 (2.30a) E - B 2q+s (2.30b) 7 F: = A2q+s, (2.30~) a, = d2q+s7 (2.30d) 0 - 77-z) even S= ( (2.31) 1; (;-m)odd and the superscripts m and n and the argument c have been suppressedfor simplicity. Then Eq. (2.20) becomes Dqaq-1+ (Eq - A)a, + Fqaq+l = 0, q 2 0. (2.32) This method of indexing allows the index q to range upward from zero in integer steps for all cases. Changing variable a4 = (Dl D2D3 l l . Dq/FoFlF2 l . . Fq-#/2bq (2.33) in Eq. (2.32) and multiplying the resulting expression by (FoFl F2 l l l Fq-JD1 D2D3 w.4) 1/ 2, we will obtain l (DqFq-1)“2bq-l + (Eq - X)b, + (Dq+1Fq)1’2bq+l = 0, q 2 0. (2.34) This set of equations may be written in matrix form as E. - X (Dl Fo)li2 0 0 0 ... (D1Fo)‘12 El - X (D2Fl >112 0 0 0 (D2Fl)lj2 E2 - - X (D3F2)lj2 0 1:: 0 (D3fi ) V2 E3 - X (D4F3)li2 l l l . . \ 15. - SPHEROIDAL RADIAL HARMONICS 27 t b0 0 h 0 b2 0 (2.35) b3 0 . . . . . b l So A can then be obtained by determining the eigenvalues of this real tridi- agonal symmetric matrix. In practice, it is necessary to truncate this matrix to, say, an N x N matrix. For the source code, N has been truncated to 200. After the eigenvalues are obtained, the same recursion relations in Eq. (2.20) and the normalization scheme in Eqs. (2.29a) and (2.29b) can be used to obtain c” (c). 2.5 SPHEROIDAL RADIAL HARMONICS 2.5.1 Series Representation in Terms of Spherical Bessel Functions The radial spheroidal functions of the first to fourth kinds can be expressed bY 42 c2 -1 (2m + r)! ( c2 > 00 . c qr+m-nqan . ‘Yzi!r (g), tc)c2”,t (2.36a) r=O, 1 ml2 R$;(-ic, it) = r=O, 1 .c /jr+m-nqy-iC) r=O, c2”,:$Q,(@), 1 ‘I! l (2.36b) where zi0 (z) is the ith kind of spherical Bessel functions of order n [i.e.,$)(z) = jn(x), Zig' = n,(x), Zig' = h:)(x), and z;~'(x) = h?)(x), n - j n (x) + in n (x) and ht2)(x) - jn(x) - in n (x) 9 respectively] . Since h(l) (x) - n - thus
16. 28 SPHEROIDAL COORDINATES AND WAVE FUNCTIONS for the prolate functions, and I$:;(-ic, it) = R$,i$(-ic, it) + iR:;(-ic, it>, (2.38a) Rg;(-& it) = R$,$(-ic, ic) - iR$f;(-ic, it) (2.38b) for the oblate functions. The radial functions of the first kind can also be derived from the free-space Green’s function PI as ‘i”-“+‘d~“(c)P,“+,(O)jm+, [C(c2 - 1)li2] 7 (m- n) even, ‘im-n+rd~n(c)P~~T(0)jm+, [c(E2 - 1)112] , (m - n) odd; (2.39a) Smn(-iCj 0) 00 l ‘i”-“+‘drn(-ic)P,m,, O)jm+r [-iC(t2 + 1)li2] , x r=O ( m- 72)even, . p-n+rd~n( -ic)P,“;, O)jm+r [-iC(c2 + 1)lj2] , ‘T;7 r=l (m- n) odd. (2.39b) For G$-+ 00, the radial functions have the asymptotic forms 1 n+l R( l ) mn -+ - cos(c< - 7”)’ (2.40a) 4 1 n+l R( 2) mn + - sin(c< - --j----r), (2.40b) cc 1 n+l R(3) mn 4 - exp[i(cc - ~31~ (2.40~) ti 1 n+l R( *) mn -+ 2 exP[-i(ct - -y+]; (2.40d) 17. SPHEROIDAL RADIAL HARMONICS 29 and for c[ --+ 0 they have the asymptotic forms 2*n!(c (2.41b) 2nn!(cf;n+1’ It is checked numerically that although summation of Eqs. (2.36a) and (2.36b) of the first kind (i = 1) is rapidly convergent, summation for the second kind (i = 2) converges very slowly. Summation of Eqs. (2.39a) or (2.39b) even converges much more slowly than Eqs. (2.36a) or (2.36b). It is observed from numerical evaluation that the spherical Bessel function of the second kind nC2) (4) becomes very large when its coefficient becomes m+r very small. However, their product varies quite slowly and remains almost constant when T is large. Therefore, Eqs. (2.36a) and (2.36b) of the second kind fail in convergence and are not recommended for the calculation of the radial harmonics of the second kind. 2.5.2 Proportional Relations of Angular and Radial Functions Numerically, the radial functions R$$(c, 5) and &L n -‘Y 18. 30 SPHEROIDAL COORDINATES AND WAVE FUNCTIONS and I 2 (2m - l)m!(n + m)!cmB1 r=O (n- m) even, tJ2) (0) mn = (2.44) n+m+l !(fZm+l(e> - 2 m - l)m!(n + m + 1)!cme2 r=l i ( n-m) odd, where l stands for either c or -ic. It is found numerically that Eqs. (2.42~) and (2.42d) are easily computed if prolate and oblate angular functions are obtained. It is also seenthat in the computations, only the intermediate parameters en(c) and cn(-ic) (where r starts from -2m for even n-m and from -2m+ 1 for odd n-m) are needed. Therefore, Eqs. (2.42a) and (2.42d) are highly recommended in the numerical calculation. 2.5.3 Power and Legendre Functional Series Representations 2.5.3.1 Power Series of Radial Functions of the First Kind Besides the expres- sions above, another representation of the spheroidal radial functions is the series expression in terms of its powers of t2 - 1 in prolate coordinates and of t2 + 1 in oblate coordinates, given respectively by R;;(c,c) = (‘2- ‘) m/2 ~(-l)%~n(C)(~2 - 1)‘” d2&) k=O 1, (n - m) even, (2.45a) [, (n -m) odd, R$$(-ic, it) = 1, (n - m) even, X (2.45b) it, (n - m) odd, where the parameters cykn with real and imaginary arguments c and -ic are given respectively by Eq. (2.15) in terms of the coefficients en(c) and dr”(-ic), while F&,$(C) and K$,$-ic) are given by Eq. (2.43).
19. SPHEROIDAL RADIAL HARMONICS 31 Compared with Eqs. (2.36a) and (2.36b), the expressions in Eqs. (2.45a) and (2.45b) are not so straightforward and require longer computational time. Therefore, it is not highly recommended in the numerical implementation. 2.5.3.2 Power Series of Radial Functions of the Second Kind The radial func- tions of the second kind can be also represented in power series, that is, + !?mn(c, 0, (2.46a) R\$J-ic,i (log5 - ;) + 9mn(-ic, it>* (2.46b) The intermediate functions Qmn(c) and Qkn(--ic) are defined by 2 Qmn(c> = X (2 - 2r)! m (2m-2r+l)! (n- m) even, (n-m) odd, (2.47a) Kmn . 2 QLHc> - ( 1)- m [ ( )I (1) C --ZC m x a?Y-id r![2m-r(m - r)!]” r=o X (2 -- 2r + l)! m 2r)! (2m ( n- m) even, (n - m) odd, (2.47b) mn a, (4 = g(-l)%~n(e)zk k=O (2.48) The other intermediate functions gmn(c) and gmn(-ic) are defined by 9mn(C,
20. 32 SPHEROIDAL COORDINATES AND WAVE FUNCTIONS where r(r + m + 1/2)(2m + 2r - l)! X 2m-1 (m - 1)!(2r + l)! [ - r(r + l)(r + 7-n- 1/2)(r + m + 1/2)(2m + 2r - 2)! 2m-1 ( m - 2)!(2r + 2)! 1 [(m+2r---1)/Z] + 4r - 4k - + 2r - 2k - k=O 2mm!(2k + l)(m + 2r - k)(2r - 2k - l)! + >: (2m 1)(2m l)! - (2r - l)! 2 qz”,(@) r=m+l 2mm!(2r - 2m - l)! (n- m) even, (2.50a) b;“(e) = -- l ~GW) 422(.) { r=O x (r + l)(r + m + l/2)(2772 + 2r)! [ 2”-r (m - 1)!(2r + 2)! - (r + l)(r + m + 1/2)[2m + r(r + m + 3/2)] 2”-l ( m- 2)’. (2m + 2r - l)! ’ (2r+3)! [(m+WPl (2m + 4r - 4k + 1)(2m + 2r - 2k)! + x 2mm!(m + 2r - k + 1)(2r - 2k)! k=O 00 - (2r - 2)! E Ppl2nr-1 (l > 2”m!(2r - 2m - 2)! r=m+ 1 (n - m) odd, (2.50b) and the symbol l stands for either c or -ic. 2.5.3.3 Legendre Functional Series of Radial Functions of the Second Kind Since &&,(1)