17.4 A Worked Example:Spheroidal Harmonics 773 Here m is an integer,c is the"oblateness parameter,"and A is the eigenvalue.Despite the notation,c2 can be positive or negative.For c2>0 the functions are called “prolate,.”while if c2<0 they are called“oblate.”The equation has singular points at x=+1 and is to be solved subject to the boundary conditions that the solution be regular at x =+1.Only for certain values of A,the eigenvalues,will this be possible. If we consider first the spherical case,where c=0.we recognize the differential equation for Legendre functions P().In this case the eigenvalues are Amn- n(n +1),n =m,m+1,....The integer n labels successive eigenvalues for fixed m:When n =m we have the lowest eigenvalue,and the corresponding eigenfunction has no nodes in the interval-1<x<1;when n =m+1 we have 8 the next eigenvalue,and the eigenfunction has one node inside(-1,1);and so on. A similar situation holds for the general case c20.We write the eigenvalues of (17.4.1)as Amn(c)and the eigenfunctions as Smn(r;c).For fixed m,n m,m +1,...labels the successive eigenvalues. ICAL The computation ofmn(c)and Smn(;c)traditionally has been quite difficult. Complicated recurrence relations,power series expansions,etc.,can be found in references [1-31.Cheap computing makes evaluation by direct solution of the RECIPES differential equation quite feasible. The first step is to investigate the behavior of the solution near the singular 9 points z =+1.Substituting a power series expansion of the form S=(1±x)ak(1士x) (17.4.2) 起g合9 9 k=0 in equation (17.4.1),we find that the regular solution has a =m/2.(Without loss of generality we can take m >0 since m --m is a symmetry of the equation.) We get an equation that is numerically more tractable if we factor out this behavior. Accordingly we set S=(1-x2)m/2y (17.4.3) We then find from(17.4.1)that y satisfies the equation Numerica 10621 (1-x2) 2-2m+1)z Py dx +(4-c2x2)y=0 (17.4.4) where μ≡入-m(m+1) (17.4.5) Both equations (17.4.1)and (17.4.4)are invariant under the replacement --Thus the functions S and y must also be invariant,except possibly for an overall scale factor.(Since the equations are linear,a constant multiple of a solution is also a solution.Because the solutions will be normalized,the scale factor can only be +1.Ifn-m is odd,there are an odd number ofzeros in the interval (-1,1). Thus we must choose the antisymmetric solution y(-)=-y(x)which has a zero at =0.Conversely,if n-m is even we must have the symmetric solution.Thus ymn(-x）=（-1)-mymn(x) (17.4.617.4 A Worked Example: Spheroidal Harmonics 773 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). Here m is an integer, c is the “oblateness parameter,” and λ is the eigenvalue. Despite the notation, c2 can be positive or negative. For c2 > 0 the functions are called “prolate,” while if c2 < 0 they are called “oblate.” The equation has singular points at x = ±1 and is to be solved subject to the boundary conditions that the solution be regular at x = ±1. Only for certain values of λ, the eigenvalues, will this be possible. If we consider first the spherical case, where c = 0, we recognize the differential equation for Legendre functions P m n (x). In this case the eigenvalues are λmn = n(n + 1), n = m, m + 1,... . The integer n labels successive eigenvalues for fixed m: When n = m we have the lowest eigenvalue, and the corresponding eigenfunction has no nodes in the interval −1 <x< 1; when n = m + 1 we have the next eigenvalue, and the eigenfunction has one node inside (−1, 1); and so on. A similar situation holds for the general case c2 = 0. We write the eigenvalues of (17.4.1) as λmn(c) and the eigenfunctions as Smn(x; c). For fixed m, n = m, m + 1,... labels the successive eigenvalues. The computation of λmn(c) and Smn(x; c) traditionally has been quite difficult. Complicated recurrence relations, power series expansions, etc., can be found in references [1-3]. Cheap computing makes evaluation by direct solution of the differential equation quite feasible. The first step is to investigate the behavior of the solution near the singular points x = ±1. Substituting a power series expansion of the form S = (1 ± x) α∞ k=0 ak(1 ± x) k (17.4.2) in equation (17.4.1), we find that the regular solution has α = m/2. (Without loss of generality we can take m ≥ 0 since m → −m is a symmetry of the equation.) We get an equation that is numerically more tractable if we factor out this behavior. Accordingly we set S = (1 − x2) m/2y (17.4.3) We then find from (17.4.1) that y satisfies the equation (1 − x2) d2y dx2 − 2(m + 1)x dy dx + (µ − c2x2)y =0 (17.4.4) where µ ≡ λ − m(m + 1) (17.4.5) Both equations (17.4.1) and (17.4.4) are invariant under the replacement x → −x. Thus the functions S and y must also be invariant, except possibly for an overall scale factor. (Since the equations are linear, a constant multiple of a solution is also a solution.) Because the solutions will be normalized, the scale factor can only be ±1. If n− m is odd, there are an odd number of zeros in the interval (−1, 1). Thus we must choose the antisymmetric solution y(−x) = −y(x) which has a zero at x = 0. Conversely, if n − m is even we must have the symmetric solution. Thus ymn(−x)=(−1)n−mymn(x) (17.4.6)