《数字信号处理》教学参考资料（Numerical Recipes in C，The Art of Scientific Computing Second Edition）Chapter 06.8 Special Functions 6.8 Spherical Harmonics

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252 Chapter 6. Special Functions 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 machinereadable 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). CITED REFERENCES AND FURTHER READING: Barnett, A.R., Feng, D.H., Steed, J.W., and Goldfarb, L.J.B. 1974, Computer Physics Communications, vol. 8, pp. 377–395. [1] Temme, N.M. 1976, Journal of Computational Physics, vol. 21, pp. 343–350 [2]; 1975, op. cit., vol. 19, pp. 324–337. [3] Thompson, I.J., and Barnett, A.R. 1987, Computer Physics Communications, vol. 47, pp. 245– 257. [4] Barnett, A.R. 1981, Computer Physics Communications, vol. 21, pp. 297–314. Thompson, I.J., and Barnett, A.R. 1986, Journal of Computational Physics, vol. 64, pp. 490–509. Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathematics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by Dover Publications, New York), Chapter 10. 6.8 Spherical Harmonics Spherical harmonics occur in a large variety of physical problems, for example, whenever a wave equation, or Laplace’s equation, is solved by separation of variables in spherical coordinates. The spherical harmonic Ylm(θ, φ), −l ≤ m ≤ l, is a function of the two coordinates θ, φ on the surface of a sphere. The spherical harmonics are orthogonal for different l and m, and they are normalized so that their integrated square over the sphere is unity: 2π 0 dφ 1 −1 d(cos θ)Ylm*(θ, φ)Ylm(θ, φ) = δllδmm (6.8.1) Here asterisk denotes complex conjugation. Mathematically, the spherical harmonics are related to associated Legendre polynomials by the equation Ylm(θ, φ) = 2l + 1 4π (l − m)! (l + m)!P m l (cos θ)eimφ (6.8.2) By using the relation Yl,−m(θ, φ)=(−1)mYlm*(θ, φ) (6.8.3) we can always relate a spherical harmonic to an associated Legendre polynomial with m ≥ 0. With x ≡ cos θ, these are defined in terms of the ordinary Legendre polynomials (cf. §4.5 and §5.5) by P m l (x)=(−1)m(1 − x2) m/2 dm dxm Pl(x) (6.8.4)

6.8 Spherical Harmonics 253 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 machinereadable 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). The first few associated Legendre polynomials, and their corresponding normalized spherical harmonics, are P0 0 (x)= 1 Y00 = 1 4π P1 1 (x) = − (1 − x2)1/2 Y11 = − 3 8π sin θeiφ P0 1 (x) = x Y10 = 3 4π cos θ P2 2 (x) = 3 (1 − x2) Y22 = 1 4 15 2π sin2 θe2iφ P1 2 (x) = −3 (1 − x2)1/2x Y21 = − 15 8π sin θ cos θeiφ P0 2 (x) = 1 2 (3x2 − 1) Y20 = 5 4π ( 3 2 cos2 θ − 1 2 ) (6.8.5) There are many bad ways to evaluate associated Legendre polynomials numerically. For example, there are explicit expressions, such as P m l (x) = (−1)m(l + m)! 2mm!(l − m)! (1 − x2) m/2 1 − (l − m)(m + l + 1) 1!(m + 1) 1 − x 2 + (l − m)(l − m − 1)(m + l + 1)(m + l + 2) 2!(m + 1)(m + 2) 1 − x 2 2 −··· (6.8.6) where the polynomial continues up through the term in (1 − x)l−m. (See [1] for this and related formulas.) This is not a satisfactory method because evaluation of the polynomial involves delicate cancellations between successive terms, which alternate in sign. For large l, the individual terms in the polynomial become very much larger than their sum, and all accuracy is lost. In practice, (6.8.6) can be used only in single precision (32-bit) for l up to 6 or 8, and in double precision (64-bit) for l up to 15 or 18, depending on the precision required for the answer. A more robust computational procedure is therefore desirable, as follows: The associated Legendre functions satisfy numerous recurrence relations, tabulated in [1-2]. These are recurrences on l alone, on m alone, and on both l and m simultaneously. Most of the recurrences involving m are unstable, and so dangerous for numerical work. The following recurrence on l is, however, stable (compare 5.5.1): (l − m)P m l = x(2l − 1)P m l−1 − (l + m − 1)P m l−2 (6.8.7) It is useful because there is a closed-form expression for the starting value, P m m = (−1)m(2m − 1)!!(1 − x2) m/2 (6.8.8) (The notation n!! denotes the product of all odd integers less than or equal to n.) Using (6.8.7) with l = m + 1, and setting P m m−1 = 0, we find P m m+1 = x(2m + 1)P m m (6.8.9) Equations (6.8.8) and (6.8.9) provide the two starting values required for (6.8.7) for general l. The function that implements this is

6.9 Fresnel Integrals, Cosine and Sine Integrals 255 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 machinereadable 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). 6.9 Fresnel Integrals, Cosine and Sine Integrals Fresnel Integrals The two Fresnel integrals are defined by C(x) = x 0 cos π 2 t 2 dt, S(x) = x 0 sin π 2 t 2 dt (6.9.1) The most convenient way of evaluating these functions to arbitrary precision is to use power series for small x and a continued fraction for large x. The series are C(x) = x − π 2 2 x5 5 · 2! + π 2 4 x9 9 · 4! −··· S(x) = π 2 x3 3 · 1! − π 2 3 x7 7 · 3! + π 2 5 x11 11 · 5! −··· (6.9.2) There is a complex continued fraction that yields both S(x) and C(x) simultaneously: C(x) + iS(x) = 1 + i 2 erf z, z = √π 2 (1 − i)x (6.9.3) where ez2 erfc z = 1 √π 1 z + 1/2 z + 1 z + 3/2 z + 2 z + ··· = 2z √π 1 2z2 + 1 − 1 · 2 2z2 + 5 − 3 · 4 2z2 + 9 − ··· (6.9.4) In the last line we have converted the “standard” form of the continued fraction to its “even” form (see §5.2), which converges twice as fast. We must be careful not to evaluate the alternating series (6.9.2) at too large a value of x; inspection of the terms shows that x = 1.5 is a good point to switch over to the continued fraction. Note that for large x C(x) ∼ 1 2 + 1 πx sin π 2 x2 , S(x) ∼ 1 2 − 1 πx cos π 2 x2 (6.9.5) Thus the precision of the routine frenel may be limited by the precision of the library routines for sine and cosine for large x

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