252 Chapter 6.Special Functions CITED REFERENCES AND FURTHER READING: Barnett,A.R..Feng.D.H.,Steed,J.W.,and Goldfarb,L.J.B.1974,Computer Physics Commu- nications,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 Computationa/Physics,vol.64,pp.490-509. Abramowitz,M.,and Stegun,I.A.1964,Handbook of Mathematical Functions,Applied Mathe- matics Series,Volume 55(Washington:National Bureau of Standards;reprinted 1968 by Dover Publications,New York),Chapter 10. NUMERICAL 6.8 Spherical Harmonics 豆三。 令 Spherical harmonics occur in a large variety of physical problems,for ex- ample,whenever a wave equation,or Laplace's equation,is solved by separa- Press. tion of variables in spherical coordinates.The spherical harmonic Yim(,), -1<m <I,is a function of the two coordinates 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: SCIENTIFIC do d(cos0)Yvm'*(0,)Yim(0,=6v16m'm (6.8.1) Here asterisk denotes complex conjugation. Mathematically,the spherical harmonics are related to associated Legendre polynomials by the equation 香 Numerica 10621 4312 Ym(0,p)= 4 m)P"(cos)etme 2l+1(l-m)！ (6.8.2) (outside Recipes By using the relation North Y,-m(0,p)=(-1)mYm*(0,) (6.8.3) we can always relate a spherical harmonic to an associated Legendre polynomial with m >0.With x cos0,these are defined in terms of the ordinary Legendre polynomials (cf.$4.5 and $5.5)by P(a)=(-1)m(1-x2me domP(r) (6.8.4)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 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). CITED REFERENCES AND FURTHER READING: Barnett, A.R., Feng, D.H., Steed, J.W., and Goldfarb, L.J.B. 1974, Computer Physics Commu￾nications, 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 Mathe￾matics 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 ex￾ample, whenever a wave equation, or Laplace’s equation, is solved by separa￾tion 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)Yl
m
*(θ, φ)Ylm(θ, φ) = δl
lδm
m (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)