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Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).*bi=rootx*(rk/PI+2.0*ONOVRT*ri);bessik(z,TWOTHR,&ri,&rk,&rip,&rkp);*aip = -x*ONOVRT*rk/PI;*bip=x*(rk/PI+2.0*ONOVRT*ri);} else if (x < 0.0) {bessjy(z,THIRD,&rj,&ry,&rjp,&ryp);*ai=0.5*rootx*(rj-ONOVRT*ry);*bi = -0.5*rootx*(ry+ONOVRT*rj);bessjy(z,TWOTHR,&rj,&ry,&rjp,&ryp);*aip=0.5*absx*(ONOVRT*ry+rj);*bip=0.5*absx*(ONOVRT*rj-ry);} else {Case x = 0.*ai=0.35502805;*bi=(*ai)/ONOVRT;*aip = -0.25881940;*bip = -(*aip)/ONOVRT;}252Chapter 6.Special FunctionsCITED 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]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 byDover Publications, New York), Chapter 10.6.8 Spherical HarmonicsSpherical 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 arenormalized so that their integrated square over the sphere is unity:ZZ2π1dφ−10d(cos θ)Yl0 m0 *(θ, φ)Ylm (θ, φ) = δl0 l δm0 m(6.8.1)Here asterisk denotes complex conjugation.Mathematically, the spherical harmonics are related to associated Legendrepolynomials by the equationsYlm (θ, φ) =2l + 1 (l − m)! mP (cos θ)eimφ4π (l + m)! l(6.8.2)By using the relationYl,−m (θ, φ) = (−1)m Ylm *(θ, φ)(6.8.3)we can always relate a spherical harmonic to an associated Legendre polynomialwith m ≥ 0.

With x ≡ cos θ, these are defined in terms of the ordinary Legendrepolynomials (cf. §4.5 and §5.5) byPlm (x) = (−1)m (1 − x2 )m/2dmPl (x)dxm(6.8.4)Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.

Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).Thompson, I.J., and Barnett, A.R. 1987, Computer Physics Communications, vol. 47, pp. 245–257. [4].

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