TAILIEUCHUNG - Special Functions part 9

252 Chapter 6. Special Functions CITED REFERENCES AND FURTHER READING: Barnett, ., Feng, ., Steed, ., and Goldfarb, . 1974, Computer Physics Communications, vol. 8, pp. 377–395. [1] Temme, . 1976, Journal of Computational Physics, vol. 21, pp. 343–350 [2]; 1975, op. cit., vol. 19, pp. 324–337. [3] Thompson, ., and Barnett, . 1987, Computer Physics Communications, vol. 47, pp. 245– 257. [4] Barnett, . 1981, Computer Physics Communications, vol. 21, pp. 297–314. Thompson, ., and Barnett, . 1986, Journal of Computational Physics, vol. 64, pp. 490–509. Abramowitz, M., and Stegun, . 1964, Handbook of Mathematical Functions, Applied Mathematics Series, Volume 55 (Washington:. | 252 Chapter 6. Special Functions CITED REFERENCES AND FURTHER READING Barnett . Feng . Steed . and Goldfarb . 1974 Computer PhysicsCommunications vol. 8 pp. 377-395. 1 Temme . 1976 Journal of Computational Physics vol. 21 pp. 343-350 2 1975 op. cit. vol. 19 pp. 324-337. 3 Thompson . and Barnett . 1987 Computer Physics Communications vol. 47 pp. 245257. 4 Barnett . 1981 Computer PhysicsCommunications vol. 21 pp. 297-314. Thompson . and Barnett . 1986 Journal ofComputational Physics vol. 64 pp. 490-509. Abramowitz M. and Stegun . 1964 Handbook of Mathematical Functions Applied Mathematics Series Volume 55 Washington National Bureau of Standards reprinted 1968 by Dover Publications New York Chapter 10. 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 6 -l m l is a function of the two coordinates 6 ÿ 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 2 zd Jo L 1 1 d cos ff Yym 6 Yim 6 fi Syi6m m Here asterisk denotes complex conjugation. Mathematically the spherical harmonics are related to associated Legendre polynomials by the equation V in 21 1 l m pm a im Yim e üTmPi cos6 e By using the relation Yi-m 6 ft -1 mYim 6 ft Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 we can always relate a spherical harmonic to an associated Legendre polynomial with m 0. With x cos 9 these are defined in terms of the ordinary Legendre polynomials cf. and by pm x -i m i - x2 m 2 dm Pi x Spherical Harmonics 253 The first few associated Legendre polynomials and their corresponding normalized spherical harmonics are P x 1 y Ji Pi x - 1 - x2 1 2 Y11 - sin ee Pio x x Yto i .

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