| 释义 |
Spherical HarmonicThe spherical harmonics  are the angular portion of the solution to Laplace's Equation inSpherical Coordinates where azimuthal symmetry is not present. Some care must be taken in identifying the notationalconvention being used. In the below equations,  is taken as the azimuthal (longitudinal) coordinate, and  asthe polar (latitudinal) coordinate (opposite the notation of Arfken 1985).
 | (1) |
 ,  , ..., 0, ...,  and the normalization is chosen such that
 | (2) |
 is the Kronecker Delta. Sometimes, the Condon-Shortley Phase  is prepended to thedefinition of the spherical harmonics.
Integrals of the spherical harmonics are given by  | |  | (3) |
where  is a Wigner 3j-Symbol (which is related to theClebsch-Gordan Coefficients). The spherical harmonics obey
where  is a Legendre Polynomial.
The above illustrations show  (top) and  and  (bottom).The first few spherical harmonics are
Written in terms of Cartesian Coordinates,
so
These can be separated into their Real and Imaginary Parts
 | (16) |
 | (17) |
The Zonal Harmonics are defined to be those of the form
 | (18) |
 | (19) |
 | (20) |
 . The Sectorial Harmonics are of the form
 | (21) |
 | (22) |
The spherical harmonics form a Complete Orthonormal Basis, so an arbitraryReal function  can be expanded in terms of Complex sphericalharmonics
 | (23) |
 | (24) |
References
 Spherical HarmonicsArfken, G. ``Spherical Harmonics.'' §12.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 680-685, 1985. Ferrers, N. M. An Elementary Treatise on Spherical Harmonics and Subjects Connected with Them. London: Macmillan, 1877. Groemer, H. Geometric Applications of Fourier Series and Spherical Harmonics. New York: Cambridge University Press, 1996. Hobson, E. W. The Theory of Spherical and Ellipsoidal Harmonics. New York: Chelsea, 1955. MacRobert, T. M. and Sneddon, I. N. Spherical Harmonics: An Elementary Treatise on Harmonic Functions, with Applications, 3rd ed. rev. Oxford, England: Pergamon Press, 1967. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Spherical Harmonics.'' §6.8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 246-248, 1992. Sansone, G. ``Harmonic Polynomials and Spherical Harmonics,'' ``Integral Properties of Spherical Harmonics and the Addition Theorem for Legendre Polynomials,'' and ``Completeness of Spherical Harmonics with Respect to Square Integrable Functions.'' §3.18-3.20 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 253-272, 1991. Sternberg, W. and Smith, T. L. The Theory of Potential and Spherical Harmonics, 2nd ed. Toronto: University of Toronto Press, 1946.
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