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1: 14.30 Spherical and Spheroidal Harmonics
Y l , m ( θ , ϕ ) are known as spherical harmonics. … … The special class of spherical harmonics Y l , m ( θ , ϕ ) , defined by (14.30.1), appear in many physical applications. … In the quantization of angular momentum the spherical harmonics Y l , m ( θ , ϕ ) are normalized solutions of the eigenvalue equation …
2: 34.3 Basic Properties: 3 j Symbol
§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics
For the polynomials P l see §18.3, and for the function Y l , m see §14.30. …
34.3.20 Y l 1 , m 1 ( θ , ϕ ) Y l 2 , m 2 ( θ , ϕ ) = l , m ( ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l + 1 ) 4 π ) 1 2 ( l 1 l 2 l m 1 m 2 m ) Y l , m ( θ , ϕ ) ¯ ( l 1 l 2 l 0 0 0 ) ,
34.3.22 0 2 π 0 π Y l 1 , m 1 ( θ , ϕ ) Y l 2 , m 2 ( θ , ϕ ) Y l 3 , m 3 ( θ , ϕ ) sin θ d θ d ϕ = ( ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) 4 π ) 1 2 ( l 1 l 2 l 3 0 0 0 ) ( l 1 l 2 l 3 m 1 m 2 m 3 ) .
3: 1.17 Integral and Series Representations of the Dirac Delta
Spherical Harmonics14.30)
1.17.25 δ ( cos θ 1 - cos θ 2 ) δ ( ϕ 1 - ϕ 2 ) = = 0 m = - Y , m ( θ 1 , ϕ 1 ) Y , m ( θ 2 , ϕ 2 ) ¯ .
4: 18.38 Mathematical Applications
Zonal Spherical Harmonics
Ultraspherical polynomials are zonal spherical harmonics. …
5: 29.18 Mathematical Applications
§29.18(iii) Spherical and Ellipsoidal Harmonics
6: Errata
  • Section 14.30


    In regard to the definition of the spherical harmonics Y l , m , the domain of the integer m originally written as 0 m l has been replaced with the more general | m | l . Because of this change, in the sentence just below (14.30.2), “tesseral for m < l and sectorial for m = l ” has been replaced with “tesseral for | m | < l and sectorial for | m | = l ”. Furthermore, in (14.30.4), m has been replaced with | m | .

    Reported by Ching-Li Chai on 2019-10-05

  • 7: 10.73 Physical Applications
    With the spherical harmonic Y , m ( θ , ϕ ) defined as in §14.30(i), the solutions are of the form f = g ( k ρ ) Y , m ( θ , ϕ ) with g = j , y , h ( 1 ) , or h ( 2 ) , depending on the boundary conditions. …
    8: Bibliography M
  • T. M. MacRobert (1967) Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.
  • 9: Bibliography B
  • W. J. Braithwaite (1973) Associated Legendre polynomials, ordinary and modified spherical harmonics. Comput. Phys. Comm. 5 (5), pp. 390–394.
  • 10: Bibliography C
  • B. C. Carlson and G. S. Rushbrooke (1950) On the expansion of a Coulomb potential in spherical harmonics. Proc. Cambridge Philos. Soc. 46, pp. 626–633.