Spherical harmonics: Summation (formula 05.10.23.0008)

SphericalHarmonicY

$Mathematica Notation$

Polynomials

SphericalHarmonicY[n,m,theta,phi]

Summation

Finite summation

Involving Clebsch-Gordan functions

http://functions.wolfram.com/05.10.23.0008.01

Input Form

Sum[ClebschGordan[{Subscript[n, 1], k}, {Subscript[n, 2], M - k}, {L, M}] SphericalHarmonicY[Subscript[n, 1], k, \[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[Subscript[n, 2], M - k, \[CurlyTheta], \[CurlyPhi]], {k, Max[M - Subscript[n, 2], -Subscript[n, 1]], Min[M + Subscript[n, 2], Subscript[n, 1]]}] == Sqrt[((2 Subscript[n, 1] + 1) (2 Subscript[n, 2] + 1))/(4 Pi (2 L + 1))] ClebschGordan[{Subscript[n, 1], 0}, {Subscript[n, 2], 0}, {L, 0}] SphericalHarmonicY[L, M, \[CurlyTheta], \[CurlyPhi]] /; Element[Subscript[n, 1], Integers] && Subscript[n, 1] >= 0 && Element[Subscript[n, 2], Integers] && Subscript[n, 2] >= 0 && Element[L, Integers] && Element[M, Integers] && Abs[Subscript[n, 1] - Subscript[n, 2]] <= L <= Subscript[n, 1] + Subscript[n, 2] && -L <= M <= L

Standard Form

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MathML Form

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Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29

SphericalHarmonicY[lambda,mu,theta,phi]