Spherical Harmonics  

Scalar Functions for S2
* Definition: If Plm are the associated Legendre functions,

Ylm(θ, φ) = {[(2l + 1)/4π] [(lm)!/(l + m)!]}1/2 ei Plm(cosθ) .

* Properties: Relationship between Ylm and Yl,–m, orthonormality and completeness,

Yl,–m = (–1)m Ylm* ;    dΩ Y*l'm'Ylm = δll' δmm' ;

l=0m=–ll Y*lm(θ', φ') Ylm(θ, φ) = δ(φφ') δ(cosθ–cosθ') .

* Use in expansions: A useful formula is

| xx' |–1 = 4π ∑lm (2l+1)–1 (r</r>l+1) Ylm*(θ', φ') Ylm(θ, φ) .

@ General references: Schmid AJP(58)oct [explicit expressions, and group-representation theory]; Pérez Saborid a0806/AJP [Maxwell-Thomson-Tait coordinate-free approach].
@ Related topics: Coster & Hart AJP(91)apr [addition theorem]; Ma & Yan a1203 [rotationally invariant products of three spherical harmonics].
> Online resources: Wikipedia page.

Generalizations and Other Spaces > s.a. manifolds [superspace]; multipoles.
* Tensor spherical harmonics:
* For S3: The eigenfunctions of L2, belonging to representations of SO(4), given by

ψnlm(α, θ, φ) = il Ylm(θ, φ) Ml–1 sinlα (dl+1cosα / d(cosα)l+1) ,

Ml = [n2 (n2–12) ··· (n2l2)]1/2 .

@ Vector spherical harmonics: Hill AJP(54)apr; Novitsky a0803 [and Maxwell theory].
@ On S3: Fock ZP(35); in Lifshitz & Khalatnikov AiP(63); Meremianin JPA(06)mp/05; Lindblom et al GRG(17)-a1709 [scalar, vector and tensor harmonics].
@ Spin-weighted: Scanio AJP(77)feb [and electromagnetic fields]; Straumann a1403 [as vector-valued functions on the total space SO(3) of the Hopf bundle]; Shah & Whiting GRG(16)-a1503 [spin-weighted spheroidal harmonics, raising and lowering operators]; Boyle JMP(16)-a1604 [geometry and definition].
@ In higher dimensions: Frye & Efthimiou a1205; Gundlach et al CQG(13) [for the wave equation, summation-by-parts methods].
@ In superspace: Zhang & Zou JMP(05)m.RT/06 [homogeneous superspaces]; De Bie & Sommen JPA(07)-a0705 [and integration].
@ Related topics: Dolginov JETP(56) [pseudo-euclidean]; Hughes JMP(94) [higher spin]; Ramgoolam NPB(01) [fuzzy spheres]; Coelho & Amaral JPA(02)gq/01 [conical spaces]; Mweene qp/02; Cotăescu & Visinescu MPLA(04)ht/03 [euclidean Taub-NUT]; Mulindwa & Mweene qp/05 [l = 2]; Hunter & Emami-Razavi qp/05/JPA [fermionic, half-integer l and m]; Bouzas JPA(11), JPA(11) [spin spherical harmonics, addition theorems].

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