Spherical Harmonics| Spherical Harmonics |
Scalar Functions for S2
* Definition: If
Plm
are the associated Legendre functions,
Ylm(θ, φ) = {[(2l + 1)/4π] [(l – m)!/(l + m)!]}1/2 eimφ 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=0∞ ∑m=−ll Y*lm(θ', φ') Ylm(θ, φ) = δ(φ−φ') δ(cosθ−cosθ') .
* Use in expansions: A useful formula is
| x−x' |−1 = 4π ∑l ∑m (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];
Weitzman & Freericks CondMP(18)-a1805 [calculations without derivatives].
@ 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) ··· (n2 − l2)]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].
@ Tensor spherical harmonics: Mandrilli et al GRG(20) [correspondence between tensorial spin-s and spin-weighted].
@ 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];
Alessio & Arzano a1901 [non-commutative deformation].
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send feedback and suggestions to bombelli at olemiss.edu – modified 30 jun 2020