Spherical
Harmonics| Spherical
Harmonics |
Scalar Functions for S2
* Definition: If Plm are
the associated Legendre functions,
* Properties: Relationship between Ylm and Yl,–m, orthonormality and completeness,
Yl,–m = (–1)mYlm*
; 
d
Y*l'm'Ylm = 
ll' mm'
;
l=0infty m=–ll
Y*lm(
',
')
Ylm(,)
= (–')
(cos–cos')
.
* Use in expansions: A useful formula is
| x–x' |–1 =
4
l m
(2l+1)–1 (r</r>l+1)
Ylm*(',')
Ylm(,)
.
@ References: Pérez Saborid a0806 [Maxwell-Thomson-Tait coordinate-free approach].
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
nlm(
,,)
= il Ylm(,)
Ml–1 sinl (dl+1cos /
d(cos)l+1)
,
Ml = [n2 (n2–12) ··· (n2–l2)]1/2 .
@ Vector spherical harmonics: Novitsky a0803 [and
Maxwell theory].
@ On S3: Fock ZP(35);
in Lifshitz & Khalatnikov
AiP(63);
Meremianin JPA(06)mp/05.
@ In superspace:
Zhang & Zou JMP(05)m.RT/06 [homogeneous
superspaces]; De Bie & Sommen JPA(07)
[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;
Cotaescu & 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].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
23 jun 2008