spherical harmonics

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1: 14.30 Spherical and Spheroidal Harmonics

… ► $Y_{l,m}(\theta ,\varphi )$ are known as spherical harmonics. … … ►The special class of spherical harmonics $Y_{l,m}(\theta ,\varphi )$, defined by (14.30.1), appear in many physical applications. … ►In the quantization of angular momentum the spherical harmonics $Y_{l,m}(\theta ,\varphi )$ are normalized solutions of the eigenvalue equations … ►

2: 34.3 Basic Properties: $\mathit{3}j$ Symbol

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§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

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, : $\mathit{3}j$ symbol and $l,l_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.3.E20

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the sphericalharmonic in the sum over $l,m$ has been explicity marked up as a complex conjugate.

See also:

Annotations for §34.3(vii), §34.3 and Ch.34

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34.3.22

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\int$: integral, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, : $\mathit{3}j$ symbol and $l,l_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§14.30(ii), §34.3(vii)

Permalink:

http://dlmf.nist.gov/34.3.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §34.3(vii), §34.3 and Ch.34

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3: 1.17 Integral and Series Representations of the Dirac Delta

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Spherical Harmonics (§14.30)

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1.17.25 $$\delta \left(\cos {\theta }_1-\cos {\theta }_2\right)\delta \left({\varphi }_1-{\varphi }_2\right)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}\sum _{m=-\mathrm{\ell }}^{\mathrm{\ell }}{Y}_{\mathrm{\ell },m}({\theta }_1,{\varphi }_1)\overline{Y_{\mathrm{\ell },m}({\theta }_2,{\varphi }_2)}.$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\overline{z}$: complex conjugate, $\cos z$: cosine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic and $m$: nonnegative integer

Referenced by:

§1.17(iii), §1.17(iv)

Permalink:

http://dlmf.nist.gov/1.17.E25

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the second sphericalharmonic in the sum over $\mathrm{\ell }$ has been explicity marked up as a complex conjugate.

See also:

Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

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4: 18.38 Mathematical Applications

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Zonal Spherical Harmonics

►Ultraspherical polynomials are zonal spherical harmonics. …

5: 29.18 Mathematical Applications

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§29.18(iii) Spherical and Ellipsoidal Harmonics

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6: 10.73 Physical Applications

… ►With the spherical harmonic $Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\mathrm{\ell }}(k\rho )Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ with $g_{\mathrm{\ell }}={𝗃}_{\mathrm{\ell }}$, ${𝗒}_{\mathrm{\ell }}$, ${𝗁}_{\mathrm{\ell }}^{(1)}$, or ${𝗁}_{\mathrm{\ell }}^{(2)}$, depending on the boundary conditions. …

7: Errata

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

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8: Bibliography M

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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.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

Ch.14.

Encodings:

BibTeX

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9: Bibliography B

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Associated Legendre polynomials, ordinary and modified spherical harmonics. Comput. Phys. Comm. 5 (5), pp. 390–394.

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Notes:

Single-precision Fortran, maximum accuracy 16S

External Links:

Document, Code

Referenced by:

2nd item.

Encodings:

BibTeX

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10: Bibliography C

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On the expansion of a Coulomb potential in spherical harmonics. Proc. Cambridge Philos. Soc. 46, pp. 626–633.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§34.12.

Encodings:

BibTeX

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