# spherical harmonics

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##### 1: 14.30 Spherical and Spheroidal Harmonics
$Y_{{l},{m}}\left(\theta,\phi\right)$ are known as spherical harmonics. … … The special class of spherical harmonics $Y_{{l},{m}}\left(\theta,\phi\right)$, defined by (14.30.1), appear in many physical applications. … In the quantization of angular momentum the spherical harmonics $Y_{{l},{m}}\left(\theta,\phi\right)$ are normalized solutions of the eigenvalue equation …
##### 2: 34.3 Basic Properties: $\mathit{3j}$ Symbol
###### §34.3(vii) Relations to Legendre Polynomials and SphericalHarmonics
For the polynomials $P_{l}$ see §18.3, and for the function $Y_{{l},{m}}$ see §14.30. …
34.3.20 $Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)Y_{{l_{2}},{m_{2}}}\left(\theta,% \phi\right)=\sum_{l,m}\left(\frac{(2l_{1}+1)(2l_{2}+1)(2l+1)}{4\pi}\right)^{% \frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l\\ m_{1}&m_{2}&m\end{pmatrix}\overline{Y_{{l},{m}}\left(\theta,\phi\right)}\begin% {pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix},$
34.3.22 $\int_{0}^{2\pi}\!\int_{0}^{\pi}Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)Y_{{% l_{2}},{m_{2}}}\left(\theta,\phi\right)Y_{{l_{3}},{m_{3}}}\left(\theta,\phi% \right)\sin\theta\mathrm{d}\theta\mathrm{d}\phi=\left(\frac{(2l_{1}+1)(2l_{2}+% 1)(2l_{3}+1)}{4\pi}\right)^{\frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.$
##### 3: 1.17 Integral and Series Representations of the Dirac Delta
###### SphericalHarmonics (§14.30)
1.17.25 $\delta\left(\cos\theta_{1}-\cos\theta_{2}\right)\delta\left(\phi_{1}-\phi_{2}% \right)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}Y_{{\ell},{m}}\left(\theta_% {1},\phi_{1}\right)\overline{Y_{{\ell},{m}}\left(\theta_{2},\phi_{2}\right)}.$
##### 4: 18.38 Mathematical Applications
###### Zonal SphericalHarmonics
Ultraspherical polynomials are zonal spherical harmonics. …
##### 6: Errata
• Section 14.30

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

• ##### 7: 10.73 Physical Applications
With the spherical harmonic $Y_{\ell,m}\left(\theta,\phi\right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\ell}(k\rho)Y_{\ell,m}\left(\theta,\phi\right)$ with $g_{\ell}=\mathsf{j}_{\ell}$, $\mathsf{y}_{\ell}$, ${\mathsf{h}^{(1)}_{\ell}}$, or ${\mathsf{h}^{(2)}_{\ell}}$, depending on the boundary conditions. …
##### 8: Bibliography M
• T. M. MacRobert (1967) 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.
• ##### 9: Bibliography B
• W. J. Braithwaite (1973) Associated Legendre polynomials, ordinary and modified spherical harmonics. Comput. Phys. Comm. 5 (5), pp. 390–394.
• ##### 10: Bibliography C
• B. C. Carlson and G. S. Rushbrooke (1950) On the expansion of a Coulomb potential in spherical harmonics. Proc. Cambridge Philos. Soc. 46, pp. 626–633.