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 of $x$ , $\int$: 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: 37.11 Spherical Harmonics

§37.11 Spherical Harmonics

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Zonal Functions

… ► … ►

Jacobi Polynomials as Spherical Harmonics

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Fourier Transform Involving Spherical Harmonics

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4: 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)

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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5: 29.18 Mathematical Applications

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§29.18(iii) Spherical and Ellipsoidal Harmonics

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6: 18.39 Applications in the Physical Sciences

… ►The eigenfunctions of ${\mathrm{L}}^2$ are the spherical harmonics $Y_{l,m_l}(\theta ,\varphi )$ with eigenvalues ${\mathrm{\hslash }}^{2}l(l+1)$, each with degeneracy $2l+1$ as $m_l=-l,-l+1,\mathrm{\dots },l$. … … ►

18.39.24 $${\mathrm{\Psi }}_{n,l,m_l}(r,\theta ,\varphi )=R_{n,l}(r)Y_{l,m_l}(\theta ,\varphi ).$$

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Symbols:

$Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $l$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.39(ii)

Permalink:

http://dlmf.nist.gov/18.39.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(ii), §18.39 and Ch.18

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7: 18.38 Mathematical Applications

… ►

Zonal Spherical Harmonics

►Ultraspherical polynomials are zonal spherical harmonics. …

8: 37.12 Orthogonal Polynomials on Quadratic Surfaces

… ► … ►Let ${\{Y_{\mathrm{\ell }}^m\}}_{\mathrm{\ell }=1}^{N_m^d}$ be an orthonormal basis of spherical harmonics in ${\mathscr{H}}_m^{0,d}$; for example, an enumeration of the basis in (37.11.18). … ►

37.12.2 $$S_{m,\mathrm{\ell }}^n(x,t)=q_{n-m}^{(m)}(t)\varphi {(t)}^m{Y}_{\mathrm{\ell }}^m\left(\frac{𝐱}{\varphi (t)}\right),$$ $0\le m\le n,\mathrm{\hspace{0.17em}\hspace{0.17em}1}\le \mathrm{\ell }\le N_m^d$.

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Symbols:

$(a,b)$: open interval, $m$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Proved:

Olver and Xu (2020, Theorem 4.3)(proved)

Referenced by:

§37.12(i), §37.12(i), §37.12(ii), §37.12(iii)

Permalink:

http://dlmf.nist.gov/37.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.12(i), §37.12(i), §37.12, Part 37.III and Ch.37

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§37.12(ii) Jacobi Polynomials on the Conic Surface

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9: 37.13 General Orthogonal Polynomials of $d$ Variables

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37.13.7 $${𝒱}_n^d=\underset{k=0}{\overset{\lfloor \frac{1}{2}n\rfloor }{⨁}}{\{R_{Y,k,n}\}}_{Y\in {\mathscr{H}}_{n-2k}^{0,d}},$$

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Symbols:

$\in$: element of, $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$ and ${𝒱}_n^d$: vector space of $d$-variable polynomials of degree $n$

Referenced by:

§37.15(iii)

Permalink:

http://dlmf.nist.gov/37.13.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13(i), §37.13, Part 37.III and Ch.37

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37.13.8 $$R_{Y,k,n}(r\xi )=p_k^{(n-2k+\frac{1}{2}d-1)}(r^2)r^{n-2k}Y(\xi ),$$ $0\le k\le \lfloor \frac{1}{2}n\rfloor$, $Y\in {\mathscr{H}}_{n-2k}^{0,d}$, $r\ge 0$, $\xi \in {𝕊}^{d-1}$.

ⓘ

Symbols:

$\in$: element of, $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$

Referenced by:

§37.13(i), §37.15(iii)

Permalink:

http://dlmf.nist.gov/37.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13(i), §37.13, Part 37.III and Ch.37

►If the spherical harmonics $Y$ in (37.13.8) are chosen as orthogonal bases of ${\mathscr{H}}_{n-2k}^{0,d}$ ($0\le k\le \frac{1}{2}n$) then the corresponding polynomials $R_{Y,k,n}$ form an orthogonal basis of ${𝒱}_n^d$. … ►

37.13.9 $${\int }_{{ℝ}^d}{R}_{Y_1,k,n}(𝐱)R_{Y_2,k,n}(𝐱)W(𝐱)d𝐱=\frac{1}{2}{\omega }_d{\int }_0^{\mathrm{\infty }}{\left(p_k^{(n-2k+\frac{1}{2}d-1)}(x)\right)}^{2}w(x)x^{n-2k+\frac{1}{2}d-1}dx{⟨Y_1,Y_2⟩}_{{𝕊}^{d-1}},$$ $0\le k\le \lfloor \frac{1}{2}n\rfloor$, $Y_1,Y_2\in {\mathscr{H}}_{n-2k}^{0,d}$.

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $ℝ$: real line, $k$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ , ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$ , ${\omega }_d$: surface area of unit sphere ${𝕊}^{d-1}$ , ${⟨f,g⟩}_{{𝕊}^{d-1}}$: inner product on $(d-1)$-dimensional sphere, $W$: weight function in $d$-variables and $x$: real variable

Permalink:

http://dlmf.nist.gov/37.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13(i), §37.13, Part 37.III and Ch.37

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10: 37.17 Hermite Polynomials on ${ℝ}^d$

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37.17.7 $${𝒱}_n({ℝ}^d)=\underset{k=0}{\overset{\lfloor \frac{1}{2}n\rfloor }{⨁}}{\{S_{Y,k,n}\}}_{Y\in {\mathscr{H}}_{n-2k}^{0,d}},$$

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Symbols:

$\in$: element of, $\lfloor x\rfloor$: floor of $x$ , $ℝ$: real line, $k$: nonnegative integer, $n$: nonnegative integer, ${ℝ}^d$: $d$-dimensional Euclidean space and ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$

Permalink:

http://dlmf.nist.gov/37.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(ii), §37.17, Part 37.III and Ch.37

►

37.17.8 $$S_{Y,k,n}(r\xi )=L_k^{(n-2k+\frac{1}{2}d-1)}\left(r^2\right)r^{n-2k}Y(\xi ),$$ $0\le k\le \lfloor \frac{1}{2}n\rfloor$, $Y\in {\mathscr{H}}_{n-2k}^{0,d}$, $r\ge 0$, $\xi \in {𝕊}^{d-1}$.

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\in$: element of, $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$

Referenced by:

§37.17(v)

Permalink:

http://dlmf.nist.gov/37.17.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(ii), §37.17, Part 37.III and Ch.37

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37.17.9 $$⟨S_{Y_1,k,n},S_{Y_2,k,n}⟩=\frac{{\left(\frac{1}{2}d\right)}_{n-k}}{k!}{⟨Y_1,Y_2⟩}_{{𝕊}^{d-1}},$$ $0\le k\le \lfloor \frac{1}{2}n\rfloor$, $Y_1,Y_2\in {\mathscr{H}}_{n-2k}^{0,d}$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ , ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$ , ${⟨f,g⟩}_{{𝕊}^{d-1}}$: inner product on $(d-1)$-dimensional sphere and $⟨f,g⟩$: inner product for $d$-variable Hermite polynomials

Source:

Dunkl and Xu (2014, p. 140)

Permalink:

http://dlmf.nist.gov/37.17.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(ii), §37.17, Part 37.III and Ch.37

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37.17.13 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{2k}{R}_{Y,k,n}^{\alpha }({\alpha }^{-\frac{1}{2}}𝐱)={(-1)}^{k}k!S_{Y,k,n}(𝐱),$$ $0\le k\le \lfloor \frac{1}{2}n\rfloor$, $Y\in {\mathscr{H}}_{n-2k}^{0,d}$.

ⓘ

Symbols:

$\in$: element of, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer, $n$: nonnegative integer, $𝐱$: vector in $d$-dimensional Euclidean space and ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$

Proof sketch:

Use (37.15.7) and (18.7.22).

Permalink:

http://dlmf.nist.gov/37.17.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(v), §37.17, Part 37.III and Ch.37

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