14 Legendre and Related Functions Applications 14.29 Generalizations 14.31 Other Applications

§14.30 Spherical and Spheroidal Harmonics

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Keywords:: Ferrers functions, applications, spherical harmonics, spheroidal harmonics
Referenced by:: §1.17(iii), §18.38(ii), §34.3(vii), Erratum (V1.0.25) for Section 14.30
Permalink:: http://dlmf.nist.gov/14.30
See also:: Annotations for Ch.14

§14.30(i) Definitions
§14.30(ii) Basic Properties
§14.30(iii) Sums
§14.30(iv) Applications

§14.30(i) Definitions

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Keywords:: definitions, oblate, prolate, sectorial harmonics, spherical harmonics, spheroidal harmonics, surface harmonics of the first kind, tesseral harmonics
Notes:: See Edmonds (1974, p. 24).
Referenced by:: §10.73(ii)
Permalink:: http://dlmf.nist.gov/14.30.i
Correction (effective with 1.0.25):: The domain of the integer originally written as $0\leq m\leq l$ has been replaced with $|m|\leq l$ . Because of this change, in the sentence just below (14.30.2), “tesseral for and sectorial for ” has been replaced with “tesseral for and sectorial for ”.
Suggested 2019-10-15 by Ching-Li Chai
See also:: Annotations for §14.30 and Ch.14

With and integers such that $|m|\leq l$ , and $\theta$ and $\phi$ angles such that $0\leq\theta\leq\pi$ , $0\leq\phi\leq 2\pi$ ,

14.30.1 $Y_{{l},{m}}\left(\theta,\phi\right)=\left(\frac{(l-m)!(2l+1)}{4\pi(l+m)!}% \right)^{1/2}e^{im\phi}\mathsf{P}^{m}_{l}\left(\cos\theta\right),$

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Defines:: $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic
Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $\mathrm{i}$ : imaginary unit, : integer and : nonnegative integer
Referenced by:: §14.30(i), §14.30(ii), §14.30(ii), §14.30(iv)
Permalink:: http://dlmf.nist.gov/14.30.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §14.30(i), §14.30 and Ch.14

14.30.2 $Y_{l}^{m}\left(\theta,\phi\right)=\cos\left(m\phi\right)\mathsf{P}^{m}_{l}% \left(\cos\theta\right)\text{ or }\sin\left(m\phi\right)\mathsf{P}^{m}_{l}% \left(\cos\theta\right).$

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Defines:: $Y_{\NVar{l}}^{\NVar{m}}\left(\NVar{\theta},\NVar{\phi}\right)$ : surface harmonic of the first kind
Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : integer and : nonnegative integer
Referenced by:: §14.30(i), Erratum (V1.0.25) for Section 14.30
Permalink:: http://dlmf.nist.gov/14.30.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §14.30(i), §14.30 and Ch.14

$Y_{{l},{m}}\left(\theta,\phi\right)$ are known as spherical harmonics. $Y_{l}^{m}\left(\theta,\phi\right)$ are known as surface harmonics of the first kind: tesseral for and sectorial for . Sometimes $Y_{{l},{m}}\left(\theta,\phi\right)$ is denoted by $i^{-l}\mathfrak{D}_{lm}(\theta,\phi)$ ; also the definition of $Y_{{l},{m}}\left(\theta,\phi\right)$ can differ from (14.30.1), for example, by inclusion of a factor $(-1)^{m}$ .

$P^{m}_{n}\left(x\right)$ and $Q^{m}_{n}\left(x\right)$ () are often referred to as the prolate spheroidal harmonics of the first and second kinds, respectively. $P^{m}_{n}\left(ix\right)$ and $Q^{m}_{n}\left(ix\right)$ () are known as oblate spheroidal harmonics of the first and second kinds, respectively. Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics $R_{n}^{m}(x)=e^{-i\pi n/2}P^{m}_{n}\left(ix\right)$ and $T_{n}^{m}(x)=ie^{i\pi n/2}Q^{m}_{n}\left(ix\right)$ which are real when and $n=0,1,2,\dots$ .

§14.30(ii) Basic Properties

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Keywords:: basic properties, spherical harmonics
Notes:: For (14.30.3), (14.30.6), and (14.30.8) see Edmonds (1974, pp. 20–21). (Note that Edmonds’ $P_{l}^{m}(x)$ differs by a factor $(-1)^{m}$ from $\mathsf{P}^{m}_{l}\left(x\right)$ .) (14.30.4) may be derived from (14.8.1), (14.8.2), and (14.30.1). (14.30.5) may be derived from (14.5.1) and (14.30.1). (14.30.7) may be derived from (14.7.17) and (14.30.1). (14.30.3) also follows from (14.30.1) and (14.7.10).
Referenced by:: Erratum (V1.0.19) for Notation, Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/14.30.ii
Addition (effective with 1.1.0):: Paragraph Herglotz generating function was added.
Suggested 2020-10-21 by Anupam Garg
See also:: Annotations for §14.30 and Ch.14

Most mathematical properties of $Y_{{l},{m}}\left(\theta,\phi\right)$ can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter.

Explicit Representation

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See also:: Annotations for §14.30(ii), §14.30 and Ch.14

14.30.3 $Y_{{l},{m}}\left(\theta,\phi\right)=\frac{(-1)^{l+m}}{2^{l}l!}\left(\frac{(l-m% )!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\left(\sin\theta\right)^{m}\*\left% (\frac{\mathrm{d}}{\mathrm{d}(\cos\theta)}\right)^{l+m}\left(\sin\theta\right)% ^{2l}.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, : integer and : nonnegative integer
Referenced by:: §14.30(ii)
Permalink:: http://dlmf.nist.gov/14.30.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

Special Values

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See also:: Annotations for §14.30(ii), §14.30 and Ch.14

14.30.4 $Y_{{l},{m}}\left(0,\phi\right)=\begin{cases}\left(\dfrac{2l+1}{4\pi}\right)^{1% /2},&m=0,\\ 0,&|m|=1,2,3,\dots,\end{cases}$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, : integer and : nonnegative integer
Referenced by:: §14.30(ii), Erratum (V1.0.25) for Section 14.30
Permalink:: http://dlmf.nist.gov/14.30.E4
Encodings:: TeX, pMML, png
Correction (effective with 1.0.25):: Previously this equation was only given for $m\geq 0$ . In this update we have extended the definition of spherical harmonics to include negative integer values such that $|m|\leq l$ . Hence in the second part of this equation, we have replaced with .
Suggested 2019-10-15 by Ching-Li Chai
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

14.30.5 $Y_{{l},{m}}\left(\tfrac{1}{2}\pi,\phi\right)=\begin{cases}\dfrac{(-1)^{(l+m)/2% }e^{im\phi}}{2^{l}\left(\frac{1}{2}l-\frac{1}{2}m\right)!\left(\frac{1}{2}l+% \frac{1}{2}m\right)!}\left(\dfrac{(l-m)!(l+m)!(2l+1)}{4\pi}\right)^{1/2},&% \frac{1}{2}l+\frac{1}{2}m\in\mathbb{Z},\\[15.0pt] 0,&\frac{1}{2}l+\frac{1}{2}m\notin\mathbb{Z}.\end{cases}$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $\notin$ : not an element of, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, : integer and : nonnegative integer
Referenced by:: §14.30(ii)
Permalink:: http://dlmf.nist.gov/14.30.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

Symmetry

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See also:: Annotations for §14.30(ii), §14.30 and Ch.14

14.30.6 $Y_{{l},{-m}}\left(\theta,\phi\right)=(-1)^{m}\overline{Y_{{l},{m}}\left(\theta% ,\phi\right)}.$

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, : integer and : nonnegative integer
Referenced by:: §14.30(ii)
Permalink:: http://dlmf.nist.gov/14.30.E6
Encodings:: TeX, pMML, png
Notational Change (effective with 1.0.19):: As a notational clarification, the spherical harmonic on the right-hand side has been explicity marked up as a complex conjugate.
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

Parity Operation

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See also:: Annotations for §14.30(ii), §14.30 and Ch.14

14.30.7 $Y_{{l},{m}}\left(\pi-\theta,\phi+\pi\right)=(-1)^{l}Y_{{l},{m}}\left(\theta,% \phi\right).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, : integer and : nonnegative integer
Referenced by:: §14.30(ii)
Permalink:: http://dlmf.nist.gov/14.30.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

Orthogonality

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See also:: Annotations for §14.30(ii), §14.30 and Ch.14

14.30.8 $\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{1}},{m_{1}}}\left(\theta,% \phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)\sin\theta\,\mathrm{d}% \theta\,\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}}.$

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, : integer and : nonnegative integer
Referenced by:: §14.30(ii)
Permalink:: http://dlmf.nist.gov/14.30.E8
Encodings:: TeX, pMML, png
Notational Change (effective with 1.0.19):: As a notational clarification, the first spherical harmonic in the integral over $\theta$ been explicity marked up as a complex conjugate.
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

See also (34.3.22), and for further related integrals see Askey et al. (1986).

Herglotz generating function

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Keywords:: Herglotz generating function, basic properties, spherical harmonics
Referenced by:: §14.30(ii), Erratum (V1.1.0) for Additions
Addition (effective with 1.1.0):: This paragraph was added.
Suggested 2020-10-21 by Anupam Garg
See also:: Annotations for §14.30(ii), §14.30 and Ch.14

The following is the Herglotz generating function

14.30.8_5 ${\mathrm{e}}^{t{\mathbf{a}}\cdot{\mathbf{x}}}=\sqrt{4\pi}\sum_{n=0}^{\infty}% \sum_{m=-n}^{n}\frac{t^{n}r^{n}\lambda^{m}Y_{{n},{m}}\left(\theta,\phi\right)}% {\sqrt{(2n+1)(n+m)!(n-m)!}},$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, : nonnegative integer and : integer
Source:: Courant and Hilbert (1953, §VII.7.3)
Permalink:: http://dlmf.nist.gov/14.30.E8_5
Encodings:: TeX, pMML, png
Addition (effective with 1.1.0):: This equation was added.
Suggested 2019-09-26 by Anupam Garg
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

where $\mathbf{a}=\left(\tfrac{1}{2\lambda}-\tfrac{\lambda}{2},-\tfrac{\mathrm{i}}{2% \lambda}-\tfrac{\mathrm{i}\lambda}{2},1\right)$ and $\mathbf{x}=\left(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta\right)$ .

§14.30(iii) Sums

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Keywords:: spherical harmonics, sums
Notes:: See Edmonds (1974, p. 63).
Permalink:: http://dlmf.nist.gov/14.30.iii
See also:: Annotations for §14.30 and Ch.14

Distributional Completeness

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Keywords:: distributional completeness, spherical harmonics
See also:: Annotations for §14.30(iii), §14.30 and Ch.14

For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii).

Addition Theorem

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Keywords:: addition theorem, spherical harmonics
See also:: Annotations for §14.30(iii), §14.30 and Ch.14

14.30.9 $\mathsf{P}_{l}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}% \cos\left(\phi_{1}-\phi_{2}\right)\right)=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}% \overline{Y_{{l},{m}}\left(\theta_{1},\phi_{1}\right)}Y_{{l},{m}}\left(\theta_% {2},\phi_{2}\right).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\cos\NVar{z}$ : cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, : integer and : nonnegative integer
Referenced by:: §18.18(ii)
Permalink:: http://dlmf.nist.gov/14.30.E9
Encodings:: TeX, pMML, png
Notational Change (effective with 1.0.19):: As a notational clarification, the first spherical harmonic in the sum over has been explicity marked up as a complex conjugate.
See also:: Annotations for §14.30(iii), §14.30(iii), §14.30 and Ch.14

§14.30(iv) Applications

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Keywords:: Ferrers functions, Laplace’s equation, angular momentum operator, applications, geophysics, homogeneous harmonic polynomials, reduced Planck’s constant, spherical coordinates, spherical harmonics
Notes:: See Edmonds (1974, pp. 21–22).
Referenced by:: §1.5(ii)
Permalink:: http://dlmf.nist.gov/14.30.iv
Addition (effective with 1.1.0):: This section has been modified to include the addition of Equations (14.30.11_5) and (14.30.13), which are the eigenvalue equation and the definition of the component of the angular momentum operator in spherical coordinates.
See also:: Annotations for §14.30 and Ch.14

In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. See Andrews et al. (1999, Chapter 9). The special class of spherical harmonics $Y_{{l},{m}}\left(\theta,\phi\right)$ , defined by (14.30.1), appear in many physical applications. As an example, Laplace’s equation $\nabla^{2}W=0$ in spherical coordinates (§1.5(ii)):

14.30.10 ${\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^{2}\frac{\partial W% }{\partial\rho}\right)+\frac{1}{\rho^{2}\sin\theta}\frac{\partial}{\partial% \theta}\left(\sin\theta\frac{\partial W}{\partial\theta}\right)}+\frac{1}{\rho% ^{2}{\sin}^{2}\theta}\frac{{\partial}^{2}W}{{\partial\phi}^{2}}=0,$

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Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , $\sin\NVar{z}$ : sine function and $W(\rho,\theta,\phi)$ : solution
Permalink:: http://dlmf.nist.gov/14.30.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §14.30(iv), §14.30 and Ch.14

has solutions $W(\rho,\theta,\phi)=\rho^{l}Y_{{l},{m}}\left(\theta,\phi\right)$ , which are everywhere one-valued and continuous.

In the quantization of angular momentum the spherical harmonics $Y_{{l},{m}}\left(\theta,\phi\right)$ are normalized solutions of the eigenvalue equations

14.30.11 $\mathrm{L}^{2}Y_{{l},{m}}=\hbar^{2}l(l+1)Y_{{l},{m}},$ $l=0,1,2,\dots$ ,

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Symbols:: $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, : integer, : nonnegative integer and $\mathrm{L}$ : angular momentum operator
Referenced by:: §18.39(ii)
Permalink:: http://dlmf.nist.gov/14.30.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §14.30(iv), §14.30 and Ch.14

and

14.30.11_5 $\mathrm{L}_{z}Y_{{l},{m}}=\hbar mY_{{l},{m}},$ $m=-l,-1+1,\dots,0,\dots,l-1,l$ ,

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Symbols:: $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, : integer, : nonnegative integer and $\mathrm{L}_{z}$ : component of the angular momentum operator
Referenced by:: §14.30(iv), Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/14.30.E11_5
Encodings:: TeX, pMML, png
Addition (effective with 1.1.0):: This equation was added.
See also:: Annotations for §14.30(iv), §14.30 and Ch.14

where $\hbar$ is the reduced Planck’s constant. Here, in spherical coordinates, $\mathrm{L}^{2}$ is the squared angular momentum operator:

14.30.12 $\mathrm{L}^{2}=-\hbar^{2}\left(\frac{1}{\sin\theta}\frac{\partial}{\partial% \theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{{\sin}^% {2}\theta}\frac{{\partial}^{2}}{{\partial\phi}^{2}}\right),$

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