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11: 14.30 Spherical and Spheroidal Harmonics

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§14.30(i) Definitions

… ►Sometimes $Y_{l,m}(\theta ,\varphi )$ is denoted by ${\mathrm{i}}^{-l}{𝔇}_{lm}(\theta ,\varphi )$; also the definition of $Y_{l,m}(\theta ,\varphi )$ can differ from (14.30.1), for example, by inclusion of a factor ${(-1)}^m$. … ►Most mathematical properties of $Y_{l,m}(\theta ,\varphi )$ can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter. … ►

14.30.4

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii), Erratum (V1.0.25) for Section 14.30

Permalink:

http://dlmf.nist.gov/14.30.E4

Encodings:

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Correction (effective with 1.0.25):

Previously this equation was only given for $m\ge 0$. In this update we have extended the definition of spherical harmonics to include negative integer values such that $|m|\le l$. Hence in the second part of this equation, we have replaced $m$ with $|m|$.

Suggested 2019-10-15 by Ching-Li Chai

See also:

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

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12: 23.20 Mathematical Applications

§23.20 Mathematical Applications

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§23.20(ii) Elliptic Curves

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13: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A more precise mathematical discussion then follows in §1.18(ix). … ►

§1.18(ix) Mathematical Background

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Self-adjoint extensions of a symmetric Operator

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14: 3.11 Approximation Techniques

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§3.11(vi) Splines

… ►A special applications area of Bézier curves is mathematical typography and the design of type fonts. …

15: 31.17 Physical Applications

… ►More applications—including those of generalized spheroidal wave functions and confluent Heun functions in mathematical physics, astrophysics, and the two-center problem in molecular quantum mechanics—can be found in Leaver (1986) and Slavyanov and Lay (2000, Chapter 4). …

16: 18.39 Applications in the Physical Sciences

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18.39.22 $$T_e\equiv \frac{-{\mathrm{\hslash }}^2}{2m}{\nabla }^2=-\frac{{\mathrm{\hslash }}^2}{2m}\frac{1}{r^2}\frac{d}{dr}{r}^2\frac{d}{dr}+\frac{{\mathrm{L}}^2}{2m{r}^2},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\equiv$: equals by definition and $m$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.39.E22

Encodings:

pMML, png, TeX

See also:

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

… ►In what follows the radial and spherical radial eigenfunctions corresponding to (18.39.27) are found in four different notations, with identical eigenvalues, all of which appear in the current and past mathematical and theoretical physics and chemistry literatures, regarding this central problem. … ►Mathematical underpinnings appear in Ismail (2009, §5.8), and Ismail and Koelink (2011). …