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11: 8 Incomplete Gamma and Related
Functions

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12: 28 Mathieu Functions and Hill’s Equation

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13: 14.30 Spherical and Spheroidal Harmonics

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

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Symbols:

$Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer, $l$: nonnegative integer and $L$: angular momentum operator

Referenced by:

§18.39(ii)

Permalink:

http://dlmf.nist.gov/14.30.E11

Encodings:

pMML, png, TeX

See also:

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

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14.30.11_5 $${\mathrm{L}}_z{Y}_{l,m}=\mathrm{\hslash }m{Y}_{l,m},$$ $m=-l,-1+1,\mathrm{\dots },0,\mathrm{\dots },l-1,l$,

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Symbols:

$Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer, $l$: nonnegative integer and ${\mathrm{L}}_z$: $z$ 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:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

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

… ►Here, in spherical coordinates, $L^2$ is the squared angular momentum operator: …and ${\mathrm{L}}_z$ is the $z$ component of the angular momentum operator ►

14.30.13 $${\mathrm{L}}_z=-\mathrm{i}\mathrm{\hslash }\frac{\partial }{\partial \varphi };$$

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Defines:

${\mathrm{L}}_z$: $z$ component of the angular momentum operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $z$: complex variable

Referenced by:

§14.30(iv), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/14.30.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

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

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14: 11.6 Asymptotic Expansions

… ►For the corresponding expansions for ${𝐇}_{\nu }\left(z\right)$ and ${𝐋}_{\nu }\left(z\right)$ combine (11.6.1), (11.6.2) with (11.2.5), (11.2.6), (10.17.4), and (10.40.1). … ►

11.6.5 $${𝐇}_{\nu }\left(z\right),{𝐋}_{\nu }\left(z\right)\sim \frac{z}{\pi \nu \sqrt{2}}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{\nu },$$ $|\mathrm{ph}\nu |\le \pi -\delta$.

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Referenced by:

(11.11.7)

Permalink:

http://dlmf.nist.gov/11.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(ii), §11.6 and Ch.11

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$c_3(\lambda )=20{\lambda }^{-6}-4{\lambda }^{-4},$

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11.6.9 $${𝐋}_{\nu }\left(\lambda \nu \right)\sim I_{\nu }\left(\lambda \nu \right),$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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15: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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§1.18(ii) $L^2$ spaces on intervals in $ℝ$

… ►Assume that ${\left\{{\varphi }_n\right\}}_{n=0}^{\mathrm{\infty }}$ is an orthonormal basis of $L^2\left(X\right)$. …where the limit has to be understood in the sense of $L^2$ convergence in the mean: … ►The eigenfunctions form a complete orthogonal basis in $L^2\left(X\right)$, and we can take the basis as orthonormal: … ►Eigenfunctions corresponding to the continuous spectrum are non-$L^2$ functions. …

16: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

17: 23 Weierstrass Elliptic and Modular
Functions

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18: 36 Integrals with Coalescing Saddles

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19: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

20: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …