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21—30 of 113 matching pages

21: 4.24 Inverse Trigonometric Functions: Further Properties

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§4.24(iii) Addition Formulas

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22: 7.17 Inverse Error Functions

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7.17.2 $$\mathrm{inverf}x=t+\frac{1}{3}{t}^3+\frac{7}{30}{t}^5+\frac{127}{630}{t}^7+\mathrm{\cdots }=\sum _{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1},$$ ,

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Symbols:

$\mathrm{inverf}x$: inverse error function and $x$: real variable

Referenced by:

Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/7.17.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

The equality “$={\sum }_{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1}$” was added to this equation.

See also:

Annotations for §7.17(ii), §7.17 and Ch.7

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7.17.2_5 $$a_{m+1}=\frac{1}{2m+3}\sum _{n=0}^m\frac{2n+1}{m-n+1}{a}_n{a}_{m-n},$$ $m=0,1,2,\mathrm{\dots }$.

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Symbols:

$n$: nonnegative integer

Referenced by:

§7.17(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/7.17.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §7.17(ii), §7.17 and Ch.7

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23: 10.60 Sums

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§10.60(i) Addition Theorems

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24: 14.18 Sums

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§14.18(ii) Addition Theorems

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25: 23.10 Addition Theorems and Other Identities

§23.10 Addition Theorems and Other Identities

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§23.10(i) Addition Theorems

… ►For further addition-type identities for the $\sigma$-function see Lawden (1989, §6.4). …

26: 4.21 Identities

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§4.21(i) Addition Formulas

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4.21.1_5 $$A\cos u+B\sin u=\sqrt{A^2+B^2}\cos \left(u-\mathrm{ph}\left(A+B\mathrm{i}\right)\right),$$ $A,B\in ℝ$,

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Symbols:

$\cos z$: cosine function, $\in$: element of, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $ℝ$: real line and $\sin z$: sine function

Referenced by:

§4.21(i), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/4.21.E1_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

Suggested 2018-08-14 by Ted Ersek

See also:

Annotations for §4.21(i), §4.21 and Ch.4

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27: 21.6 Products

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§21.6(ii) Addition Formulas

… ►For addition formulas for classical theta functions see §20.7(ii).

28: 34.7 Basic Properties: $\mathit{9}j$ Symbol

… ►It constitutes an addition theorem for the $\mathit{9}j$ symbol. …

29: 22.8 Addition Theorems

§22.8 Addition Theorems

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30: 14.30 Spherical and Spheroidal Harmonics

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Addition Theorem

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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

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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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