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1: 10 Bessel Functions

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2: 34.6 Definition: $\mathit{9}j$ Symbol

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34.6.1

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Defines:

: $\mathit{9}j$ symbol

Symbols:

: $\mathit{3}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half., $r$: nonnegative integer and $s$: integer

Permalink:

http://dlmf.nist.gov/34.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §34.6 and Ch.34

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34.6.2

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Symbols:

: $\mathit{9}j$ symbol, : $\mathit{6}j$ symbol and $j,j_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§34.13

Permalink:

http://dlmf.nist.gov/34.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §34.6 and Ch.34

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3: 11 Struve and Related Functions

Chapter 11 Struve and Related Functions

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4: 34.7 Basic Properties: $\mathit{9}j$ Symbol

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34.7.1

ⓘ

Symbols:

: $\mathit{9}j$ symbol, : $\mathit{6}j$ symbol and $j,j_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §34.7(i), §34.7 and Ch.34

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34.7.2

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, : $\mathit{9}j$ symbol and $j,j_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §34.7(iv), §34.7 and Ch.34

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34.7.3

ⓘ

Symbols:

: $\mathit{9}j$ symbol and $j,j_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§34.9

Permalink:

http://dlmf.nist.gov/34.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §34.7(vi), §34.7 and Ch.34

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34.7.4

ⓘ

Symbols:

: $\mathit{9}j$ symbol, : $\mathit{3}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $r$: nonnegative integer

Referenced by:

Erratum (V1.0.10) for Equation (34.7.4)

Permalink:

http://dlmf.nist.gov/34.7.E4

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Originally the third $\mathit{3}j$ symbol in the summation was written incorrectly as

Reported 2015-01-19 by Yan-Rui Liu

See also:

Annotations for §34.7(vi), §34.7 and Ch.34

►

34.7.5

ⓘ

Symbols:

: $\mathit{9}j$ symbol, : $\mathit{6}j$ symbol and $j,j_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §34.7(vi), §34.7 and Ch.34

5: 26.9 Integer Partitions: Restricted Number and Part Size

… ► $p_k\left(n\right)$ denotes the number of partitions of $n$ into at most $k$ parts. See Table 26.9.1. … ►It follows that $p_k\left(n\right)$ also equals the number of partitions of $n$ into parts that are less than or equal to $k$. ► $p_k(\le m,n)$ is the number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$. …

6: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
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$11\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$	$\sqrt{2}(\sqrt{3}+1)$	$-\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$
…

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4.17.1 $$\underset{z\to 0}{lim}\frac{\sin z}{z}=1,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.74

Permalink:

http://dlmf.nist.gov/4.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

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4.17.2 $$\underset{z\to 0}{lim}\frac{\tan z}{z}=1.$$

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Symbols:

$\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.75

Permalink:

http://dlmf.nist.gov/4.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

►

4.17.3 $$\underset{z\to 0}{lim}\frac{1-\cos z}{z^2}=\frac{1}{2}.$$

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Symbols:

$\cos z$: cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

7: Publications

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Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71.

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Web-Based 3D Visualization in a Digital Library of Mathematical Functions, Proceedings of the Web3D Symposium, Bangor, UK, March 29–April 1, 2005.

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From B-Spline Mesh Generation to Effective Visualizations for the NIST Digital Library of Mathematical Functions, in Curve and Surface Design, Proceedings of the Sixth International Conference on Curves and Surfaces, Avignon, France June 29–July 5, 2006, pp. 235–243.

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8: Staff

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Richard B. Paris, University of Abertay, Chaps. 8, 11

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Hans Volkmer, University of Wisconsin, Milwaukee, Chaps. 29, 30

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Richard B. Paris, University of Abertay Dundee, for Chaps. 8, 11 (deceased)

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Hans Volkmer, University of Wisconsin–Milwaukee, for Chaps. 29, 30

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9: 29 Lamé Functions

Chapter 29 Lamé Functions

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10: 27.2 Functions

… ►where $p_1,p_2,\mathrm{\dots },p_{\nu \left(n\right)}$ are the distinct prime factors of $n$, each exponent $a_r$ is positive, and $\nu \left(n\right)$ is the number of distinct primes dividing $n$. … ►Note that ${\sigma }_0\left(n\right)=d\left(n\right)$. …Note that $J_1\left(n\right)=\varphi \left(n\right)$. ►In the following examples, $a_1,\mathrm{\dots },a_{\nu \left(n\right)}$ are the exponents in the factorization of $n$ in (27.2.1). … ►Table 27.2.1 lists the first 100 prime numbers $p_n$. …