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21: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols, different asymptotic forms are obtained depending on which parameters are large. … ►For approximations for the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.

22: 4.25 Continued Fractions

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4.25.2 $$\tan \left(az\right)=\frac{a\tan z}{1+}\frac{(1-a^2){\tan }^{2}z}{3+}\frac{(4-a^2){\tan }^{2}z}{5+}\frac{(9-a^2){\tan }^{2}z}{7+}\mathrm{\cdots },$$ , $az\ne \pm \frac{1}{2}\pi ,\pm \frac{3}{2}\pi ,\mathrm{\dots }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $\tan z$: tangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.3.95

Permalink:

http://dlmf.nist.gov/4.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

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4.25.3 $$\frac{\arcsin z}{\sqrt{1-z^2}}=\frac{z}{1-}\frac{1\cdot 2z^2}{3-}\frac{1\cdot 2z^2}{5-}\frac{3\cdot 4z^2}{7-}\frac{3\cdot 4z^2}{9-}\mathrm{\cdots },$$

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Symbols:

$\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.44

Permalink:

http://dlmf.nist.gov/4.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

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4.25.4 $$\arctan z=\frac{z}{1+}\frac{z^2}{3+}\frac{4z^2}{5+}\frac{9z^2}{7+}\frac{16z^2}{9+}\mathrm{\cdots },$$

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Symbols:

$\arctan z$: arctangent function and $z$: complex variable

A&S Ref:

4.4.43

Referenced by:

§3.10(ii)

Permalink:

http://dlmf.nist.gov/4.25.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

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4.25.5 $${\mathrm{e}}^{2a\arctan \left(1/z\right)}=1+\frac{2a}{z-a+}\frac{a^2+1}{3z+}\frac{a^2+4}{5z+}\frac{a^2+9}{7z+}\mathrm{\cdots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\arctan z$: arctangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.42

Permalink:

http://dlmf.nist.gov/4.25.E5

Encodings:

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See also:

Annotations for §4.25 and Ch.4

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23: 18 Orthogonal Polynomials

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24: Leonard C. Maximon

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34 3j, 6j, 9j Symbols

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25: 14.33 Tables

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Abramowitz and Stegun (1964, Chapter 8) tabulates ${𝖯}_n\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 5–8D; ${𝖯}_n^{\prime }\left(x\right)$ for $n=1(1)4,9,10$, $x=0(.01)1$, 5–7D; ${𝖰}_n\left(x\right)$ and ${𝖰}_n^{\prime }\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 6–8D; $P_n\left(x\right)$ and $P_n^{\prime }\left(x\right)$ for $n=0(1)5,9,10$, $x=1(.2)10$, 6S; $Q_n\left(x\right)$ and $Q_n^{\prime }\left(x\right)$ for $n=0(1)3,9,10$, $x=1(.2)10$, 6S. (Here primes denote derivatives with respect to $x$.)

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26: 26.3 Lattice Paths: Binomial Coefficients

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Table 26.3.1: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$.

► ►►►►

$m$	$n$
	0	1	2	3	4	5	6	7	8	9	10
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9	1	9	36	84	126	126	84	36	9	1
…

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Table 26.3.2: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)$ for lattice paths.

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$m$	$n$
	…
1	1	2	3	4	5	6	7	8	9
…
8	1	9	45	165	495	1287	3003	6435	12870

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27: 4.39 Continued Fractions

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4.39.2 $$\frac{\mathrm{arcsinh}z}{\sqrt{1+z^2}}=\frac{z}{1+}\frac{1\cdot 2z^2}{3+}\frac{1\cdot 2z^2}{5+}\frac{3\cdot 4z^2}{7+}\frac{3\cdot 4z^2}{9+}\mathrm{\cdots },$$

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Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function and $z$: complex variable

A&S Ref:

4.6.36

Permalink:

http://dlmf.nist.gov/4.39.E2

Encodings:

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See also:

Annotations for §4.39 and Ch.4

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4.39.3 $$\mathrm{arctanh}z=\frac{z}{1-}\frac{z^2}{3-}\frac{4z^2}{5-}\frac{9z^2}{7-}\mathrm{\cdots },$$

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Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.6.35

Permalink:

http://dlmf.nist.gov/4.39.E3

Encodings:

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See also:

Annotations for §4.39 and Ch.4

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28: 4.9 Continued Fractions

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4.9.1 $$\ln \left(1+z\right)=\frac{z}{1+}\frac{z}{2+}\frac{z}{3+}\frac{4z}{4+}\frac{4z}{5+}\frac{9z}{6+}\frac{9z}{7+}\mathrm{\cdots },$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

4.1.39

Permalink:

http://dlmf.nist.gov/4.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(i), §4.9 and Ch.4

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4.9.2 $$\ln \left(\frac{1+z}{1-z}\right)=\frac{2z}{1-}\frac{z^2}{3-}\frac{4z^2}{5-}\frac{9z^2}{7-}\frac{16z^2}{9-}\mathrm{\cdots },$$

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Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.40

Permalink:

http://dlmf.nist.gov/4.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(i), §4.9 and Ch.4

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29: Diego Dominici

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9 Airy and Related Functions

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30: Mourad E. H. Ismail

… ►Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive Approximation; Journal of Approximation Theory; and Integral Transforms and Special Functions. …