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11: 20.16 Software

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§20.16(ii) Real Argument and Parameter

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§20.16(iii) Complex Argument and/or Parameter

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12: 22.22 Software

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§22.22(ii) Real Argument

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§22.22(iii) Complex Argument

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13: 13.19 Asymptotic Expansions for Large Argument

§13.19 Asymptotic Expansions for Large Argument

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13.19.1 $$M_{\kappa ,\mu }\left(x\right)\sim \frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}x}{x}^{-\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\mu +\kappa \right)}_s{\left(\frac{1}{2}+\mu +\kappa \right)}_s}{s!}{x}^{-s},$$ $\mu -\kappa \ne -\frac{1}{2},-\frac{3}{2},\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $s$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

… ►provided that both $\mu \mp \kappa \ne -\frac{1}{2},-\frac{3}{2},\mathrm{\dots }$. … ►

13.19.3 $$W_{\kappa ,\mu }\left(z\right)\sim {\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

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14: 16.24 Physical Applications

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§16.24(iii) $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ Symbols

►The $\mathit{3}j$ symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. They can be expressed as ${}_3{}^{}F_2^{}$ functions with unit argument. …These are balanced ${}_4{}^{}F_3^{}$ functions with unit argument. Lastly, special cases of the $\mathit{9}j$ symbols are ${}_5{}^{}F_4^{}$ functions with unit argument. …

15: 10.40 Asymptotic Expansions for Large Argument

§10.40 Asymptotic Expansions for Large Argument

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Products

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§10.40(ii) Error Bounds for Real Argument and Order

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§10.40(iii) Error Bounds for Complex Argument and Order

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16: 34.13 Methods of Computation

§34.13 Methods of Computation

►Methods of computation for $\mathit{3}j$ and $\mathit{6}j$ symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). …

17: 10.17 Asymptotic Expansions for Large Argument

§10.17 Asymptotic Expansions for Large Argument

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§10.17(ii) Asymptotic Expansions of Derivatives

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§10.17(iii) Error Bounds for Real Argument and Order

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§10.17(iv) Error Bounds for Complex Argument and Order

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§10.17(v) Exponentially-Improved Expansions

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18: 4.46 Tables

… ►Extensive numerical tables of all the elementary functions for real values of their arguments appear in Abramowitz and Stegun (1964, Chapter 4). … ►(These roots are zeros of the Bessel function $J_{3/2}\left(x\right)$; see §10.21.) …

19: Software Index

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	Open Source													With Book						Commercial
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16.27(ii) Real Arguments	✓		✓		✓	✓						a	✓								✓	✓	✓
16.27(iii) Complex Arguments	✓											a	✓								✓	✓	✓
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22.22(ii) Real Argument	✓		✓		✓	✓		✓				✓	✓	✓		✓	✓		✓	✓	✓	✓	✓	✓
22.22(iii) Complex Argument	✓					✓						✓	✓	✓						✓	✓	✓	a	✓
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23.24(ii) Real Argument																					✓	✓	a
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20: 19.14 Reduction of General Elliptic Integrals

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19.14.1 $${\int }_1^x\frac{dt}{\sqrt{t^3-1}}=3^{-1/4}F(\varphi ,k),$$ $\cos \varphi ={\displaystyle \frac{\sqrt{3}+1-x}{\sqrt{3}-1+x}}$, $k^2={\displaystyle \frac{2-\sqrt{3}}{4}}$.

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.14(i)

Permalink:

http://dlmf.nist.gov/19.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.2 $${\int }_x^1\frac{dt}{\sqrt{1-t^3}}=3^{-1/4}F(\varphi ,k),$$ $\cos \varphi ={\displaystyle \frac{\sqrt{3}-1+x}{\sqrt{3}+1-x}}$, $k^2={\displaystyle \frac{2+\sqrt{3}}{4}}$.

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\varphi$: real or complex argument and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.5 $${\sin }^2\varphi =\frac{\gamma -\alpha }{U^2+\gamma },$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

… ►The classical method of reducing (19.2.3) to Legendre’s integrals is described in many places, especially Erdélyi et al. (1953b, §13.5), Abramowitz and Stegun (1964, Chapter 17), and Labahn and Mutrie (1997, §3). …If no such branch point is accessible from the interval of integration (for example, if the integrand is ${(t(3-t)(4-t))}^{-3/2}$ and the interval is [1,2]), then no method using this assumption succeeds. …