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21—30 of 602 matching pages

21: 19.9 Inequalities

… ►

19.9.3

9+\frac{k^{2}{k^{\prime}}^{2}}{8}<\frac{(8+k^{2})K\left(k\right)}{\ln\left(4/k% ^{\prime}\right)}<9.096.

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus
Referenced by:: §19.9(i), §19.9(i)
Permalink:: http://dlmf.nist.gov/19.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.9(i), §19.9 and Ch.19

… ►Further inequalities for

K\left(k\right)

and

E\left(k\right)

can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996). … ►Sharper inequalities for

F\left(\phi,k\right)

are: … ►Inequalities for both

F\left(\phi,k\right)

and

E\left(\phi,k\right)

involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). … ►Other inequalities for

F\left(\phi,k\right)

can be obtained from inequalities for

R_{F}\left(x,y,z\right)

given in Carlson (1966, (2.15)) and Carlson (1970) via (19.25.5).

22: 8.4 Special Values

… ►For

\operatorname{erf}\left(z\right)

\operatorname{erfc}\left(z\right)

, and

F\left(z\right)

, see §§7.2(i), 7.2(ii). For

E_{n}\left(z\right)

see §8.19(i). … ►

8.4.4

\Gamma\left(0,z\right)=\int_{z}^{\infty}t^{-1}e^{-t}\,\mathrm{d}t=E_{1}\left(z% \right),

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral and $z$ : complex variable
A&S Ref:: 6.5.15
Permalink:: http://dlmf.nist.gov/8.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §8.4 and Ch.8

… ►

8.4.13

\Gamma\left(1-n,z\right)=z^{1-n}E_{n}\left(z\right),

ⓘ

Symbols:: $E_{\NVar{p}}\left(\NVar{z}\right)$ : generalized exponential integral, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §8.19(iv)
Permalink:: http://dlmf.nist.gov/8.4.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §8.4 and Ch.8

… ►

8.4.15

\Gamma\left(-n,z\right)=\frac{(-1)^{n}}{n!}\left(E_{1}\left(z\right)-e^{-z}% \sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{z^{k+1}}\right)=\frac{(-1)^{n}}{n!}\left(% \psi\left(n+1\right)-\ln z\right)-z^{-n}\sum_{\begin{subarray}{c}k=0\\ k\neq n\end{subarray}}^{\infty}\frac{(-z)^{k}}{k!(k-n)}.

ⓘ

Symbols:: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $!$ : factorial (as in $n!$ ), $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\ln\NVar{z}$ : principal branch of logarithm function, $z$ : complex variable, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 6.5.19
Referenced by:: §8.19(iii), §8.19(iv), §8.4
Permalink:: http://dlmf.nist.gov/8.4.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §8.4 and Ch.8

23: Software Index

… ► ►►►►►►►

	Open Source										With Book				Commercial
…
8.28(vii) $E_{p}\left(z\right)$ , $z,p\in\mathbb{C}$	✓								✓	✓						✓	✓	✓
9 Airy and Related Functions
…
19.39(iii) $F\left(\phi,k\right)$ , $E\left(\phi,k\right)$ , $\phi,k\in\mathbb{R}$	✓		✓	✓	✓	✓	✓		✓	✓		✓				✓	✓	a	✓
…
24.21(ii) $B_{n}$ , $B_{n}\left(x\right)$ , $E_{n}$ , $E_{n}\left(x\right)$	✓	✓				✓		✓	a	✓	✓		✓	✓		✓	✓	✓		Derive, MuPAD
…
34 3j, 6j, 9j Symbols
…

…

24: 16.7 Relations to Other Functions

… ►For

\mathit{3j}

\mathit{6j}

\mathit{9j}

symbols see Chapter 34. Further representations of special functions in terms of

{{}_{p}F_{q}}

functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of

{{}_{q+1}F_{q}}

functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).

25: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols, different asymptotic forms are obtained depending on which parameters are large. … ►For approximations for the

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.

26: 24.21 Software

… ►

§24.21(ii) $B_{n}$ , $B_{n}\left(x\right)$ , $E_{n}$ , and $E_{n}\left(x\right)$

…

27: 19.5 Maclaurin and Related Expansions

… ►where

{{}_{2}F_{1}}

is the Gauss hypergeometric function (§§15.1 and 15.2(i)). …where

{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)

is an Appell function (§16.13). … ►Coefficients of terms up to

\lambda^{49}

are given in Lee (1990), along with tables of fractional errors in

K\left(k\right)

and

E\left(k\right)

0.1\leq k^{2}\leq 0.9999

, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). … ►Series expansions of

F\left(\phi,k\right)

and

E\left(\phi,k\right)

are surveyed and improved in Van de Vel (1969), and the case of

F\left(\phi,k\right)

is summarized in Gautschi (1975, §1.3.2). For series expansions of

\Pi\left(\phi,\alpha^{2},k\right)

when

|\alpha^{2}|<1

see Erdélyi et al. (1953b, §13.6(9)). …

28: 3.5 Quadrature

… ►If

f\in C^{2m+2}[a,b]

, then the remainder

E_{n}(f)

in (3.5.2) can be expanded in the form … ►About

2^{9}=512

function evaluations are needed. … ►with weight function

w(x)

, is one for which

E_{n}(f)=0

whenever

f

is a polynomial of degree

\leq n-1

. The nodes

x_{1},x_{2},\dots,x_{n}

are prescribed, and the weights

w_{k}

and error term

E_{n}(f)

are found by integrating the product of the Lagrange interpolation polynomial of degree

n-1

and

w(x)

. … ►where

E_{n}(f)=0

f(\zeta)

is a polynomial of degree

\leq 2n-1

1/\zeta

. …

29: 28.6 Expansions for Small $q$

… ►Leading terms of the of the power series for

m=7,8,9,\dots

are: … ►Numerical values of the radii of convergence

\rho_{n}^{(j)}

of the power series (28.6.1)–(28.6.14) for

n=0,1,\dots,9

are given in Table 28.6.1. … ►where

k

is the unique root of the equation

2E\left(k\right)=K\left(k\right)

in the interval

(0,1)

, and

k^{\prime}=\sqrt{1-k^{2}}

. For

E\left(k\right)

and

K\left(k\right)

see §19.2(ii). … ►

28.6.22

\operatorname{ce}_{1}\left(z,q\right)=\cos z-\tfrac{1}{8}q\cos 3z+\tfrac{1}{12% 8}q^{2}\left(\tfrac{2}{3}\cos 5z-2\cos 3z-\cos z\right)-\tfrac{1}{1024}q^{3}% \left(\tfrac{1}{9}\cos 7z-\tfrac{8}{9}\cos 5z-\tfrac{1}{3}\cos 3z+2\cos z% \right)+\cdots,

ⓘ

Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $q=h^{2}$ : parameter and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/28.6.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(ii), §28.6 and Ch.28

…

30: 3.6 Linear Difference Equations

… ►The Weber function

\mathbf{E}_{n}\left(1\right)

satisfies …Thus the asymptotic behavior of the particular solution

\mathbf{E}_{n}\left(1\right)

is intermediate to those of the complementary functions

J_{n}\left(1\right)

and

Y_{n}\left(1\right)

; moreover, the conditions for Olver’s algorithm are satisfied. We apply the algorithm to compute

\mathbf{E}_{n}\left(1\right)

to 8S for the range

n=1,2,\dots,10

, beginning with the value

\mathbf{E}_{0}\left(1\right)=-0.56865\;\,663

obtained from the Maclaurin series expansion (§11.10(iii)). … ►The values of

w_{n}

for

n=1,2,\dots,10

are the wanted values of

\mathbf{E}_{n}\left(1\right)

. (It should be observed that for

n>10

, however, the

w_{n}

are progressively poorer approximations to

\mathbf{E}_{n}\left(1\right)

: the underlined digits are in error.) …

.%E5%B9%BF%E5%B7%9E%E5%B9%B4%E4%B8%96%E7%95%8C%E6%9D%AF%E7%9C%8B%E7%90%83%E9%85%92%E5%90%A7_%E3%80%8Ewn4.com_%E3%80%8F%E4%B8%96%E7%95%8C%E6%9D%AF%E6%9C%89%E5%A5%B3%E8%B6%B3%E5%90%97_w6n2c9o_2022%E5%B9%B411%E6%9C%8829%E6%97%A57%E6%97%B615%E5%88%8646%E7%A7%92_p5nl57rxz_gov_hk

21—30 of 602 matching pages

21: 19.9 Inequalities

22: 8.4 Special Values

23: Software Index

24: 16.7 Relations to Other Functions

25: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

26: 24.21 Software

§24.21(ii) B n , B n ⁡ ( x ) , E n , and E n ⁡ ( x )

27: 19.5 Maclaurin and Related Expansions

28: 3.5 Quadrature

29: 28.6 Expansions for Small q

30: 3.6 Linear Difference Equations

§24.21(ii) $B_{n}$ , $B_{n}\left(x\right)$ , $E_{n}$ , and $E_{n}\left(x\right)$

29: 28.6 Expansions for Small $q$