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31—40 of 590 matching pages

31: 24.10 Arithmetic Properties

… ►Let

B_{2n}=\ifrac{N_{2n}}{D_{2n}}

, with

N_{2n}

and

D_{2n}

relatively prime and

D_{2n}>0

. … ►

24.10.7

(b^{2n}-1)N_{2n}\equiv{2nb^{2n-1}D_{2n}\sum_{k=1}^{M-1}k^{2n-1}\left\lfloor% \frac{kb}{M}\right\rfloor\pmod{M}},

ⓘ

Symbols:: $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\equiv$ : modular equivalence, $k$ : integer, $n$ : integer, $M$ : integer and $b$ : integer
Permalink:: http://dlmf.nist.gov/24.10.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(iii), §24.10 and Ch.24

… ►

24.10.9

E_{2n}\equiv\begin{cases}0\pmod{p^{\ell}}&\text{if }p\equiv 1\pmod{4},\\ 2\pmod{p^{\ell}}&\text{if }p\equiv 3\pmod{4},\end{cases}

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\equiv$ : modular equivalence, $\ell$ : integer, $n$ : integer and $p$ : prime
Referenced by:: §24.10(iv)
Permalink:: http://dlmf.nist.gov/24.10.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(iv), §24.10 and Ch.24

…

32: 24.19 Methods of Computation

… ►

D_{2n}=\prod_{p-1\mathbin{|}2n}p,

►

B_{2n}=\dfrac{N_{2n}}{D_{2n}}.

… ►For number-theoretic applications it is important to compute

B_{2n}\pmod{p}

for

2n\leq p-3

; in particular to find the irregular pairs

(2n,p)

for which

B_{2n}\equiv 0\pmod{p}

. …

33: 1.6 Vectors and Vector-Valued Functions

… ►The divergence of a differentiable vector-valued function

\mathbf{F}=F_{1}\mathbf{i}+F_{2}\mathbf{j}+F_{3}\mathbf{k}

is … ►The line integral of a vector-valued function

\mathbf{F}=F_{1}\mathbf{i}+F_{2}\mathbf{j}+F_{3}\mathbf{k}

along

\mathbf{c}

is given by … ►with

(u,v)\in D

, an open set in the plane. … ►If

\boldsymbol{{\Phi}}_{1}

and

\boldsymbol{{\Phi}}_{2}

are both orientation preserving or both orientation reversing parametrizations of

S

defined on open sets

D_{1}

and

D_{2}

respectively, then ►

1.6.56

\iint_{\boldsymbol{{\Phi}}_{1}(D_{1})}\mathbf{F}\cdot\,\mathrm{d}\mathbf{S}=% \iint_{\boldsymbol{{\Phi}}_{2}(D_{2})}\mathbf{F}\cdot\,\mathrm{d}\mathbf{S};

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\boldsymbol{{\Phi}}(x,y,z)$ : parameterization and $D$ : open set in the plane
Permalink:: http://dlmf.nist.gov/1.6.E56
Encodings:: pMML, png, TeX
See also:: Annotations for §1.6(v), §1.6 and Ch.1

…

35: 12.1 Special Notation

… ►An older notation, due to Whittaker (1902), for

U\left(a,z\right)

D_{\nu}\left(z\right)

. The notations are related by

U\left(a,z\right)=D_{-a-\frac{1}{2}}\left(z\right)

. Whittaker’s notation

D_{\nu}\left(z\right)

is useful when

\nu

is a nonnegative integer (Hermite polynomial case).

36: 3.10 Continued Fractions

… ►if the expansion of its

n

th convergent

C_{n}

in ascending powers of

z

agrees with (3.10.7) up to and including the term in

z^{n-1}

n=1,2,3,\dots

. … ►We say that it is associated with the formal power series

f(z)

in (3.10.7) if the expansion of its

n

th convergent

C_{n}

in ascending powers of

z

, agrees with (3.10.7) up to and including the term in

z^{2n-1}

n=1,2,3,\dots

. … ►The

n

th partial sum

t_{0}+t_{1}+\dots+t_{n-1}

equals the

n

th convergent of (3.10.13),

n=1,2,3,\dots

. … ►

D_{n}=\frac{1}{D_{n-1}a_{n}+b_{n}},

… ►For further information on the preceding algorithms, including convergence in the complex plane and methods for accelerating convergence, see Blanch (1964) and Lorentzen and Waadeland (1992, Chapter 3). …

37: 28.28 Integrals, Integral Representations, and Integral Equations

… ►With the notations of §28.4 for

A_{m}^{n}(q)

and

B_{m}^{n}(q)

, §28.14 for

c_{n}^{\nu}(q)

, and (28.23.1) for

\mathcal{C}_{\mu}^{(j)}

j=1,2,3,4

, … ►where again

\varepsilon_{0}=2

and

\varepsilon_{\ell}=1

\ell=1,2,3,\dotsc

. … ►where

m-n=2p+1

p\in\mathbb{Z}

;

m,n=1,2,3,\dots

. … ►where

m-n=2p+1

p\in\mathbb{Z}

;

m=0,1,2,\dots

n=1,2,3,\dots

. … ►where

n-m=2p

p\in\mathbb{Z}

;

m=0,1,2,\dots

n=1,2,3,\dots

. …

38: 34.6 Definition: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ 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

►

34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ 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

…

39: 19.30 Lengths of Plane Curves

… ►

19.30.3

s/a=E\left(\phi,k\right)={R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{% D}\left(c-1,c-k^{2},c\right)},

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $s$ : arclength
Referenced by:: §19.30(i)
Permalink:: http://dlmf.nist.gov/19.30.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(i), §19.30 and Ch.19

… ►

19.30.9

s=\tfrac{1}{2}I(\mathbf{e}_{1})=-\tfrac{1}{3}a^{2}b^{2}R_{D}\left(r,r+b^{2}+a^% {2},r+b^{2}\right)+y\sqrt{\frac{r+b^{2}+a^{2}}{r+b^{2}}},

r=b^{4}/y^{2}

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $I(\mathbf{m})$ : integral and $s$ : arclength
Permalink:: http://dlmf.nist.gov/19.30.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(ii), §19.30 and Ch.19

►For

s

in terms of

E\left(\phi,k\right)

F\left(\phi,k\right)

, and an algebraic term, see Byrd and Friedman (1971, p. 3). … ►For other plane curves with arclength representable by an elliptic integral see Greenhill (1892, p. 190) and Bowman (1953, pp. 32–33). …

40: 19.36 Methods of Computation

… ►For both

R_{D}

and

R_{J}

the factor

(r/4)^{-1/6}

in Carlson (1995, (2.18)) is changed to

(r/5)^{-1/8}

when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms: … ►Because of cancellations in (19.26.21) it is advisable to compute

R_{G}

from

R_{F}

and

R_{D}

by (19.21.10) or else to use §19.36(ii). … ►Accurate values of

F\left(\phi,k\right)-E\left(\phi,k\right)

for

k^{2}

near 0 can be obtained from

R_{D}

by (19.2.6) and (19.25.13). … ►

E\left(\phi,k\right)

can be evaluated by using (19.25.7), and

R_{D}

by using (19.21.10), but cancellations may become significant. … ►Near these points there will be loss of significant figures in the computation of

R_{J}

R_{D}

. …

抢庄牛牛3D游戏大厅,网上抢庄牛牛3D游戏规则,【复制打开网址：33kk55.com】正规博彩平台,在线赌博平台,抢庄牛牛3D游戏玩法介绍,真人抢庄牛牛3D游戏规则,网上真人棋牌游戏平台,真人博彩游戏平台网址YBsCyXMBsMAyNMBM

31—40 of 590 matching pages

31: 24.10 Arithmetic Properties

32: 24.19 Methods of Computation

33: 1.6 Vectors and Vector-Valued Functions

34: 3.3 Interpolation

35: 12.1 Special Notation

36: 3.10 Continued Fractions

37: 28.28 Integrals, Integral Representations, and Integral Equations

38: 34.6 Definition: 9 ⁢ j Symbol

39: 19.30 Lengths of Plane Curves

40: 19.36 Methods of Computation

38: 34.6 Definition: $\mathit{9j}$ Symbol