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11: 27.2 Functions

… ►where

p_{1},p_{2},\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)=\phi\left(n\right)

. ►In the following examples,

a_{1},\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}

. …

12: 3.11 Approximation Techniques

… ►Beginning with

u_{n+1}=0

u_{n}=c_{n}

, we apply … ►With

b_{0}=1

, the last

q

equations give

b_{1},\dots,b_{q}

as the solution of a system of linear equations. … ►(3.11.29) is a system of

n+1

linear equations for the coefficients

a_{0},a_{1},\dots,a_{n}

. … ►With this choice of

a_{k}

and

f_{j}=f(x_{j})

, the corresponding sum (3.11.32) vanishes. … ►Two are endpoints:

(x_{0},y_{0})

and

(x_{3},y_{3})

; the other points

(x_{1},y_{1})

and

(x_{2},y_{2})

are control points. …

13: 24.2 Definitions and Generating Functions

… ►

B_{2n+1}=0

… ►

24.2.4

B_{n}=B_{n}\left(0\right),

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.2
Referenced by:: (25.11.7)
Permalink:: http://dlmf.nist.gov/24.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(i), §24.2 and Ch.24

… ►

Table 24.2.4: Euler numbers

E_{n}

► ►►

$n$	$E_{n}$
…

► ►

Table 24.2.5: Coefficients

b_{n,k}

of the Bernoulli polynomials

B_{n}\left(x\right)=\sum_{k=0}^{n}b_{n,k}x^{k}

► ►►

	$k$
…

► ►

Table 24.2.6: Coefficients

e_{n,k}

of the Euler polynomials

E_{n}\left(x\right)=\sum_{k=0}^{n}e_{n,k}x^{k}

► ►►

	$k$
…

►

14: 3.7 Ordinary Differential Equations

… ►The path is partitioned at

P+1

points labeled successively

z_{0},z_{1},\dots,z_{P}

, with

z_{0}=a

z_{P}=b

. … ►Write

\tau_{j}=z_{j+1}-z_{j}

j=0,1,\dots,P

, expand

w(z)

and

w^{\prime}(z)

in Taylor series (§1.10(i)) centered at

z=z_{j}

, and apply (3.7.2). … ►If, for example,

\beta_{0}=\beta_{1}=0

, then on moving the contributions of

w(z_{0})

and

w(z_{P})

to the right-hand side of (3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of

\mathbf{A}_{P}

that lie below the main diagonal and its two adjacent diagonals. … ►The values

\lambda_{k}

are the eigenvalues and the corresponding solutions

w_{k}

of the differential equation are the eigenfunctions. … ►where

h=z_{n+1}-z_{n}

and …

15: 23.9 Laurent and Other Power Series

… ►Let

z_{0}(\neq 0)

be the nearest lattice point to the origin, and define …Explicit coefficients

c_{n}

in terms of

c_{2}

and

c_{3}

are given up to

c_{19}

in Abramowitz and Stegun (1964, p. 636). ►For

j=1,2,3

, and with

e_{j}

as in §23.3(i), … ►where

a_{0,0}=1

a_{m,n}=0

if either

m

n<0

, and …For

a_{m,n}

with

m=0,1,\dots,12

and

n=0,1,\dots,8

, see Abramowitz and Stegun (1964, p. 637).

16: 3.2 Linear Algebra

… ►where

u_{j}=c_{j}

j=1,2,\dots,n-1

d_{1}=b_{1}

, and …Forward elimination for solving

\mathbf{A}\mathbf{x}=\mathbf{f}

then becomes

y_{1}=f_{1}

, …and back substitution is

x_{n}=y_{n}/d_{n}

, followed by … ►Define the Lanczos vectors

\mathbf{v}_{j}

and coefficients

\alpha_{j}

and

\beta_{j}

\mathbf{v}_{0}=\boldsymbol{{0}}

, a normalized vector

\mathbf{v}_{1}

(perhaps chosen randomly),

\alpha_{1}=\mathbf{v}_{1}^{\rm T}\mathbf{A}\mathbf{v}_{1}

\beta_{1}=0

, and for

j=1,2,\ldots,n-1

by the recursive scheme … ►Start with

\mathbf{v}_{0}=\boldsymbol{{0}}

, vector

\mathbf{v}_{1}

such that

\mathbf{v}_{1}^{\rm T}\mathbf{S}\mathbf{v}_{1}=1

\alpha_{1}=\mathbf{v}_{1}^{\rm T}\mathbf{A}\mathbf{v}_{1}

\beta_{1}=0

. …

17: 32.7 Bäcklund Transformations

… ►Let

w_{j}(z_{j})=w(z_{j};\alpha_{j},\beta_{j},\gamma_{j},\delta_{j})

j=0,1,2

, be solutions of

\mbox{P}_{\mbox{\scriptsize V}}

with … ►satisfies

\mbox{P}_{\mbox{\scriptsize V}}

with … ►Let

w_{j}(z_{j})=w_{j}(z_{j};\alpha_{j},\beta_{j},\gamma_{j},\delta_{j})

j=0,1,2,3

, be solutions of

\mbox{P}_{\mbox{\scriptsize VI}}

with … ►

\mbox{P}_{\mbox{\scriptsize VI}}

also has quadratic and quartic transformations. …Also, …

18: 17.6 ${{}_{2}\phi_{1}}$ Function

§17.6 ${{}_{2}\phi_{1}}$ Function

… ►

§17.6(ii) ${{}_{2}\phi_{1}}$ Transformations

… ►

Three-Term ${{}_{2}\phi_{1}}$ Transformations

… ►(17.6.27) reduces to the hypergeometric equation (15.10.1) with the substitutions

a\to q^{a}

b\to q^{b}

c\to q^{c}

, followed by

\lim_{q\to 1-}

. … ►For continued-fraction representations of the

{{}_{2}\phi_{1}}

function, see Cuyt et al. (2008, pp. 395–399).

19: 3.9 Acceleration of Convergence

… ►A transformation of a convergent sequence

\{s_{n}\}

with limit

\sigma

into a sequence

\{t_{n}\}

is called limit-preserving if

\{t_{n}\}

converges to the same limit

\sigma

. … ►This transformation is accelerating if

\{s_{n}\}

is a linearly convergent sequence, i. … ►Then the transformation of the sequence

\{s_{n}\}

into a sequence

\{t_{n,2k}\}

is given by … ►Then

t_{n,2k}=\varepsilon_{2k}^{(n)}

. … ►We give a special form of Levin’s transformation in which the sequence

s=\{s_{n}\}

of partial sums

s_{n}=\sum_{j=0}^{n}a_{j}

is transformed into: …

20: 3.4 Differentiation

… ►The

B_{k}^{n}

are the differentiated Lagrangian interpolation coefficients: … ►where

\xi_{0}

and

\xi_{1}\in I

. ►For the values of

n_{0}

and

n_{1}

used in the formulas below … ►For partial derivatives we use the notation

u_{t,s}=u(x_{0}+th,y_{0}+sh)

. …

.1966年世界杯决赛悬案_『wn4.com_』国际象棋世界杯第十轮_w6n2c9o_2022年11月28日19时19分39秒_swou6u2qo_cc

11—20 of 782 matching pages

11: 27.2 Functions

12: 3.11 Approximation Techniques

13: 24.2 Definitions and Generating Functions

14: 3.7 Ordinary Differential Equations

15: 23.9 Laurent and Other Power Series

16: 3.2 Linear Algebra

17: 32.7 Bäcklund Transformations

18: 17.6 ϕ 1 2 Function

§17.6 ϕ 1 2 Function

§17.6(ii) ϕ 1 2 Transformations

Three-Term ϕ 1 2 Transformations

19: 3.9 Acceleration of Convergence

20: 3.4 Differentiation

18: 17.6 ${{}_{2}\phi_{1}}$ Function

§17.6 ${{}_{2}\phi_{1}}$ Function

§17.6(ii) ${{}_{2}\phi_{1}}$ Transformations

Three-Term ${{}_{2}\phi_{1}}$ Transformations