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32: 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. …

33: 23.7 Quarter Periods

… ►

23.7.1

\wp\left(\tfrac{1}{2}\omega_{1}\right)=e_{1}+\sqrt{(e_{1}-e_{3})(e_{1}-e_{2})}% =e_{1}+\omega_{1}^{-2}(K\left(k\right))^{2}k^{\prime},

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►

23.7.2

\wp\left(\tfrac{1}{2}\omega_{2}\right)=e_{2}-i\sqrt{(e_{1}-e_{2})(e_{2}-e_{3})% }=e_{2}-i\omega_{1}^{-2}(K\left(k\right))^{2}kk^{\prime},

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►

23.7.3

\wp\left(\tfrac{1}{2}\omega_{3}\right)=e_{3}-\sqrt{(e_{1}-e_{3})(e_{2}-e_{3})}% =e_{3}-\omega_{1}^{-2}(K\left(k\right))^{2}k,

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $k$ : modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

…

34: 10.71 Integrals

… ►In the following equations

f_{\nu},g_{\nu}

is any one of the four ordered pairs given in (10.63.1), and

\widehat{f}_{\nu},\widehat{g}_{\nu}

is either the same ordered pair or any other ordered pair in (10.63.1). … ►

10.71.3

\int x(f_{\nu}\widehat{g}_{\nu}-g_{\nu}\widehat{f}_{\nu})\,\mathrm{d}x=\frac{x% }{2\sqrt{2}}\left(\widehat{f}_{\nu}(f_{\nu+1}+g_{\nu+1})-\widehat{g}_{\nu}(f_{% \nu+1}-g_{\nu+1})-f_{\nu}(\widehat{f}_{\nu+1}+\widehat{g}_{\nu+1})+g_{\nu}(% \widehat{f}_{\nu+1}-\widehat{g}_{\nu+1})\right)=\tfrac{1}{2}x(f_{\nu}^{\prime}% \widehat{f}_{\nu}-f_{\nu}\widehat{f}_{\nu}^{\prime}+g_{\nu}^{\prime}\widehat{g% }_{\nu}-g_{\nu}\widehat{g}^{\prime}_{\nu}),

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function, $g_{\nu}$ a Kelvin function, $\widehat{f}_{\nu}$ a Kelvin function and $\widehat{g}_{\nu}$ a Kelvin function
A&S Ref:: 9.9.23
Permalink:: http://dlmf.nist.gov/10.71.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §10.71(i), §10.71 and Ch.10

… ►

10.71.5

\int x(f_{\nu}^{2}+g_{\nu}^{2})\,\mathrm{d}x=x(f_{\nu}g_{\nu}^{\prime}-f_{\nu}% ^{\prime}g_{\nu})=-\frac{x}{\sqrt{2}}(f_{\nu}f_{\nu+1}+g_{\nu}g_{\nu+1}-f_{\nu% }g_{\nu+1}+f_{\nu+1}g_{\nu}),

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function and $g_{\nu}$ a Kelvin function
A&S Ref:: 9.9.25
Permalink:: http://dlmf.nist.gov/10.71.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §10.71(i), §10.71 and Ch.10

►

10.71.6

\int xf_{\nu}g_{\nu}\,\mathrm{d}x=\tfrac{1}{4}x^{2}\left(2f_{\nu}g_{\nu}-f_{% \nu-1}g_{\nu+1}-f_{\nu+1}g_{\nu-1}\right),

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function and $g_{\nu}$ a Kelvin function
A&S Ref:: 9.9.26
Permalink:: http://dlmf.nist.gov/10.71.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §10.71(i), §10.71 and Ch.10

… ►where

M_{\nu}\left(x\right)

and

N_{\nu}\left(x\right)

are the modulus functions introduced in §10.68(i). …

35: 19.19 Taylor and Related Series

… ►where the summation extends over all nonnegative integers

m_{1},\dots,m_{n}

whose sum is

N

. … ►Define the elementary symmetric function

E_{s}(\mathbf{z})

by …where

M=\sum_{j=1}^{n}m_{j}

and the summation extends over all nonnegative integers

m_{1},\dots,m_{n}

such that

\sum_{j=1}^{n}jm_{j}=N

. ►This form of

T_{N}

can be applied to (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) if we use …The number of terms in

T_{N}

can be greatly reduced by using variables

\mathbf{Z}=\boldsymbol{{1}}-(\mathbf{z}/A)

with

A

chosen to make

E_{1}(\mathbf{Z})=0

. …

36: 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 …

38: 10.75 Tables

… ►

•

Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

… ►

•

Zhang and Jin (1996, p. 199) tabulates the real and imaginary parts of the first 15 conjugate pairs of complex zeros of $Y_{0}\left(z\right)$ , $Y_{1}\left(z\right)$ , $Y_{1}'\left(z\right)$ and the corresponding values of $Y_{1}\left(z\right)$ , $Y_{0}\left(z\right)$ , $Y_{1}\left(z\right)$ , respectively, 10D.

… ►

•

Abramowitz and Stegun (1964, Chapter 11) tabulates $\int_{0}^{x}J_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 10D; $\int_{0}^{x}t^{-1}(1-J_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 8D.

►

•

Zhang and Jin (1996, p. 270) tabulates $\int_{0}^{x}J_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}t^{-1}(1-J_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{0}^{x}Y_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)1(.5)20$ , 8D.

… ►

•

Achenbach (1986) tabulates $I_{0}\left(x\right)$ , $I_{1}\left(x\right)$ , $K_{0}\left(x\right)$ , $K_{1}\left(x\right)$ , $x=0(.1)8$ , 19D or 19–21S.

…

39: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}

, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod

m

), where

m

is the product of the moduli. … ►Choose four relatively prime moduli

m_{1},m_{2},m_{3}

, and

m_{4}

of five digits each, for example

2^{16}-3

2^{16}-1

2^{16}+1

, and

2^{16}+3

. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod

m_{1}

), (mod

m_{2}

), (mod

m_{3}

), and (mod

m_{4}

), respectively. Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers

a_{1}\pmod{m_{1}}

a_{2}\pmod{m_{2}}

a_{3}\pmod{m_{3}}

, and

a_{4}\pmod{m_{4}}

, where each

a_{j}

has no more than five digits. …

40: 17.11 Transformations of $q$ -Appell Functions

… ►

17.11.1

\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)=\frac{\left(a,bx,b^{\prime}y;q% \right)_{\infty}}{\left(c,x,y;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({c/a,x,y% \atop bx,b^{\prime}y};q,a\right),

ⓘ

Symbols:: $\Phi^{(1)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c};\NVar{q};\NVar{x}% ,\NVar{y}\right)$ : first $q$ -Appell function, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $x$ : real variable and $y$ : real variable
Referenced by:: §17.11, Erratum (V1.0.10) for Section 17.1
Permalink:: http://dlmf.nist.gov/17.11.E1
Encodings:: pMML, png, TeX
Correction (effective with 1.0.10):: The notation for $\Phi^{(1)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.
See also:: Annotations for §17.11 and Ch.17

►

17.11.2

\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)=\frac{\left(b,ax;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a,b% ^{\prime};q\right)_{n}\left(c/b,x;q\right)_{r}b^{r}y^{n}}{\left(q,c^{\prime};q% \right)_{n}\left(q;q\right)_{r}\left(ax;q\right)_{n+r}},

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\Phi^{(2)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c},\NVar{c^{\prime}}% ;\NVar{q};\NVar{x},\NVar{y}\right)$ : second $q$ -Appell function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer, $r$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Referenced by:: Erratum (V1.0.10) for Section 17.1, Erratum (V1.1.3) for Equation (17.11.2)
Permalink:: http://dlmf.nist.gov/17.11.E2
Encodings:: pMML, png, TeX
Correction (effective with 1.1.3):: The factor ${\left(q\right)_{r}}$ originally used in the denominator has been corrected to be $\left(q;q\right)_{r}$ .
Correction (effective with 1.0.10):: The notation for $\Phi^{(2)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.
See also:: Annotations for §17.11 and Ch.17

►

17.11.3

\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)=\frac{\left(a,bx;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a^{% \prime},b^{\prime};q\right)_{n}\left(x;q\right)_{r}\left(c/a;q\right)_{n+r}a^{% r}y^{n}}{\left(q,c/a;q\right)_{n}\left(q,bx;q\right)_{r}}.

ⓘ

Symbols:: $\Phi^{(3)}\left(\NVar{a},\NVar{a^{\prime}};\NVar{b},\NVar{b^{\prime}};\NVar{c}% ;\NVar{q};\NVar{x},\NVar{y}\right)$ : third $q$ -Appell function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer, $r$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Referenced by:: §17.11, Erratum (V1.0.10) for Section 17.1
Permalink:: http://dlmf.nist.gov/17.11.E3
Encodings:: pMML, png, TeX
Correction (effective with 1.0.10):: The notation for $\Phi^{(3)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.
See also:: Annotations for §17.11 and Ch.17

… ►

17.11.4

\sum_{m_{1},\dots,m_{n}\geqq 0}\frac{\left(a;q\right)_{m_{1}+m_{2}+\cdots+m_{n% }}\left(b_{1};q\right)_{m_{1}}\left(b_{2};q\right)_{m_{2}}\cdots\left(b_{n};q% \right)_{m_{n}}x_{1}^{m_{1}}x_{2}^{m_{2}}\cdots x_{n}^{m_{n}}}{\left(q;q\right% )_{m_{1}}\left(q;q\right)_{m_{2}}\cdots\left(q;q\right)_{m_{n}}\left(c;q\right% )_{m_{1}+m_{2}+\cdots+m_{n}}}=\frac{\left(a,b_{1}x_{1},b_{2}x_{2},\dots,b_{n}x% _{n};q\right)_{\infty}}{\left(c,x_{1},x_{2},\dots,x_{n};q\right)_{\infty}}{{}_% {n+1}\phi_{n}}\left({c/a,x_{1},x_{2},\dots,x_{n}\atop b_{1}x_{1},b_{2}x_{2},% \dots,b_{n}x_{n}};q,a\right).

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.11 and Ch.17

.今年世界杯对阵__『wn4.com_』_巴西世界杯梅西金球奖_w6n2c9o_2022年11月30日8时7分_lvhhtouai.com

31—40 of 783 matching pages

31: 17.12 Bailey Pairs

32: 3.11 Approximation Techniques

33: 23.7 Quarter Periods

34: 10.71 Integrals

35: 19.19 Taylor and Related Series

36: 3.7 Ordinary Differential Equations

37: 3.10 Continued Fractions

38: 10.75 Tables

39: 27.15 Chinese Remainder Theorem

40: 17.11 Transformations of $q$ -Appell Functions

.今年世界杯对阵__『wn4.com_』_巴西世界杯梅西金球奖_w6n2c9o_2022年11月30日8时7分_lvhhtouai.com

31—40 of 783 matching pages

31: 17.12 Bailey Pairs

32: 3.11 Approximation Techniques

33: 23.7 Quarter Periods

34: 10.71 Integrals

35: 19.19 Taylor and Related Series

36: 3.7 Ordinary Differential Equations

37: 3.10 Continued Fractions

38: 10.75 Tables

39: 27.15 Chinese Remainder Theorem

40: 17.11 Transformations of q -Appell Functions

40: 17.11 Transformations of $q$ -Appell Functions