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31: 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). …

32: 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

. …

33: 3.10 Continued Fractions

… ►For instance, if none of the

u_{n}

vanish, then we can define … ►The first two columns in this table are defined by …where the

c_{n}

(

\neq 0

) appear in (3.10.7). … ►The

A_{n}

and

B_{n}

of (3.10.2) can be computed by means of three-term recurrence relations (1.12.5). … ►Then

u_{0}=C_{n}

. …

34: 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. …

35: 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

36: 10.58 Zeros

… ►For

n\geq 0

the

m

th positive zeros of

\mathsf{j}_{n}\left(x\right)

\mathsf{j}_{n}'\left(x\right)

\mathsf{y}_{n}\left(x\right)

, and

\mathsf{y}_{n}'\left(x\right)

are denoted by

a_{n,m}

a^{\prime}_{n,m}

b_{n,m}

, and

b^{\prime}_{n,m}

, respectively, except that for

n=0

we count

x=0

as the first zero of

\mathsf{j}_{0}'\left(x\right)

. … ►

a_{n,m}=j_{n+\frac{1}{2},m},

►

b_{n,m}=y_{n+\frac{1}{2},m},

… ►Hence properties of

a_{n,m}

and

b_{n,m}

are derivable straightforwardly from results given in §§10.21(i)–10.21(iii), 10.21(vi)–10.21(viii), and 10.21(x). …For some properties of

a^{\prime}_{n,m}

and

b^{\prime}_{n,m}

, including asymptotic expansions, see Olver (1960, pp. xix–xxi). …

37: 28.1 Special Notation

… ►The functions

{\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)

and

{\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)

are also known as the radial Mathieu functions. … ►

g_{\mathit{e},n}(h),

►

g_{\mathit{o},n}(h),

►

f_{\mathit{e},n}(h),

… ►The radial functions

{\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)

and

{\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)

are denoted by

{\operatorname{Mc}^{(j)}_{n}}\left(z,q\right)

and

{\operatorname{Ms}^{(j)}_{n}}\left(z,q\right)

, respectively.

38: 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, …

39: 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. …

40: 22.7 Landen Transformations

… ►

22.7.2

\operatorname{sn}\left(z,k\right)=\frac{(1+k_{1})\operatorname{sn}\left(z/(1+k% _{1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}% \right)},

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.2
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

… ►

22.7.4

\operatorname{dn}\left(z,k\right)=\frac{{\operatorname{dn}}^{2}\left(z/(1+k_{1% }),k_{1}\right)-(1-k_{1})}{1+k_{1}-{\operatorname{dn}}^{2}\left(z/(1+k_{1}),k_% {1}\right)}.

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.4
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

… ►

k_{2}=\frac{2\sqrt{k}}{1+k},

►

k_{2}^{\prime}=\frac{1-k}{1+k},

… ►

22.7.8

\operatorname{dn}\left(z,k\right)=\frac{(1-k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)+k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}.

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus, $k_{2}$ : change of variable and $k_{2}^{\prime}$ : change of variable
A&S Ref:: 16.14.4
Permalink:: http://dlmf.nist.gov/22.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(ii), §22.7 and Ch.22

…

.韩国兰芝岛世界杯公园_『网址:68707.vip』篮球世界杯竞猜_b5p6v3_pr7bnb1nb.com

31—40 of 765 matching pages

31: 10.71 Integrals

32: 19.19 Taylor and Related Series

33: 3.10 Continued Fractions

34: 27.15 Chinese Remainder Theorem

35: 17.11 Transformations of q -Appell Functions

36: 10.58 Zeros

37: 28.1 Special Notation

38: 32.7 Bäcklund Transformations

39: 3.11 Approximation Techniques

40: 22.7 Landen Transformations

35: 17.11 Transformations of $q$ -Appell Functions