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21: 1.12 Continued Fractions

… ►when

p_{k}\not=0

k=1,2,3,\dots

. …when

c_{k}\not=0

k=1,2,3,\dots

. … ►Define … ►The continued fraction

\displaystyle{\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}}

converges when … ►Then the convergents

C_{n}

satisfy …

22: 19.32 Conformal Map onto a Rectangle

… ►with

x_{1},x_{2},x_{3}

real constants, has differential … ►

19.32.3

x_{1}>x_{2}>x_{3},

ⓘ

Permalink:: http://dlmf.nist.gov/19.32.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.32 and Ch.19

… ►

z(x_{2})=z(x_{1})+z(x_{3}),

►

z(x_{3})=R_{F}\left(x_{3}-x_{1},x_{3}-x_{2},0\right)=-iR_{F}\left(0,x_{1}-x_{3% },x_{2}-x_{3}\right).

►As

p

proceeds along the entire real axis with the upper half-plane on the right,

z

describes the rectangle in the clockwise direction; hence

z(x_{3})

is negative imaginary. …

23: 34.10 Zeros

§34.10 Zeros

►In a

\mathit{3j}

symbol, if the three angular momenta

j_{1},j_{2},j_{3}

do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the

\mathit{3j}

symbol is zero. Similarly the

\mathit{6j}

symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four

\mathit{3j}

symbols in the summation. …However, the

\mathit{3j}

and

\mathit{6j}

symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …

24: 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}

► ►►

	$k$
…

►

25: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

§35.8(iii) ${{}_{3}F_{2}}$ Case

►

Kummer Transformation

►Let

c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}

. … ►Let

a_{1}+a_{2}+a_{3}+\frac{1}{2}(m+1)=b_{1}+b_{2}

; one of the

a_{j}

be a negative integer;

\Re\left(b_{1}-a_{1}\right)

\Re\left(b_{1}-a_{2}\right)

\Re\left(b_{1}-a_{3}\right)

\Re\left(b_{1}-a_{1}-a_{2}-a_{3}\right)>\frac{1}{2}(m-1)

. … ►Again, let

c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}

. …

26: 23.2 Definitions and Periodic Properties

… ►If

\omega_{1}

and

\omega_{3}

are nonzero real or complex numbers such that

\Im\left(\omega_{3}/\omega_{1}\right)>0

, then the set of points

2m\omega_{1}+2n\omega_{3}

, with

m,n\in\mathbb{Z}

, constitutes a lattice

\mathbb{L}

with

2\omega_{1}

and

2\omega_{3}

lattice generators. … ►then

2\omega_{2}

2\omega_{3}

are generators, as are

2\omega_{2}

2\omega_{1}

. …where

a, b, c, d

are integers, then

2\chi_{1}

2\chi_{3}

are generators of

\mathbb{L}

iff … ►If

2\omega_{1}

2\omega_{3}

is any pair of generators of

\mathbb{L}

, and

\omega_{2}

is defined by (23.2.1), then … ►where …

27: 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)

. …

28: 24.3 Graphs

… ►

See accompanying text — Figure 24.3.1: Bernoulli polynomials $B_{n}\left(x\right)$ , $n=2,3,\dots,6$ . Magnify

►

29: 1.11 Zeros of Polynomials

… ►

f(z)=z^{3}-6z^{2}+6z-2

g(w)=w^{3}-6w-6

A=3\sqrt[3]{4}

B=3\sqrt[3]{2}

. Roots of

f(z)=0

are

2+\sqrt[3]{4}+\sqrt[3]{2}

2+\sqrt[3]{4}\rho+\sqrt[3]{2}\rho^{2}

2+\sqrt[3]{4}\rho^{2}+\sqrt[3]{2}\rho

. … ►For the roots

\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}

g(w)=0

and the roots

\theta_{1},\theta_{2},\theta_{3}

of the resolvent cubic equation … ►Resolvent cubic is

z^{3}+12z^{2}+20z+9=0

with roots

\theta_{1}=-1

\theta_{2}=-\tfrac{1}{2}(11+\sqrt{85})

\theta_{3}=-\tfrac{1}{2}(11-\sqrt{85})

, and

\sqrt{-\theta_{1}}=1

\sqrt{-\theta_{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})

\sqrt{-\theta_{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})

. So

2\alpha_{1}=1+\sqrt{17}

2\alpha_{2}=1-\sqrt{17}

2\alpha_{3}=-1+\sqrt{5}

2\alpha_{4}=-1-\sqrt{5}

, and the roots of

f(z)=0

are

\tfrac{1}{2}(3\pm\sqrt{17})

\tfrac{1}{2}(1\pm\sqrt{5})

. …

30: 16.12 Products

… ►

16.12.1

{{}_{0}F_{1}}\left(-;a;z\right){{}_{0}F_{1}}\left(-;b;z\right)={{}_{2}F_{3}}% \left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop a,b,a+b-1};4z\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: (10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8
Permalink:: http://dlmf.nist.gov/16.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

… ►

16.12.2

\left({{}_{2}F_{1}}\left({a,b\atop a+b+\frac{1}{2}};z\right)\right)^{2}={{}_{3% }F_{2}}\left({2a,2b,a+b\atop a+b+\frac{1}{2},2a+2b};z\right).

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.12, §17.9(iii), Erratum (V1.1.12) for Subsection 17.9(iii)
Permalink:: http://dlmf.nist.gov/16.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

… ►

16.12.3

\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)\right)^{2}=\sum_{k=0}^{\infty}% \frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}}{\left(c-\frac{1}{2}\right)_{k% }}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}k!}{{}_{4}F_{3}}\left({-\frac{1% }{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}\atop a+\frac{1}{2},b+% \frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k},

|z|<1

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.12
Permalink:: http://dlmf.nist.gov/16.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

…

.中国韩国世界杯3月__『wn4.com_』_邯郸看世界杯的好地方_w6n2c9o_2022年11月28日21时26分39秒_znlrjd1fz.com

21—30 of 817 matching pages

21: 1.12 Continued Fractions

22: 19.32 Conformal Map onto a Rectangle

23: 34.10 Zeros

§34.10 Zeros

24: 24.2 Definitions and Generating Functions

25: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8(iii) F 2 3 Case

Kummer Transformation

26: 23.2 Definitions and Periodic Properties

27: 3.4 Differentiation

28: 24.3 Graphs

29: 1.11 Zeros of Polynomials

30: 16.12 Products

§35.8(iii) ${{}_{3}F_{2}}$ Case