.put世界杯相声汇_『网址:687.vii』世界杯决赛乌龙球_b5p6v3_2022年11月30日4时58分48秒_ucgsamhsa

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11: 19.21 Connection Formulas

… ►The complete cases of

R_{F}

and

R_{G}

have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … ►The complete case of

R_{J}

can be expressed in terms of

R_{F}

and

R_{D}

: … ►Because

R_{G}

is completely symmetric,

x, y, z

can be permuted on the right-hand side of (19.21.10) so that

(x-z)(y-z)\leq 0

if the variables are real, thereby avoiding cancellations when

R_{G}

is calculated from

R_{F}

and

R_{D}

(see §19.36(i)). … ►Connection formulas for

R_{-a}\left(\mathbf{b};\mathbf{z}\right)

are given in Carlson (1977b, pp. 99, 101, and 123–124). ►

§19.21(iii) Change of Parameter of $R_{J}$

…

12: 19.16 Definitions

… ►It should be noted that the integrals (19.16.1)–(19.16.2_5) have been normalized so that

R_{F}\left(1,1,1\right)=R_{J}\left(1,1,1,1\right)=R_{G}\left(1,1,1\right)=1

. … ►and

R_{D}

is a degenerate case of

R_{J}

, so is

R_{J}

a degenerate case of the hyperelliptic integral, … ►Thus

R_{-a}\left(\mathbf{b};\mathbf{z}\right)

is symmetric in the variables

z_{j}

and

z_{\ell}

if the parameters

b_{j}

and

b_{\ell}

are equal. … ►

§19.16(iii) Various Cases of $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

… ►The only cases that are integrals of the third kind are those in which at least one

b_{j}

is a positive integer. …

13: 18.40 Methods of Computation

… ►The quadrature points and weights can be put to a more direct and efficient use. …

14: 19.5 Maclaurin and Related Expansions

… ►

19.5.1

K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer and $k$ : real or complex modulus
Source:: Carlson (1961b, (2.4))
Referenced by:: §19.36(iv), §19.5
Permalink:: http://dlmf.nist.gov/19.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.5 and Ch.19

►where

{{}_{2}F_{1}}

is the Gauss hypergeometric function (§§15.1 and 15.2(i)). …where

{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)

is an Appell function (§16.13). … ►where

k_{0}=k

and ►

19.5.11

k_{m+1}=\frac{1-\sqrt{1-k_{m}^{2}}}{1+\sqrt{1-k_{m}^{2}}},

m=0,1,\dots

ⓘ

Symbols:: $m$ : nonnegative integer and $k$ : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.5.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §19.5 and Ch.19

…

15: 19.18 Derivatives and Differential Equations

… ►Let

\partial_{j}=\ifrac{\partial}{\partial z_{j}}

, and

\mathbf{e}_{j}

be an

n

-tuple with 1 in the

j

th place and 0’s elsewhere. Also define … ►More concisely, if

v=R_{-a}\left(\mathbf{b};\mathbf{z}\right)

, then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation: …If

n=2

, then elimination of

\partial_{2}v

between (19.18.11) and (19.18.12), followed by the substitution

(b_{1},b_{2},z_{1},z_{2})=(b,c-b,1-z,1)

, produces the Gauss hypergeometric equation (15.10.1). ►The next four differential equations apply to the complete case of

R_{F}

and

R_{G}

in the form

R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)

(see (19.16.20) and (19.16.23)). …

16: 19.23 Integral Representations

… ►

19.23.4

R_{F}\left(0,y,z\right)=\frac{2}{\pi}\int_{0}^{\pi/2}R_{C}\left(y,z{\cos}^{2}% \theta\right)\,\mathrm{d}\theta=\frac{2}{\pi}\int_{0}^{\infty}R_{C}\left(y{% \cosh}^{2}t,z\right)\,\mathrm{d}t.

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function and $\int$ : integral
Referenced by:: §19.23
Permalink:: http://dlmf.nist.gov/19.23.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.23 and Ch.19

… ►

19.23.8

R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{2}{\mathrm{B}\left(b_{1},b_{2}% \right)}\int_{0}^{\pi/2}{(z_{1}{\cos}^{2}\theta+z_{2}{\sin}^{2}\theta)}^{-a}\*% (\cos\theta)^{2b_{1}-1}(\sin\theta)^{2b_{2}-1}\,\mathrm{d}\theta,

b_{1},b_{2}>0

;

\Re z_{1},\Re z_{2}>0

ⓘ

Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part and $\sin\NVar{z}$ : sine function
Referenced by:: §19.2(iv), §19.23, §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.23 and Ch.19

►With

l_{1},l_{2},l_{3}

denoting any permutation of

\sin\theta\cos\phi

\sin\theta\sin\phi

\cos\theta

, ►

19.23.9

R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{4\Gamma\left(b_{1}+b_{2}+b_{3}% \right)}{\Gamma\left(b_{1}\right)\Gamma\left(b_{2}\right)\Gamma\left(b_{3}% \right)}\int_{0}^{\pi/2}\!\!\!\!\int_{0}^{\pi/2}\left(\sum_{j=1}^{3}z_{j}l_{j}% ^{2}\right)^{-a}\*\prod_{j=1}^{3}l_{j}^{2b_{j}-1}\sin\theta\,\mathrm{d}\theta% \,\mathrm{d}\phi,

b_{j}>0

\Re z_{j}>0

ⓘ

Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $l$ : nonnegative integer and $\phi$ : real or complex argument
Referenced by:: §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.23 and Ch.19

►

19.23.10

R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{\mathrm{B}\left(a,a^{\prime}% \right)}\int_{0}^{1}u^{a-1}(1-u)^{a^{\prime}-1}\*\prod_{j=1}^{n}(1-u+uz_{j})^{% -b_{j}}\,\mathrm{d}u,

a,a^{\prime}>0

;

a+a^{\prime}=\sum_{j=1}^{n}b_{j}

;

z_{j}\in\mathbb{C}\setminus(-\infty,0]

ⓘ

Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\setminus$ : set subtraction and $n$ : nonnegative integer
Referenced by:: §19.19, §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §19.23 and Ch.19

…

17: 23.20 Mathematical Applications

… ►An algebraic curve that can be put either into the form …

18: 19.2 Definitions

… ►where

p_{j}

is a polynomial in

t

while

\rho

and

\sigma

are rational functions of

t

. … ►Here

a, b, p

are real parameters, and

k_{c}

and

x

are real or complex variables, with

p\neq 0

k_{c}\neq 0

. … ►If

1<k\leq 1/\sin\phi

, then

k_{c}

is pure imaginary. … ►

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

… ►For the special cases of

R_{C}\left(x,x\right)

and

R_{C}\left(0,y\right)

see (19.6.15). …

19: 26.2 Basic Definitions

… ►

Table 26.2.1: Partitions

p\left(n\right)

► ►►►►►►►

$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
6	11	23	1255	40	37338
…
9	30	26	2436	43	63261
…
11	56	28	3718	45	89134
…
13	101	30	5604	47	1 24754
14	135	31	6842	48	1 47273
…

►

20: 18.12 Generating Functions

… ►Note that (18.12.2_5) yields (18.12.1) by putting

\gamma=0

and (18.12.2) by replacing

z

-\gamma^{-2}z

and next letting

\gamma\to\infty

. …