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1: 19.2 Definitions

… ►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: $R_{C}\left(x,y\right)$

… ►When

x

and

y

are positive,

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

is an inverse circular function if

x<y

and an inverse hyperbolic function (or logarithm) if

x>y

: …For the special cases of

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

and

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

see (19.6.15). …

2: 15.5 Derivatives and Contiguous Functions

… ►The six functions

F\left(a\pm 1,b;c;z\right)

F\left(a,b\pm 1;c;z\right)

F\left(a,b;c\pm 1;z\right)

are said to be contiguous to

F\left(a,b;c;z\right)

. … ►

15.5.14

c\left(a+(b-c)z\right)F\left(a,b;c;z\right)-ac(1-z)F\left(a+1,b;c;z\right)+(c-% a)(c-b)zF\left(a,b;c+1;z\right)=0,

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.5.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(ii), §15.5 and Ch.15

… ►

15.5.18

c(c-1)(z-1)F\left(a,b;c-1;z\right)+{c\left(c-1-(2c-a-b-1)z\right)}F\left(a,b;c% ;z\right)+(c-a)(c-b)zF\left(a,b;c+1;z\right)=0.

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.19(iv), §15.5(ii)
Permalink:: http://dlmf.nist.gov/15.5.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(ii), §15.5 and Ch.15

►By repeated applications of (15.5.11)–(15.5.18) any function

F\left(a+k,b+\ell;c+m;z\right)

, in which

k,\ell,m

are integers, can be expressed as a linear combination of

F\left(a,b;c;z\right)

and any one of its contiguous functions, with coefficients that are rational functions of

a, b, c

, and

z

. … ►

15.5.20

z(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right)}{\mathrm{d}z}\right)=(c-a)F% \left(a-1,b;c;z\right)+(a-c+bz)F\left(a,b;c;z\right)=(c-b)F\left(a,b-1;c;z% \right)+(b-c+az)F\left(a,b;c;z\right),

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.5.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(ii), §15.5 and Ch.15

…

3: 4.42 Solution of Triangles

… ►

4.42.5

c^{2}=a^{2}+b^{2}-2ab\cos C,

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $a$ : length, $b$ : base, $c$ : length and $C$ : angle
A&S Ref:: 4.3.148
Permalink:: http://dlmf.nist.gov/4.42.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §4.42(ii), §4.42 and Ch.4

►

4.42.6

a=b\cos C+c\cos B

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $a$ : length, $b$ : base, $c$ : length, $B$ : angle and $C$ : angle
A&S Ref:: 4.3.148
Permalink:: http://dlmf.nist.gov/4.42.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §4.42(ii), §4.42 and Ch.4

►

4.42.7

\hbox{area}=\tfrac{1}{2}bc\sin A=\left(s(s-a)(s-b)(s-c)\right)^{1/2},

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function, $a$ : length, $b$ : base, $c$ : length, $s$ : semi-perimeter and $A$ : angle
A&S Ref:: 4.3.148
Permalink:: http://dlmf.nist.gov/4.42.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §4.42(ii), §4.42 and Ch.4

►where

s=\tfrac{1}{2}(a+b+c)

(the semiperimeter). … ►

4.42.8

\cos a=\cos b\cos c+\sin b\sin c\cos A,

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $A$ : angle, $a$ : arc length, $b$ : arc length and $c$ : arc length
A&S Ref:: 4.3.149
Permalink:: http://dlmf.nist.gov/4.42.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §4.42(iii), §4.42 and Ch.4

…

4: 15.13 Zeros

… ►Let

N(a,b,c)

denote the number of zeros of

F\left(a,b;c;z\right)

in the sector

|\operatorname{ph}\left(1-z\right)|<\pi

. If

a

b

c

are real,

a

b

c

c-a

c-b\neq 0,-1,-2,\dots

, and, without loss of generality,

b\geq a

c\geq a+b

(compare (15.8.1)), then ►

15.13.1

N(a,b,c)=\begin{cases}0,&a>0,\\ \left\lfloor-a\right\rfloor+\tfrac{1}{2}(1+S),&a<0,c-a>0,\\ \left\lfloor-a\right\rfloor+\tfrac{1}{2}(1+S)+\left\lfloor a-c+1\right\rfloor S% ,&a<0,c-a<0,\\ \end{cases}

ⓘ

Symbols:: $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $N(a,b,c)$ : number of zeros and $S$
Permalink:: http://dlmf.nist.gov/15.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.13 and Ch.15

►where

S=\operatorname{sign}\left(\Gamma\left(a\right)\Gamma\left(b\right)\Gamma\left% (c-a\right)\Gamma\left(c-b\right)\right)

. … ►If

a

b

c

c-a

, or

c-b\in\{0,-1,-2,\dots\}

, then

F\left(a,b;c;z\right)

is not defined, or reduces to a polynomial, or reduces to

(1-z)^{c-a-b}

times a polynomial. …

5: 15.1 Special Notation

… ► ►►►

$x$	real variable.
…
$a, b, c$	real or complex parameters.
…

… ►

15.1.1

{{}_{2}F_{1}}\left(a,b;c;z\right)=F\left(a,b;c;z\right)=F\left({a,b\atop c};z% \right),

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.1.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.1 and Ch.15

… ►

15.1.2

\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}\left(a,b;c;z% \right)=\mathbf{F}\left({a,b\atop c};z\right)={{}_{2}{\mathbf{F}}_{1}}\left(a,% b;c;z\right),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.1.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.1 and Ch.15

…

6: 15.7 Continued Fractions

… ►

t_{n}=c+n,

►

u_{2n+1}=(a+n)(c-b+n),

►

u_{2n}=(b+n)(c-a+n).

… ►

v_{n}=c+n+(b-a+n+1)z,

►

w_{n}=(b+n)(c-a+n)z.

…

7: 28.14 Fourier Series

… ►The coefficients satisfy ►

28.14.4

{qc_{2m+2}-\left(a-(\nu+2m)^{2}\right)c_{2m}+qc_{2m-2}=0},

a=\lambda_{\nu}\left(q\right),c_{2m}=c_{2m}^{\nu}(q)

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter, $a$ : parameter and $c_{2m}(q)$ : coefficients
Referenced by:: item (d), item (d)
Permalink:: http://dlmf.nist.gov/28.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

… ►

28.14.7

c_{-2m}^{-\nu}(q)=c_{2m}^{\nu}(q),

ⓘ

Symbols:: $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
Permalink:: http://dlmf.nist.gov/28.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

►

28.14.8

c_{2m}^{\nu}(-q)=(-1)^{m}c_{2m}^{\nu}(q).

ⓘ

Symbols:: $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
Permalink:: http://dlmf.nist.gov/28.14.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

… ►

c_{0}^{\nu}(0)=1,

…

8: 15.2 Definitions and Analytical Properties

… ►In general,

F\left(a,b;c;z\right)

does not exist when

c=0,-1,-2,\dots

. … ►For all values of

c

… ►

(c)

Diverges when $\Re\left(c-a-b\right)\leq-1$ .

… ►The principal branch of

\mathbf{F}\left(a,b;c;z\right)

is an entire function of

a

b

, and

c

. …The same properties hold for

F\left(a,b;c;z\right)

, except that as a function of

c

F\left(a,b;c;z\right)

in general has poles at

c=0,-1,-2,\dots

. …

9: 26.11 Integer Partitions: Compositions

… ►

c\left(n\right)

denotes the number of compositions of

n

, and

c_{m}\left(n\right)

is the number of compositions into exactly

m

parts.

c\left(\in\!T,n\right)

is the number of compositions of

n

with no 1’s, where again

T=\{2,3,4,\ldots\}

. … ►

26.11.1

c\left(0\right)=c\left(\in\!T,0\right)=1.

ⓘ

Symbols:: $\in$ : element of, $c\left(\NVar{n}\right)$ : number of compositions of $n$ , $c\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : restricted number of compositions of $n$ and $T$ : set
Permalink:: http://dlmf.nist.gov/26.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

… ►

26.11.2

c_{m}\left(0\right)=\delta_{0,m},

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $c_{\NVar{m}}\left(\NVar{n}\right)$ : number of compositions of $n$ into exactly $m$ parts and $m$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

►

26.11.3

c_{m}\left(n\right)=\genfrac{(}{)}{0.0pt}{}{n-1}{m-1},

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $c_{\NVar{m}}\left(\NVar{n}\right)$ : number of compositions of $n$ into exactly $m$ parts, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

…

10: 26.5 Lattice Paths: Catalan Numbers

… ►

C\left(n\right)

is the Catalan number. …(Sixty-six equivalent definitions of

C\left(n\right)

are given in Stanley (1999, pp. 219–229).) … ►

26.5.3

C\left(n+1\right)=\sum_{k=0}^{n}C\left(k\right)C\left(n-k\right),

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iii), §26.5 and Ch.26

►

26.5.4

C\left(n+1\right)=\frac{2(2n+1)}{n+2}C\left(n\right),

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iii), §26.5 and Ch.26

… ►

26.5.7

\lim_{n\to\infty}\frac{C\left(n+1\right)}{C\left(n\right)}=4.

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iv), §26.5 and Ch.26

.梅西和c罗第一次世界杯『网址:mxsty.cc』.时世界杯赛程-m6q3s2-qmymcyumu.com

1—10 of 394 matching pages

1: 19.2 Definitions

§19.2(iv) A Related Function: R C ⁡ ( x , y )

2: 15.5 Derivatives and Contiguous Functions

3: 4.42 Solution of Triangles

4: 15.13 Zeros

5: 15.1 Special Notation

6: 15.7 Continued Fractions

7: 28.14 Fourier Series

8: 15.2 Definitions and Analytical Properties

9: 26.11 Integer Partitions: Compositions

10: 26.5 Lattice Paths: Catalan Numbers

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