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21—30 of 673 matching pages

21: 16.7 Relations to Other Functions

… ►For

\mathit{3j}

\mathit{6j}

\mathit{9j}

symbols see Chapter 34. Further representations of special functions in terms of

{{}_{p}F_{q}}

functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of

{{}_{q+1}F_{q}}

functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).

22: 21.5 Modular Transformations

… ►Let

\mathbf{A}

\mathbf{B}

\mathbf{C}

, and

\mathbf{D}

g\times g

matrices with integer elements such that …Here

\xi(\boldsymbol{{\Gamma}})

is an eighth root of unity, that is,

(\xi(\boldsymbol{{\Gamma}}))^{8}=1

. … ►(

\mathbf{A}

invertible with integer elements.) …(

\mathbf{B}

symmetric with integer elements.) …For a

g\times g

matrix

{\bf A}

we define

\operatorname{diag}\mathbf{A}

, as a column vector with the diagonal entries as elements. …

23: 3.4 Differentiation

… ►The

B_{k}^{n}

are the differentiated Lagrangian interpolation coefficients: …where

A_{k}^{n}

is as in (3.3.10). … ►

B_{-2}^{6}=\tfrac{1}{60}(9-9t-30t^{2}+20t^{3}+5t^{4}-3t^{5}),

… ►where

C

is a simple closed contour described in the positive rotational sense such that

C

and its interior lie in the domain of analyticity of

f

, and

x_{0}

is interior to

C

. Taking

C

to be a circle of radius

r

centered at

x_{0}

, we obtain …

24: 1.2 Elementary Algebra

… ►For matrices

\mathbf{A}

\mathbf{B}

and

\mathbf{C}

of the same dimensions, … ►Multiplication of an

m\times n

matrix

\mathbf{A}

and an

m^{\prime}\times n^{\prime}

matrix

\mathbf{B}

, giving the

m\times n^{\prime}

matrix

\mathbf{C}=\mathbf{A}\mathbf{B}

is defined iff

n=m^{\prime}

. … ►If

\det(\mathbf{A})=0

then

\mathbf{A}\mathbf{B}=\mathbf{A}\mathbf{C}

does not imply that

\mathbf{B}=\mathbf{C}

; if

\det(\mathbf{A})\neq 0

, then

\mathbf{B}=\mathbf{C}

, as both sides may be multiplied by

{\mathbf{A}}^{-1}

. … ►If

\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}

the matrices

\mathbf{A}

and

\mathbf{B}

are said to commute. The difference between

\mathbf{A}\mathbf{B}

and

\mathbf{B}\mathbf{A}

is the commutator denoted as …

25: 26.10 Integer Partitions: Other Restrictions

… ►The set

\{n\geq 1\>|\>n\equiv\pm j\ \pmod{k}\}

is denoted by

A_{j,k}

. … ►Note that

p\left(\mathcal{D}^{\prime}3,n\right)\leq p\left(\mathcal{D}3,n\right)

, with strict inequality for

n\geq 9

. It is known that for

k>3

p\left(\mathcal{D}k,n\right)\geq p\left(\in\!A_{1,k+3},n\right)

, with strict inequality for

n

sufficiently large, provided that

k=2^{m}-1,m=3,4,5

, or

k\geq 32

; see Yee (2004). … ►where

I_{1}\left(x\right)

is the modified Bessel function (§10.25(ii)), and …The quantity

A_{k}(n)

is real-valued. …

26: 34.1 Special Notation

… ► ►►

$2j_{1},2j_{2},2j_{3},2l_{1},2l_{2},2l_{3}$	nonnegative integers.
…

►The main functions treated in this chapter are the Wigner

\mathit{3j},\mathit{6j},\mathit{9j}

symbols, respectively, … ►For other notations for

\mathit{3j}

\mathit{6j}

\mathit{9j}

symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).

27: 26.3 Lattice Paths: Binomial Coefficients

… ►

Table 26.3.1: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

► ►►►►►

$m$	$n$
$m$	0	1	2	3	4	5	6	7	8	9	10
…
8	1	8	28	56	70	56	28	8	1
9	1	9	36	84	126	126	84	36	9	1
…

► ►

Table 26.3.2: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m+n}{m}

for lattice paths.

► ►►►►

$m$	$n$
$m$	…
1	1	2	3	4	5	6	7	8	9
…
8	1	9	45	165	495	1287	3003	6435	12870

► …

28: 4.42 Solution of Triangles

… ►

4.42.4

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C},

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function, $a$ : length, $b$ : base, $c$ : length, $A$ : angle, $B$ : angle and $C$ : angle
A&S Ref:: 4.3.148
Permalink:: http://dlmf.nist.gov/4.42.E4
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

… ►

4.42.9

\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c},

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function, $A$ : angle, $B$ : angle, $C$ : angle, $a$ : arc length, $b$ : arc length and $c$ : arc length
A&S Ref:: 4.3.149
Permalink:: http://dlmf.nist.gov/4.42.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §4.42(iii), §4.42 and Ch.4

►

4.42.10

\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $A$ : angle, $B$ : angle, $a$ : arc length, $b$ : arc length and $c$ : arc length
Permalink:: http://dlmf.nist.gov/4.42.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §4.42(iii), §4.42 and Ch.4

… ►

4.42.12

\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, $B$ : angle, $C$ : angle and $a$ : arc length
A&S Ref:: 4.3.149
Permalink:: http://dlmf.nist.gov/4.42.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §4.42(iii), §4.42 and Ch.4

…

29: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

…

30: 27.2 Functions

… ►It is the special case

k=2

of the function

d_{k}\left(n\right)

that counts the number of ways of expressing

n

as the product of

k

factors, with the order of factors taken into account. …Note that

\sigma_{0}\left(n\right)=d\left(n\right)

. … ►Table 27.2.2 tabulates the Euler totient function

\phi\left(n\right)

, the divisor function

d\left(n\right)

(

=\sigma_{0}(n)

), and the sum of the divisors

\sigma(n)

(

=\sigma_{1}(n)

), for

n=1(1)52

. ►

Table 27.2.1: Primes.

► ►►►

$n$	$p_{n}$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
…
9	23	67	109	167	227	277	347	401	461	523
…

► ►

Table 27.2.2: Functions related to division.

► ►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
10	4	4	18	23	22	2	24	36	12	9	91	49	42	3	57
…

►

21—30 of 673 matching pages

21: 16.7 Relations to Other Functions

22: 21.5 Modular Transformations

23: 3.4 Differentiation

24: 1.2 Elementary Algebra

25: 26.10 Integer Partitions: Other Restrictions

26: 34.1 Special Notation

27: 26.3 Lattice Paths: Binomial Coefficients

28: 4.42 Solution of Triangles

29: 8 Incomplete Gamma and Related Functions

Chapter 8 Incomplete Gamma and Related Functions

30: 27.2 Functions

29: 8 Incomplete Gamma and Related
Functions