DLMF: Untitled Document

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Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\operatorname{Si}\left(x\right)$ , $-x^{-2}\operatorname{Cin}\left(x\right)$ , $x^{-1}\operatorname{Ein}\left(x\right)$ , $-x^{-1}\operatorname{Ein}\left(-x\right)$ , $x=0(.01)0.5$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=0.5(.01)2$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x=2(.1)10$ ; $x\mathrm{f}\left(x\right)$ , $x^{2}\mathrm{g}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x^{-1}=0(.005)0.1$ ; $\operatorname{Si}\left(\pi x\right)$ , $\operatorname{Cin}\left(\pi x\right)$ , $x=0(.1)10$ . Accuracy varies but is within the range 8S–11S.

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Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

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Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $ze^{z}E_{1}\left(z\right)$ , $x=-19(1)20$ , $y=0(1)20$ , 6D; $e^{z}E_{1}\left(z\right)$ , $x=-4(.5)-2$ , $y=0(.2)1$ , 6D; $E_{1}\left(z\right)+\ln z$ , $x=-2(.5)2.5$ , $y=0(.2)1$ , 6D.

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Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$ , $\pm x=0.5,1,3,5,10,15,20,50,100$ , $y=0(.5)1(1)5(5)30,50,100$ , 8S.

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Blanch and Rhodes (1955) includes $\mathit{Be}_{n}(t)$ , $\mathit{Bo}_{n}(t)$ , $t=\tfrac{1}{2}\sqrt{q}$ , $n=0(1)15$ ; 8D. The range of $t$ is 0 to 0.1, with step sizes ranging from 0.002 down to 0.00025. Notation: $\mathit{Be}_{n}(t)=a_{n}\left(q\right)+2q-(4n+2)\sqrt{q}$ , $\mathit{Bo}_{n}(t)=b_{n}\left(q\right)+2q-(4n-2)\sqrt{q}$ .

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Ince (1932) includes eigenvalues $a_{n}$ , $b_{n}$ , and Fourier coefficients for $n=0$ or $1(1)6$ , $q=0(1)10(2)20(4)40$ ; 7D. Also $\operatorname{ce}_{n}\left(x,q\right)$ , $\operatorname{se}_{n}\left(x,q\right)$ for $q=0(1)10$ , $x=1(1)90$ , corresponding to the eigenvalues in the tables; 5D. Notation: $a_{n}=\mathit{be}_{n}-2q$ , $b_{n}=\mathit{bo}_{n}-2q$ .

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Stratton et al. (1941) includes $b_{n}$ , $b_{n}^{\prime}$ , and the corresponding Fourier coefficients for $\mathrm{Se}_{n}(c,x)$ and $\mathrm{So}_{n}(c,x)$ for $n=0$ or $1(1)4$ , $c=0(.1~{}\textrm{or}~{}.2)4.5$ . Precision is mostly 5S. Notation: $c=2\sqrt{q}$ , $b_{n}=a_{n}+2q$ , $b^{\prime}_{n}=b_{n}+2q$ , and for $\mathrm{Se}_{n}(c,x)$ , $\mathrm{So}_{n}(c,x)$ see §28.1.

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Ince (1932) includes the first zero for $\operatorname{ce}_{n}$ , $\operatorname{se}_{n}$ for $n=2(1)5$ or $6$ , $q=0(1)10(2)40$ ; 4D. This reference also gives zeros of the first derivatives, together with expansions for small $q$ .

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Zhang and Jin (1996, pp. 533–535) includes the zeros (in degrees) of $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n}\left(x,10\right)$ for $n=1(1)10$ , and the first 5 zeros of ${\operatorname{Mc}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ , ${\operatorname{Ms}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ for $n=0$ or $1(1)8$ , $j=1,2$ . Precision is mostly 9S.

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… ►Also let

\xi=2\sqrt{h}\cos x

and

D_{m}\left(\xi\right)=e^{-\ifrac{\xi^{2}}{4}}\mathit{He}_{m}\left(\xi\right)

(§18.3). … ►

\sigma_{m}\sim 1+\dfrac{s}{2^{3}h}+\dfrac{4s^{2}+3}{2^{7}h^{2}}+\dfrac{19s^{3}% +59s}{2^{11}h^{3}}+\cdots,

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28.8.11

P_{m}(x)\sim 1+\dfrac{s}{2^{3}h{\cos}^{2}x}+\dfrac{1}{h^{2}}\left(\dfrac{s^{4}% +86s^{2}+105}{2^{11}{\cos}^{4}x}-\dfrac{s^{4}+22s^{2}+57}{2^{11}{\cos}^{2}x}% \right)+\cdots,

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Defines:: $P_{m}(x)$ (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, $\cos\NVar{z}$ : cosine function, $m$ : integer, $h$ : parameter and $x$ : real variable
A&S Ref:: 20.9.14 (in different form)
Permalink:: http://dlmf.nist.gov/28.8.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(iii), §28.8 and Ch.28

… ►The approximations are expressed in terms of Whittaker functions

W_{\kappa,\mu}\left(z\right)

and

M_{\kappa,\mu}\left(z\right)

with

\mu=\tfrac{1}{4}

; compare §2.8(vi). … ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions

\operatorname{me}_{\nu}\left(z,q\right)

(§28.12(ii)) and modified Mathieu functions

{\operatorname{M}^{(j)}_{\nu}}\left(z,h\right)

(§28.20(iii)). …

… ►with

b_{0}=1

and all allowable choices of

E

,

p

,

s

, and

b_{j}

. … ►Let

E_{{\rm min}}\leq E\leq E_{{\rm max}}

with

E_{{\rm min}}<0

and

E_{{\rm max}}>0

. …The integers

p

,

E_{{\rm min}}

, and

E_{{\rm max}}

are characteristics of the machine. … ►

N_{{\rm min}}\leq x\leq N_{{\rm max}}

, and …Then rounding by chopping or rounding down of

x

gives

x_{-}

, with maximum relative error

\epsilon_{M}

. …

… ►

20.7.6

{\theta_{4}}^{2}\left(0,q\right)\theta_{1}\left(w+z,q\right)\theta_{1}\left(w-% z,q\right)={\theta_{3}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right)-{% \theta_{2}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right),

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Referenced by:: §20.7(ii)
Permalink:: http://dlmf.nist.gov/20.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(ii), §20.7 and Ch.20

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20.7.7

{\theta_{4}}^{2}\left(0,q\right)\theta_{2}\left(w+z,q\right)\theta_{2}\left(w-% z,q\right)={\theta_{4}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right)-{% \theta_{1}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right),

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(ii), §20.7 and Ch.20

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20.7.8

{\theta_{4}}^{2}\left(0,q\right)\theta_{3}\left(w+z,q\right)\theta_{3}\left(w-% z,q\right)={\theta_{4}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right)-{% \theta_{1}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right),

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(ii), §20.7 and Ch.20

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20.7.9

{\theta_{4}}^{2}\left(0,q\right)\theta_{4}\left(w+z,q\right)\theta_{4}\left(w-% z,q\right)={\theta_{3}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right)-{% \theta_{2}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right).

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Referenced by:: §20.7(ii)
Permalink:: http://dlmf.nist.gov/20.7.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(ii), §20.7 and Ch.20

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20.7.10

\theta_{1}\left(2z,q\right)=2\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z% ,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q\right)}{\theta_{2}\left% (0,q\right)\theta_{3}\left(0,q\right)\theta_{4}\left(0,q\right)}.

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(iii), §20.7 and Ch.20

…

… ►For the generalized hypergeometric function

{{}_{3}F_{2}}

see (16.2.1). … ►Define operators

K_{0}

and

K_{1}

acting on symmetric Laurent polynomials by

K_{0}=L

(

L

given by (18.28.6_2)) and

(K_{1}f)(z)=(z+z^{-1})f(z)

. …commutes with

K_{0},K_{1},K_{2}

, that is

K_{j}Q=QK_{j}

, and satisfies …where

Q_{0}

is a constant with explicit expression in terms of

e_{1},e_{2},e_{3},e_{4}

and

q

given in Koornwinder (2007a, (2.8)). ►The abstract associative algebra with generators

K_{0},K_{1},K_{2}

and relations (18.38.4), (18.38.6) and with the constants

B,C_{0},C_{1},D_{0},D_{1}

in (18.38.6) not yet specified, is called the Zhedanov algebra or Askey–Wilson algebra AW(3). …

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26.12.9

\left(\prod_{h=1}^{r}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right)^{2};

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Symbols:: $r$ : positive integer, $s$ : positive integer, $t$ : positive integer, $h$ : positive integer and $j$ : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §26.12(i), §26.12 and Ch.26

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26.12.10

\left(\prod_{h=1}^{r}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right)\*\left(\prod_% {h=1}^{r+1}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right);

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Symbols:: $r$ : positive integer, $s$ : positive integer, $t$ : positive integer, $h$ : positive integer and $j$ : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §26.12(i), §26.12 and Ch.26

… ►

26.12.11

\left(\prod_{h=1}^{r+1}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right)\*\left(% \prod_{h=1}^{r}\prod_{j=1}^{s+1}\frac{h+j+t-1}{h+j-1}\right).

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Symbols:: $r$ : positive integer, $s$ : positive integer, $t$ : positive integer, $h$ : positive integer and $j$ : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §26.12(i), §26.12 and Ch.26

… ►The notation

\sum_{\pi\subseteq B(r,s,t)}

denotes the sum over all plane partitions contained in

B(r,s,t)

, and

|\pi|

denotes the number of elements in

\pi

. … ►where

\sigma_{2}(j)

is the sum of the squares of the divisors of

j

. …

… ►

\mathfrak{S}_{n}

denotes the set of permutations of

\{1,2,\ldots,n\}

.

\sigma\in\mathfrak{S}_{n}

is a one-to-one and onto mapping from

\{1,2,\ldots,n\}

to itself. … ►An element of

\mathfrak{S}_{n}

with

a_{1}

fixed points,

a_{2}

cycles of length

2,\ldots,a_{n}

cycles of length

n

, where

n=a_{1}+2a_{2}+\cdots+na_{n}

, is said to have cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

. The number of elements of

\mathfrak{S}_{n}

with cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

is given by (26.4.7). … ►A permutation with cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

can be written as a product of

a_{2}+2a_{3}+\cdots+(n-1)a_{n}=n-(a_{1}+a_{2}+\cdots+a_{n})

transpositions, and no fewer. …

… ►where

b_{n}=a_{n}

, … ►With

b_{k}

as in (1.11.1)–(1.11.3) let

c_{n}=a_{n}

and … ►where

z_{1},z_{2},\dots,z_{n}

are the zeros of

f(z)

. … ►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})

. … ►Let …

… ►

See accompanying text — Figure 30.7.4: Eigenvalues $\lambda^{10}_{n}\left(\gamma^{2}\right)$ , $n=10,11,12,13$ , $-50\leq\gamma^{2}\leq 150$ . Magnify

… ►

►

…

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31—40 of 785 matching pages

31: 6.19 Tables

32: 28.35 Tables

33: 28.8 Asymptotic Expansions for Large $q$

34: 3.1 Arithmetics and Error Measures

35: 20.7 Identities

36: 18.38 Mathematical Applications

37: 26.12 Plane Partitions

38: 26.13 Permutations: Cycle Notation

39: 1.11 Zeros of Polynomials

40: 30.7 Graphics

.94世界杯马拉多纳_『wn4.com_』2014年世界杯德国vs阿根廷_w6n2c9o_2022年11月30日9时23分57秒_yo6cq20ae

31—40 of 785 matching pages

31: 6.19 Tables

32: 28.35 Tables

33: 28.8 Asymptotic Expansions for Large q

34: 3.1 Arithmetics and Error Measures

35: 20.7 Identities

36: 18.38 Mathematical Applications

37: 26.12 Plane Partitions

38: 26.13 Permutations: Cycle Notation

39: 1.11 Zeros of Polynomials

40: 30.7 Graphics

33: 28.8 Asymptotic Expansions for Large $q$