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31—40 of 783 matching pages

31: 11.14 Tables

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Abramowitz and Stegun (1964, Chapter 12) tabulates $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , and $I_{n}\left(x\right)-\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.1)5$ , $x^{-1}=0(.01)0.2$ to 6D or 7D.

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Agrest et al. (1982) tabulates $\mathbf{H}_{n}\left(x\right)$ and $e^{-x}\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.001)5(.005)15(.01)100$ to 11D.

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Abramowitz and Stegun (1964, Chapter 12) tabulates $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t$ and $(2/\pi)\int_{x}^{\infty}t^{-1}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.1)5$ to 5D or 7D; $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , and $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ for $x^{-1}=0(.01)0.2$ to 6D.

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Agrest et al. (1982) tabulates $\int_{0}^{x}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ and $e^{-x}\int_{0}^{x}\mathbf{L}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.001)5(.005)15(.01)100$ to 11D.

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Agrest and Maksimov (1971, Chapter 11) defines incomplete Struve, Anger, and Weber functions and includes tables of an incomplete Struve function $\mathbf{H}_{n}\left(x,\alpha\right)$ for $n=0,1$ , $x=0(.2)10$ , and $\alpha=0(.2)1.4,\tfrac{1}{2}\pi$ , together with surface plots.

32: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

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§17.7(i) ${{}_{2}\phi_{2}}$ Functions

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$q$ -Analog of Bailey’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

… ►

$q$ -Analog of Gauss’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

… ►

$q$ -Analog of Dixon’s ${{}_{3}F_{2}}\left(1\right)$ Sum

… ►where

m_{1},m_{2},\dots,m_{r}

are arbitrary nonnegative integers. …

33: 28.35 Tables

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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 $q$ .

… ►For other tables prior to 1961 see Fletcher et al. (1962, §2.2) and Lebedev and Fedorova (1960, Chapter 11).

34: Bibliography G

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W. Gautschi (1966) Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

ⓘ

External Links:: ISSN 0001-0782, Document
Referenced by:: §33.26(ii).
Encodings:: BibTeX

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W. Gautschi (1959a) Exponential integral

\int_{1}^{\infty}e^{-xt}t^{-n}dt

for large values of

n

. J. Res. Nat. Bur. Standards 62, pp. 123–125.

ⓘ

External Links:: MathReview (W. Wasow), ZentralBlatt
Referenced by:: §8.11(iii), §8.20(ii).
Encodings:: BibTeX

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D. Goss (1978) Von Staudt for

{\mathbf{F}}_{q}[{T}]

. Duke Math. J. 45 (4), pp. 885–910.

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External Links:: ISSN 0012-7094, Document, MathReview (Neal Koblitz), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:: FullText, MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

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V. I. Gromak (1975) Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

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Notes:: In Russian. English translation: Differential Equations 11(11), pp. 285–287
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.7(iii), §32.7(vi).
Encodings:: BibTeX

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35: 18.8 Differential Equations

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Table 18.8.1: Classical OP’s: differential equations

A(x)f^{\prime\prime}(x)+B(x)f^{\prime}(x)+C(x)f(x)+\lambda_{n}f(x)=0

► ►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
…
11	${\mathrm{e}}^{-n^{-1}x}x^{\ell+1}L^{(2\ell+1)}_{n-\ell-1}\left(2n^{-1}x\right)$	$1$	$0$	$\frac{2}{x}-\frac{\ell(\ell+1)}{x^{2}}$	$-\frac{1}{n^{2}}$
12	$H_{n}\left(x\right)$	$1$	$-2x$	$0$	$2n$
…
14	$\mathit{He}_{n}\left(x\right)$	$1$	$-x$	$0$	$n$

► ►Item 11 of Table 18.8.1 yields (18.39.36) for

Z=1

36: 26.13 Permutations: Cycle Notation

… ►

\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. …

37: Publications

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D. W. Lozier (2003) The NIST Digital Library of Mathematical Functions Project, Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management, Vol. 38, Nos. 1–3, pp. 105–119.

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B. R. Miller and A. Youssef (2003) Technical Aspects of the Digital Library of Mathematical Functions, Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management, Vol. 38, Nos. 1–3, pp. 121–136.

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B. V. Saunders and Q. Wang (2005) Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71.

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Q. Wang and B. V. Saunders (2005) Web-Based 3D Visualization in a Digital Library of Mathematical Functions, Proceedings of the Web3D Symposium, Bangor, UK, March 29–April 1, 2005.

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B. V. Saunders and Q. Wang (2006) From B-Spline Mesh Generation to Effective Visualizations for the NIST Digital Library of Mathematical Functions, in Curve and Surface Design, Proceedings of the Sixth International Conference on Curves and Surfaces, Avignon, France June 29–July 5, 2006, pp. 235–243.

…

38: 3.1 Arithmetics and Error Measures

… ►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}

. …

39: 3.8 Nonlinear Equations

… ►The choice of

x_{0}

here is critical. … ►Let

x_{0}

and

x_{1}

be such that

f_{0}=f(x_{0})

and

f_{1}=f(x_{1})

have opposite signs. …We continue with

x_{2}

and either

x_{0}

x_{1}

, depending which of

f_{0}

and

f_{1}

is of opposite sign to

f(x_{2})

, and so on. … ►Whether or not

f_{0}

and

f_{1}

have opposite signs,

x_{2}

is computed as in (3.8.6). … ►We construct sequences

q_{j}

and

r_{j}

j=n+1,n,\dots,0

, from

q_{n+1}=r_{n+1}=0

q_{n}=r_{n}=a_{n}

, and for

j\leq n-1

, …

40: 1.11 Zeros of Polynomials

… ►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 …

.联通盒子怎么看世界杯_『wn4.com_』世界杯赌球平了怎么算_w6n2c9o_2022年11月29日5时31分38秒_m2c8qeq0s

31—40 of 783 matching pages

31: 11.14 Tables

32: 17.7 Special Cases of Higher ϕ s r Functions

§17.7(i) ϕ 2 2 Functions

q -Analog of Bailey’s F 1 2 ⁡ ( − 1 ) Sum

q -Analog of Gauss’s F 1 2 ⁡ ( − 1 ) Sum

q -Analog of Dixon’s F 2 3 ⁡ ( 1 ) Sum

33: 28.35 Tables

34: Bibliography G

35: 18.8 Differential Equations

36: 26.13 Permutations: Cycle Notation

37: Publications

38: 3.1 Arithmetics and Error Measures

39: 3.8 Nonlinear Equations

40: 1.11 Zeros of Polynomials

32: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

§17.7(i) ${{}_{2}\phi_{2}}$ Functions

$q$ -Analog of Bailey’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

$q$ -Analog of Gauss’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

$q$ -Analog of Dixon’s ${{}_{3}F_{2}}\left(1\right)$ Sum