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21—30 of 684 matching pages

21: 27.15 Chinese Remainder Theorem

… ►Their product

m

has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

22: William P. Reinhardt

… ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

… ►Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices

2,3,4

. … ►See Lawden (1989, pp. 19–20). … ►

20.7.34

\frac{\theta_{1}\left(z,q^{2}\right)\theta_{3}\left(z,q^{2}\right)}{\theta_{1}% \left(z,iq\right)}=\frac{\theta_{2}\left(z,q^{2}\right)\theta_{4}\left(z,q^{2}% \right)}{\theta_{2}\left(z,iq\right)}=i^{-1/4}\sqrt{\frac{\theta_{2}\left(0,q^% {2}\right)\theta_{4}\left(0,q^{2}\right)}{2}}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $q$ : nome
Referenced by:: §20.7(iv), Erratum (V1.0.5) for Chapters 8, 20, 36
Permalink:: http://dlmf.nist.gov/20.7.E34
Encodings:: pMML, png, TeX
Addition (effective with 1.0.5):: This equation has been added. See Walker (2012).
See also:: Annotations for §20.7(ix), §20.7 and Ch.20

…

25: 36.4 Bifurcation Sets

… ►These are real solutions

t_{j}(\mathbf{x})

1\leq j\leq j_{\max}(\mathbf{x})\leq K+1

, of … ►Swallowtail self-intersection line: … ►Swallowtail cusp lines (ribs): … ►Elliptic umbilic cusp lines (ribs): … ►Hyperbolic umbilic cusp line (rib): …

26: 10.75 Tables

… ►

•

Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

… ►

•

Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

… ►

•

Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_{n}\left(z\right)$ and $K_{n}'\left(z\right)$ , for $n=2(1)20$ , 9S.

… ►

•

Zhang and Jin (1996, p. 322) tabulates $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , $x=0(1)20$ , 7S.

… ►

•

Zhang and Jin (1996, p. 323) tabulates the first $20$ real zeros of $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , 8D.

27: Staff

… ►

William P. Reinhardt, University of Washington, Chaps. 20, 22, 23

… ►

Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23

… ►

William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23

… ►

Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23

…

28: 22.3 Graphics

… ►

§22.3(i) Real Variables: Line Graphs

►Line graphs of the functions

\operatorname{sn}\left(x,k\right)

\operatorname{cn}\left(x,k\right)

\operatorname{dn}\left(x,k\right)

\operatorname{cd}\left(x,k\right)

\operatorname{sd}\left(x,k\right)

\operatorname{nd}\left(x,k\right)

\operatorname{dc}\left(x,k\right)

\operatorname{nc}\left(x,k\right)

\operatorname{sc}\left(x,k\right)

\operatorname{ns}\left(x,k\right)

\operatorname{ds}\left(x,k\right)

, and

\operatorname{cs}\left(x,k\right)

for representative values of real

x

and real

k

illustrating the near trigonometric (

k=0

), and near hyperbolic (

k=1

) limits. … ►

§22.3(ii) Real Variables: Surfaces

►

\operatorname{sn}\left(x,k\right)

\operatorname{cn}\left(x,k\right)

, and

\operatorname{dn}\left(x,k\right)

as functions of real arguments

x

and

k

. … ►

§22.3(iii) Complex $z$ ; Real $k$

…

29: 26.3 Lattice Paths: Binomial Coefficients

… ►The number of lattice paths from

(0,0)

(m,n)

m\leq n

, that stay on or above the line

y=x

\genfrac{(}{)}{0.0pt}{}{m+n}{m}-\genfrac{(}{)}{0.0pt}{}{m+n}{m-1}.

… ►

Table 26.3.1: Binomial coefficients

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

► ►►►

$m$	$n$
$m$	…
6	1	6	15	20	15	6	1
…

► ►

Table 26.3.2: Binomial coefficients

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

for lattice paths.

► ►►►

$m$	$n$
$m$	…
3	1	4	10	20	35	56	84	120	165
…

► … ►

26.3.3

\sum_{n=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{n}x^{n}=(1+x)^{m},

m=0,1,\dotsc

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.1
Permalink:: http://dlmf.nist.gov/26.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §26.3(ii), §26.3 and Ch.26

►

26.3.4

\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt}{}{m+n}{m}x^{m}=\frac{1}{(1-x)^{n+1}},

|x|<1

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.1 (in slightly different form)
Permalink:: http://dlmf.nist.gov/26.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.3(ii), §26.3 and Ch.26

…

30: 4.15 Graphics

… ►

§4.15(i) Real Arguments

… ►Figure 4.15.7 illustrates the conformal mapping of the strip

-\tfrac{1}{2}\pi<\Re z<\tfrac{1}{2}\pi

onto the whole

w

-plane cut along the real axis from

-\infty

-1

and

1

\infty

, where

w=\sin z

and

z=\operatorname{arcsin}w

(principal value). …Lines parallel to the real axis in the

z

-plane map onto ellipses in the

w

-plane with foci at

w=\pm 1

, and lines parallel to the imaginary axis in the

z

-plane map onto rectangular hyperbolas confocal with the ellipses. In the labeling of corresponding points

r

is a real parameter that can lie anywhere in the interval

(0,\infty)

. … ►

See accompanying text — Figure 4.15.13: $\operatorname{arccsc}\left(x+iy\right)$ (principal value). There is a branch cut along the real axis from $-1$ to $1$ . Magnify 3D Help

…

along%20the%20real%20line

21—30 of 684 matching pages

21: 27.15 Chinese Remainder Theorem

22: William P. Reinhardt

23: 4.3 Graphics

§4.3(i) Real Arguments

24: 20.7 Identities

25: 36.4 Bifurcation Sets

26: 10.75 Tables

27: Staff

28: 22.3 Graphics

§22.3(i) Real Variables: Line Graphs

§22.3(ii) Real Variables: Surfaces

§22.3(iii) Complex $z$ ; Real $k$

29: 26.3 Lattice Paths: Binomial Coefficients

30: 4.15 Graphics

§4.15(i) Real Arguments

along%20the%20real%20line

21—30 of 684 matching pages

21: 27.15 Chinese Remainder Theorem

22: William P. Reinhardt

23: 4.3 Graphics

§4.3(i) Real Arguments

24: 20.7 Identities

25: 36.4 Bifurcation Sets

26: 10.75 Tables

27: Staff

28: 22.3 Graphics

§22.3(i) Real Variables: Line Graphs

§22.3(ii) Real Variables: Surfaces

§22.3(iii) Complex z ; Real k

29: 26.3 Lattice Paths: Binomial Coefficients

30: 4.15 Graphics

§4.15(i) Real Arguments

§22.3(iii) Complex $z$ ; Real $k$