DLMF: Untitled Document

§22.2 Definitions

►The nome

q

is given in terms of the modulus

k

by … ►

22.2.3

\zeta=\frac{\pi z}{2K\left(k\right)},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/22.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

… ►

22.2.11

\operatorname{pq}\left(z,k\right)=\ifrac{\theta_{p}\left(z\middle|\tau\right)}% {\theta_{q}\left(z\middle|\tau\right)},

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$ : generic Jacobian elliptic function, $q$ : nome, $z$ : complex, $k$ : modulus and $\tau$ : ratio
A&S Ref:: 16.36.3
Referenced by:: §22.2
Permalink:: http://dlmf.nist.gov/22.2.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2, §22.2 and Ch.22

… ►

22.2.12

\tau=\ifrac{\mathrm{i}{K^{\prime}}\left(k\right)}{K\left(k\right)},

ⓘ

Symbols:: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $k$ : modulus and $\tau$ : ratio
Referenced by:: §22.2
Permalink:: http://dlmf.nist.gov/22.2.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2, §22.2 and Ch.22

…

… ►A divisor

d

of

k

is called an induced modulus for

\chi

if … ►where

\chi_{0}

is a character (mod

d

) for some induced modulus

d

for

\chi

, and

\chi_{1}

is the principal character (mod

k

). …

… ►The surface area of an ellipsoid with semiaxes

a, b, c

, and volume

V=4\pi abc/3

is given by … ►and the electric capacity

C=Q/V(0)

is given by … ►Let a homogeneous magnetic ellipsoid with semiaxes

a, b, c

, volume

V=4\pi abc/3

, and susceptibility

\chi

be placed in a previously uniform magnetic field

H

parallel to the principal axis with semiaxis

c

. The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength

H/(1+L_{c}\chi)

, where

L_{c}

is the demagnetizing factor, given in cgs units by … ►

19.33.9

\rho(x,y,z)=f\left(\sqrt{(x^{2}/\alpha^{2})+(y^{2}/\beta^{2})+(z^{2}/\gamma^{2% })}\right),

ⓘ

Symbols:: $\rho(x,y,z)$ : density, $f(t)$ : density, $\alpha$ : positive constant, $\beta$ : positive constant and $\gamma$ : positive constant
Permalink:: http://dlmf.nist.gov/19.33.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(iv), §19.33 and Ch.19

…

… ►

P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright

\omega

function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §4.13, 2nd item, §4.48(iv).
Encodings:: BibTeX

… ►

D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

… ►

J. Lehner (1941) A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.

ⓘ

External Links:: Document, MathReview (R. D. James), ZentralBlatt
Referenced by:: §17.18.
Encodings:: BibTeX

… ►

D. W. Lozier (1993) An underflow-induced graphics failure solved by SLI arithmetic. In IEEE Symposium on Computer Arithmetic, E. E. Swartzlander, M. J. Irwin, and G. A. Jullien (Eds.), Washington, D.C., pp. 10–17.

ⓘ

External Links:: FullText
Referenced by:: §3.1(iv).
Encodings:: BibTeX

…

Chapter 20 Theta Functions

…

… ►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. ►Even though the lengthy calculation is repeated four times, once for each modulus, most of it only uses five-digit integers and is accomplished quickly without overwhelming the machine’s memory. …

… ►

•

The main tables in Abramowitz and Stegun (1964, Chapter 9) give $J_{0}\left(x\right)$ to 15D, $J_{1}\left(x\right)$ , $J_{2}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ to 10D, $Y_{2}\left(x\right)$ to 8D, $x=0(.1)17.5$ ; $Y_{n}\left(x\right)-(2/\pi)J_{n}\left(x\right)\ln x$ , $n=0,1$ , $x=0(.1)2$ , 8D; $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ , $n=3(1)9$ , $x=0(.2)20$ , 5D or 5S; $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ , $n=0(1)20(10)50,100$ , $x=1,2,5,10,50,100$ , 10S; modulus and phase functions $\sqrt{x}M_{n}\left(x\right)$ , $\theta_{n}\left(x\right)-x$ , $n=0,1,2$ , $\ifrac{1}{x}=0(.01)0.1$ , 8D.

►

•

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.

… ►

•

Makinouchi (1966) tabulates all values of $j_{\nu,m}$ and $y_{\nu,m}$ in the interval $(0,100)$ , with at least 29S. These are for $\nu=0(1)5$ , 10, 20; $\nu=\tfrac{3}{2}$ , $\tfrac{5}{2}$ ; $\nu=m/n$ with $m=1(1)n-1$ and $n=3(1)8$ , except for $\nu=\tfrac{1}{2}$ .

… ►

•

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.

… ►

•

The main tables in Abramowitz and Stegun (1964, Chapter 9) give $e^{-x}I_{n}\left(x\right)$ , $e^{x}K_{n}\left(x\right)$ , $n=0,1,2$ , $x=0(.1)10(.2)20$ , 8D–10D or 10S; $\sqrt{x}e^{-x}I_{n}\left(x\right)$ , $(\sqrt{x}/\pi)$ $e^{x}K_{n}\left(x\right)$ , $n=0,1,2$ , $1/x=0(.002)0.05$ ; $K_{0}\left(x\right)+I_{0}\left(x\right)\ln x$ , $x(K_{1}\left(x\right)-I_{1}\left(x\right)\ln x)$ , $x=0(.1)2$ , 8D; $e^{-x}I_{n}\left(x\right)$ , $e^{x}K_{n}\left(x\right)$ , $n=3(1)9$ , $x=0(.2)10(.5)20$ , 5S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20(10)50,100$ , $x=1,2,5,10,50,100$ , 9–10S.

…

… ►

•

Miller (1946) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ for $x=-20(.01)2;$ $\operatorname{log}_{10}\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)/\operatorname{Ai}\left(x\right)$ for $x=0(.1)25(1)75$ ; $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=-10(.1)2.5$ ; $\operatorname{log}_{10}\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)/\operatorname{Bi}\left(x\right)$ for $x=0(.1)10$ ; $M\left(x\right)$ , $N\left(x\right)$ , $\theta\left(x\right)$ , $\phi\left(x\right)$ (respectively $F(x)$ , $G(x)$ , $\chi(x)$ , $\psi(x)$ ) for $x=-80(1)-30(.1)0$ . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

►

•

Fox (1960, Table 3) tabulates $2\pi^{1/2}x^{1/4}\*\exp(\tfrac{2}{3}x^{3/2})\*\operatorname{Ai}\left(x\right)$ , $2\pi^{1/2}x^{-1/4}\*\exp(\tfrac{2}{3}x^{3/2})\*\operatorname{Ai}'\left(x\right)$ , $\pi^{1/2}x^{1/4}\*\exp(-\tfrac{2}{3}x^{3/2})\*\operatorname{Bi}\left(x\right)$ , and $\pi^{1/2}x^{-1/4}\*\exp(-\tfrac{2}{3}x^{3/2})\*\operatorname{Bi}'\left(x\right)$ for $\tfrac{3}{2}x^{-3/2}=0(.001)0.05$ , together with similar auxiliary functions for negative values of $x$ . Precision is 10D.

►

•

Zhang and Jin (1996, p. 337) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

… ►

•

Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

►

•

Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

…

… ►Let

{k^{\prime}}^{2}=1-k^{2}

and

c={\csc}^{2}\phi

with

0\leq\Re\phi\leq\ifrac{\pi}{2}

. … ►then the five nontrivial permutations of

x, y, z

that leave

R_{F}

invariant change

k^{2}

(

=(z-y)/(z-x)

) into

1/k^{2}

,

{k^{\prime}}^{2}

,

1/{k^{\prime}}^{2}

,

-k^{2}/{k^{\prime}}^{2}

,

-{k^{\prime}}^{2}/k^{2}

, and

\sin\phi

(

=\sqrt{(z-x)/z}

) into

k\sin\phi

,

-i\tan\phi

,

-ik^{\prime}\tan\phi

,

(k^{\prime}\sin\phi)/\sqrt{1-k^{2}{\sin}^{2}\phi}

,

-ik\sin\phi/\sqrt{1-k^{2}{\sin}^{2}\phi}

. Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions). … ►Let

r=1/x^{2}

. … ►where we assume

0\leq x^{2}\leq 1

if

x=\operatorname{sn}

,

\operatorname{cn}

, or

\operatorname{cd}

;

x^{2}\geq 1

if

x=\operatorname{ns}

,

\operatorname{nc}

, or

\operatorname{dc}

;

x

real if

x=\operatorname{cs}

or

\operatorname{sc}

;

k^{\prime}\leq x\leq 1

if

x=\operatorname{dn}

;

1\leq x\leq 1/k^{\prime}

if

x=\operatorname{nd}

;

x^{2}\geq{k^{\prime}}^{2}

if

x=\operatorname{ds}

;

0\leq x^{2}\leq 1/{k^{\prime}}^{2}

if

x=\operatorname{sd}

. …

… ►

§22.3(iv) Complex $k$

… ►

See accompanying text — Figure 22.3.24: $\operatorname{sn}\left(x+iy,k\right)$ for $-4\leq x\leq 4$ , $0\leq y\leq 8$ , $k=1+\tfrac{1}{2}i$ . … Magnify 3D Help

►

…

induced%20modulus

1—10 of 579 matching pages

1: 22.2 Definitions

§22.2 Definitions

2: 27.8 Dirichlet Characters

3: 19.33 Triaxial Ellipsoids

4: Bibliography L

5: 20 Theta Functions

Chapter 20 Theta Functions

6: 27.15 Chinese Remainder Theorem

7: 10.75 Tables

8: 9.18 Tables

9: 19.25 Relations to Other Functions

10: 22.3 Graphics

§22.3(iv) Complex $k$

induced%20modulus

1—10 of 579 matching pages

1: 22.2 Definitions

§22.2 Definitions

2: 27.8 Dirichlet Characters

3: 19.33 Triaxial Ellipsoids

4: Bibliography L

5: 20 Theta Functions

Chapter 20 Theta Functions

6: 27.15 Chinese Remainder Theorem

7: 10.75 Tables

8: 9.18 Tables

9: 19.25 Relations to Other Functions

10: 22.3 Graphics

§22.3(iv) Complex k

§22.3(iv) Complex $k$