lemniscate constants

SearchAdvancedHelp

(0.000 seconds)

5 matching pages

1: 19.20 Special Cases

… ►

$$R_F(0,0,z)=\mathrm{\infty }.$$

►The first lemniscate constant is given by … ►

19.20.21 $$R_D(x,x,z)=\frac{3}{z-x}\left(R_C(z,x)-\frac{1}{\sqrt{z}}\right),$$ $x\ne z$, $xz\ne 0$.

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.20.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

►The second lemniscate constant is given by …

2: Bibliography T

… ►

The lemniscate constants. Comm. ACM 18 (1), pp. 14–19.

ⓘ

Notes:

Collection of articles honoring Alston S. Householder. For corrigendum see same journal v. 18 (1975), no. 8, p. 462

External Links:

ISSN 0001-0782, Document, MathReview (L. Fox), ZentralBlatt

Referenced by:

§19.20(i), §19.20(iv), §19.36(iii).

Encodings:

BibTeX

…

3: 19.3 Graphics

… ►

►

►

Figure 19.3.6: $\mathrm{\Pi }(\varphi ,2,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 3$, . …Its value tends to $-\mathrm{\infty }$ as $k^2\to 1+$ by (19.6.6), and to the negative of the second lemniscate constant (see (19.20.22)) as $k^2(={\csc }^2\varphi )\to 2-$. Magnify 3D Help

…

4: 19.21 Connection Formulas

… ►The case $z=1$ shows that the product of the two lemniscate constants, (19.20.2) and (19.20.22), is $\pi /4$. …

5: 22.18 Mathematical Applications

… ►

Lemniscate

►In polar coordinates, $x=r\cos \varphi$, $y=r\sin \varphi$, the lemniscate is given by $r^2=\cos \left(2\varphi \right)$, $0\le \varphi \le 2\pi$. … ►

22.18.7 $$a{x}^2{y}^2+b(x^{2}y+x{y}^2)+c(x^2+y^2)+2dxy+e(x+y)+f=0,$$

ⓘ

Symbols:

$x$: real, $y$: real, $c$: real constant, $d$: real constant, $e$: real constant, $f$: real constant, $a$: real constant and $b$: real constant

Permalink:

http://dlmf.nist.gov/22.18.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.18(iii), §22.18 and Ch.22

►in which $a,b,c,d,e,f$ are real constants, can be achieved in terms of single-valued functions. …