lemniscate%20constants

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1: 22.18 Mathematical Applications

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

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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:

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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. …

2: 19.20 Special Cases

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

►The first lemniscate constant is given by ►

19.20.2 $${\int }_0^1\frac{dt}{\sqrt{1-t^4}}=R_F(0,1,2)=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{4}\right)\right)}^2}{4{(2\pi )}^{1/2}}=\mathrm{1.31102\hspace{0.33em}87771\hspace{0.33em}46059\hspace{0.33em}90523}\mathrm{\dots }.$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral

Notes:

For more digits see OEIS Sequence A085565; see also Sloane (2003).

Referenced by:

§19.20(i), §19.20(iv), §19.21(i)

Permalink:

http://dlmf.nist.gov/19.20.E2

Encodings:

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See also:

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

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

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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:

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See also:

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

►The second lemniscate constant is given by …

3: 19.30 Lengths of Plane Curves

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§19.30(iii) Bernoulli’s Lemniscate

►For $0\le \theta \le \frac{1}{4}\pi$, the arclength $s$ of Bernoulli’s lemniscate …The perimeter length $P$ of the lemniscate is given by …

4: Bibliography T

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

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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

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5: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Zhang and Jin (1996, Table 3.8) tabulates $\gamma (a,x)$ for $a=0.5,1,3,5,10,25,50,100$, $x=0(.1)1(1)3,5(5)30,50,100$ to 8D or 8S.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

6: 20 Theta Functions

Chapter 20 Theta Functions

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7: 19.3 Graphics

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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

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8: 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 $\pi /4$. …

9: 5.22 Tables

… ►Abramowitz and Stegun (1964, Chapter 6) tabulates $\mathrm{\Gamma }\left(x\right)$, $\ln \mathrm{\Gamma }\left(x\right)$, $\psi \left(x\right)$, and ${\psi }^{\prime }\left(x\right)$ for $x=1(.005)2$ to 10D; ${\psi }^{\prime \prime }\left(x\right)$ and ${\psi }^{(3)}\left(x\right)$ for $x=1(.01)2$ to 10D; $\mathrm{\Gamma }\left(n\right)$, $1/\mathrm{\Gamma }\left(n\right)$, $\mathrm{\Gamma }\left(n+\frac{1}{2}\right)$, $\psi \left(n\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{1}{3}\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{1}{2}\right)$, and ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{2}{3}\right)$ for $n=1(1)101$ to 8–11S; $\mathrm{\Gamma }\left(n+1\right)$ for $n=100(100)1000$ to 20S. Zhang and Jin (1996, pp. 67–69 and 72) tabulates $\mathrm{\Gamma }\left(x\right)$, $1/\mathrm{\Gamma }\left(x\right)$, $\mathrm{\Gamma }\left(-x\right)$, $\ln \mathrm{\Gamma }\left(x\right)$, $\psi \left(x\right)$, $\psi \left(-x\right)$, ${\psi }^{\prime }\left(x\right)$, and ${\psi }^{\prime }\left(-x\right)$ for $x=0(.1)5$ to 8D or 8S; $\mathrm{\Gamma }\left(n+1\right)$ for $n=0(1)100(10)250(50)500(100)3000$ to 51S. … ►Abramov (1960) tabulates $\ln \mathrm{\Gamma }\left(x+\mathrm{i}y\right)$ for $x=1$ ($.01$) $2$, $y=0$ ($.01$) $4$ to 6D. Abramowitz and Stegun (1964, Chapter 6) tabulates $\ln \mathrm{\Gamma }\left(x+\mathrm{i}y\right)$ for $x=1$ ($.1$) $2$, $y=0$ ($.1$) $10$ to 12D. …Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of $\mathrm{\Gamma }\left(x+\mathrm{i}y\right)$, $\ln \mathrm{\Gamma }\left(x+\mathrm{i}y\right)$, and $\psi \left(x+\mathrm{i}y\right)$ for $x=0.5,1,5,10$, $y=0(.5)10$ to 8S.

10: 25.12 Polylogarithms

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Figure 25.12.1: Dilogarithm function ${\mathrm{Li}}_2\left(x\right)$, Magnify

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Figure 25.12.2: Absolute value of the dilogarithm function $\left|{\mathrm{Li}}_2\left(x+\mathrm{i}y\right)\right|$, $-20\le x\le 20$, $-20\le y\le 20$. … Magnify 3D Help

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25.12.11 $${\mathrm{Li}}_s\left(z\right)\equiv \frac{z}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}}{{\mathrm{e}}^x-z}dx,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\int$: integral, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $x$: real variable, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Lewin (1981, (7.188), p. 236)

Referenced by:

(25.12.16), §25.12(ii)

Permalink:

http://dlmf.nist.gov/25.12.E11

Encodings:

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See also:

Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

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25.12.14 $$F_s(x)=\frac{1}{\mathrm{\Gamma }\left(s+1\right)}{\int }_0^{\mathrm{\infty }}\frac{t^s}{{\mathrm{e}}^{t-x}+1}dt,$$ $s>-1$,

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Defines:

$F_s(x)$: Fermi–Dirac integral (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $x$: real variable and $s$: complex variable

Keywords:

definition, Fermi–Dirac integral, improper integral, integral representation

Source:

Dingle (1957b, (1), p. 226)

Referenced by:

(25.12.16), 3rd item, 5th item

Permalink:

http://dlmf.nist.gov/25.12.E14

Encodings:

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See also:

Annotations for §25.12(iii), §25.12 and Ch.25

… ►Sometimes the factor $1/\mathrm{\Gamma }\left(s+1\right)$ is omitted. …