lemniscate arc length

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1: 22.18 Mathematical Applications

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§22.18(i) Lengths and Parametrization of Plane Curves

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Ellipse

… ►The arc length $l(u)$ in the first quadrant, measured from $u=0$, is … ►

Lemniscate

… ►The arc length $l(r)$, measured from $\varphi =0$, is …

2: 4.42 Solution of Triangles

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4.42.8 $$\cos a=\cos b\cos c+\sin b\sin c\cos A,$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $a$: arc length, $b$: arc length and $c$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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4.42.9 $$\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c},$$

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Symbols:

$\sin z$: sine function, $A$: angle, $B$: angle, $C$: angle, $a$: arc length, $b$: arc length and $c$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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4.42.10 $$\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $B$: angle, $a$: arc length, $b$: arc length and $c$: arc length

Permalink:

http://dlmf.nist.gov/4.42.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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4.42.11 $$\cos a\cos C=\sin a\cot b-\sin C\cot B,$$

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Symbols:

$\cos z$: cosine function, $\cot z$: cotangent function, $\sin z$: sine function, $B$: angle, $C$: angle, $a$: arc length and $b$: arc length

Permalink:

http://dlmf.nist.gov/4.42.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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4.42.12 $$\cos A=-\cos B\cos C+\sin B\sin C\cos a.$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $B$: angle, $C$: angle and $a$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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3: 19.30 Lengths of Plane Curves

§19.30 Lengths of Plane Curves

… ►The length of the ellipse is … ►

§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: 7.20 Mathematical Applications

… ►Then the arc length between the origin and $P(t)$ equals $t$, and is directly proportional to the curvature at $P(t)$, which equals $\pi t$. …

5: 28.33 Physical Applications

… ►We shall derive solutions to the uniform, homogeneous, loss-free, and stretched elliptical ring membrane with mass $\rho$ per unit area, and radial tension $\tau$ per unit arc length. … ►If the parameters of a physical system vary periodically with time, then the question of stability arises, for example, a mathematical pendulum whose length varies as $\cos \left(2\omega t\right)$. …

6: Sidebar 9.SB1: Supernumerary Rainbows

… ►The faint line below the main colored arc is a ‘supernumerary rainbow’, produced by the interference of different sun-rays traversing a raindrop and emerging in the same direction. …

7: 19.20 Special Cases

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

►The first lemniscate constant is given by … ►

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

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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:

pMML, png, TeX

See also:

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

►The second lemniscate constant is given by …

8: 32.14 Combinatorics

… ►With , $𝝅(m_1),𝝅(m_2),\mathrm{\dots },𝝅(m_n)$ is said to be an increasing subsequence of $𝝅$ of length $n$ when . Let ${\mathrm{\ell }}_N(𝝅)$ be the length of the longest increasing subsequence of $𝝅$. … ►

32.14.1 $$\underset{N\to \mathrm{\infty }}{lim}\mathrm{Prob}\left(\frac{{\mathrm{\ell }}_N(𝝅)-2\sqrt{N}}{N^{1/6}}\le s\right)=F(s),$$

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Symbols:

$𝝅(m)$: permutations, ${\mathrm{\ell }}_N(𝝅)$: length and $F(s)$: distribution function

Permalink:

http://dlmf.nist.gov/32.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.14 and Ch.32

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9: 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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10: Bibliography B

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On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (4), pp. 1119–1178.

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External Links:

ISSN 0894-0347, FullText, MathReview (David J. Aldous), ZentralBlatt

Referenced by:

§32.14.

Encodings:

BibTeX

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A monotonicity property involving ${}_3{}^{}F_2^{}$ and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32 (2), pp. 403–419.

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External Links:

ISSN 1095-7154, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§19.9(i), §19.9(i).

Encodings:

BibTeX

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