lemniscate arc length

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11: 23.23 Tables

… ►The values are tabulated on the real and imaginary $z$-axes, mostly ranging from 0 to 1 or $\mathrm{i}$ in steps of length 0. …

12: 26.13 Permutations: Cycle Notation

… ►Cycles of length one are fixed points. … ►An element of ${𝔖}_n$ with $a_1$ fixed points, $a_2$ cycles of length $2,\mathrm{\dots },a_n$ cycles of length $n$, where $n=a_1+2a_2+\mathrm{\cdots }+n{a}_n$, is said to have cycle type $\left(a_1,a_2,\mathrm{\dots },a_n\right)$. … ►A transposition is a permutation that consists of a single cycle of length two. …A permutation that consists of a single cycle of length $k$ can be written as the composition of $k-1$ two-cycles (read from right to left): …

13: 4.1 Special Notation

… ►Sometimes “arc” is replaced by the index “$-1$”, e. …

14: 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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15: 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 $\pi /4$. …

16: 27.16 Cryptography

… ►To code a message by this method, we replace each letter by two digits, say $A=01$, $B=02$, $\mathrm{\dots }$, $Z=26$, and divide the message into pieces of convenient length smaller than the public value $n=pq$. …

17: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ► $M_2$ is the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ cycles of length 1, $a_2$ cycles of length 2, $\mathrm{\dots }$, and $a_n$ cycles of length $n$: …

18: 3.1 Arithmetics and Error Measures

… ►Computer algebra systems use exact rational arithmetic with rational numbers $p/q$, where $p$ and $q$ are multi-length integers. During the calculations common divisors are removed from the rational numbers, and the final results can be converted to decimal representations of arbitrary length. …

19: 21.9 Integrable Equations

… ►Particularly important for the use of Riemann theta functions is the Kadomtsev–Petviashvili (KP) equation, which describes the propagation of two-dimensional, long-wave length surface waves in shallow water (Ablowitz and Segur (1981, Chapter 4)): …

20: 19.24 Inequalities

… ►Approximations and one-sided inequalities for $R_G(0,y,z)$ follow from those given in §19.9(i) for the length $L(a,b)$ of an ellipse with semiaxes $a$ and $b$, since ►

19.24.7 $$L(a,b)=8R_G(0,a^2,b^2).$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind and $L(a,b)$: length

Permalink:

http://dlmf.nist.gov/19.24.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.24(i), §19.24 and Ch.19

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