ellipse arc length

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21: 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 $n=pq$. …

22: Bibliography

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Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\pi$, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.

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External Links:

ISSN 0002-9890, FullText, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

BibTeX

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23: 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 $n$: …

24: 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. …

25: 18.18 Sums

… ►Let $f(z)$ be analytic within an ellipse $E$ with foci $z=\pm 1$, and …

26: 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)): …

27: 1.9 Calculus of a Complex Variable

… ►A domain $D$, say, is an open set in $ℂ$ that is connected, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. … ►

§1.9(iii) Integration

►An arc $C$ is given by $z(t)=x(t)+\mathrm{i}y(t)$, $a\le t\le b$, where $x$ and $y$ are continuously differentiable. … ►

Conformal Transformation

►Suppose $f(z)$ is analytic in a domain $D$ and $C_1,C_2$ are two arcs in $D$ passing through $z_0$. …

28: 3.11 Approximation Techniques

… ►If $x$ is replaced by a complex variable $z$ and $f(z)$ is analytic, then the expansion (3.11.11) converges within an ellipse. …

29: 1.2 Elementary Algebra

… ►A column vector of length $n$ is an $n\times 1$ matrix …and the corresponding transposed row vector of length $n$ is … ►Column vectors $𝐮$ and $𝐯$ of the same length $n$ have a scalar product … ►Special cases are the Euclidean length or $l^2$ norm … ►Unless otherwise indicated, matrices are assumed square, of order $n$; and, when vectors are combined with them, these are of length $n$. …

30: 34.4 Definition: $\mathit{6}j$ Symbol

… ►Except in degenerate cases the combination of the triangle inequalities for the four $\mathit{3}j$ symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths $j_1,j_2,j_3,l_1,l_2,l_3$; see Figure 34.4.1. …