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1: 19.8 Quadratic Transformations

… ►

§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)

… ►As

n\to\infty

,

a_{n}

and

g_{n}

converge to a common limit

M\left(a_{0},g_{0}\right)

called the AGM (Arithmetic-Geometric Mean) of

a_{0}

and

g_{0}

. …showing that the convergence of

c_{n}

to 0 and of

a_{n}

and

g_{n}

to

M\left(a_{0},g_{0}\right)

is quadratic in each case. … ►

19.8.5

K\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)},

-\infty<k^{2}<1

.

ⓘ

Symbols:: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.8.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.8(i), §19.8 and Ch.19

… ►Again,

p_{n}

and

\varepsilon_{n}

converge quadratically to

M\left(a_{0},g_{0}\right)

and 0, respectively, and

Q_{n}

converges to 0 faster than quadratically. …

2: Alexander A. Its

… ► Novokshënov), published by Springer in 1986, Algebro-geometric Approach to Nonlinear Integrable Problems (with E. …

3: 15.17 Mathematical Applications

… ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … …

4: 22.20 Methods of Computation

… ►

§22.20(ii) Arithmetic-Geometric Mean

… ►Then as

n\to\infty

sequences

\{a_{n}\}

,

\{b_{n}\}

converge to a common limit

M=M\left(a_{0},b_{0}\right)

, the arithmetic-geometric mean of

a_{0},b_{0}

. … ►The rate of convergence is similar to that for the arithmetic-geometric mean. … ►using the arithmetic-geometric mean. … ►Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute

\operatorname{am}\left(x,k\right)

. …

5: 36.14 Other Physical Applications

… ►These are the structurally stable focal singularities (envelopes) of families of rays, on which the intensities of the geometrical (ray) theory diverge. Diffraction catastrophes describe the (linear) wave amplitudes that smooth the geometrical caustic singularities and decorate them with interference patterns. …

6: 1.2 Elementary Algebra

… ►

Geometric Progression

… ►

§1.2(iv) Means

… ►The geometric mean

G

and harmonic mean

H

of

n

positive numbers

a_{1},a_{2},\dots,a_{n}

are given by ►

1.2.18

G=(a_{1}a_{2}\cdots a_{n})^{1/n},

ⓘ

Defines:: $G$ : geometric mean (locally)
Symbols:: $n$ : nonnegative integer
A&S Ref:: 3.1.12
Permalink:: http://dlmf.nist.gov/1.2.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §1.2(iv), §1.2 and Ch.1

… ►

1.2.26

\lim_{r\to 0}M(r)=G.

ⓘ

Symbols:: $G$ : geometric mean and $M(\NVar{r})$ : weighted mean
A&S Ref:: 3.1.18
Permalink:: http://dlmf.nist.gov/1.2.E26
Encodings:: pMML, png, TeX
See also:: Annotations for §1.2(iv), §1.2 and Ch.1

…

7: Alexander I. Bobenko

… ►Bobenko’s books are Algebro-geometric Approach to Nonlinear Integrable Problems (with E. …

8: 19.22 Quadratic Transformations

… ►

§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)

►The AGM,

M\left(a_{0},g_{0}\right)

, of two positive numbers

a_{0}

and

g_{0}

is defined in §19.8(i). … ►

19.22.8

\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{M\left(a_{0},g_{% 0}\right)},

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.22.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(ii), §19.22 and Ch.19

►

19.22.9

\frac{4}{\pi}R_{G}\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{M\left(a_{0},g_{% 0}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=\frac{1}{% M\left(a_{0},g_{0}\right)}\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}% \right),

ⓘ

Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.22.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(ii), §19.22 and Ch.19

… ►As

n\to\infty

,

p_{n}

and

\varepsilon_{n}

converge quadratically to

M\left(a_{0},g_{0}\right)

and 0, respectively, and

Q_{n}

converges to 0 faster than quadratically. …

9: 1.7 Inequalities

… ►

§1.7(iii) Means

… ►

1.7.7

H\leq G\leq A,

ⓘ

Symbols:: $A$ : arithmetic mean, $G$ : geometric mean and $H$ : harmonic mean
A&S Ref:: 3.2.1
Permalink:: http://dlmf.nist.gov/1.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §1.7(iii), §1.7 and Ch.1

…

10: 29.20 Methods of Computation

… ►The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as

n\to\infty

. …

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