Cesàro means

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1: 19.8 Quadratic Transformations

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

… ►As $n\to \mathrm{\infty }$, $a_n$ and $g_n$ converge to a common limit $M(a_0,g_0)$ 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 $M(a_0,g_0)$ is quadratic in each case. … ►

19.8.5 $$K\left(k\right)=\frac{\pi }{2M(1,k^{\prime })},$$ .

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: 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

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Annotations for §19.8(i), §19.8 and Ch.19

… ►Again, $p_n$ and ${\epsilon }_n$ converge quadratically to $M(a_0,g_0)$ and 0, respectively, and $Q_n$ converges to 0 faster than quadratically. …

2: 1.15 Summability Methods

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Cesàro Summability

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General Cesàro Summability

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Cesàro (or (C,1)) Means

… ►If $f(\theta )$ is periodic and integrable on $[0,2\pi ]$, then as $n\to \mathrm{\infty }$ the Abel means $A(r,\theta )$ and the (C,1) means ${\sigma }_n(\theta )$ converge to … ►

Cesàro Summability

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3: 1.2 Elementary Algebra

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§1.2(iv) Means

►The arithmetic mean of $n$ numbers $a_1,a_2,\mathrm{\dots },a_n$ is … ►The geometric mean $G$ and harmonic mean $H$ of $n$ positive numbers $a_1,a_2,\mathrm{\dots },a_n$ are given by … ►If $r$ is a nonzero real number, then the weighted mean $M(r)$ of $n$ nonnegative numbers $a_1,a_2,\mathrm{\dots },a_n$, and $n$ positive numbers $p_1,p_2,\mathrm{\dots },p_n$ with … ►

$M(1)=A,$

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4: 1.7 Inequalities

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§1.7(iii) Means

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1.7.7 $$H\le G\le A,$$

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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

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Annotations for §1.7(iii), §1.7 and Ch.1

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1.7.8 $$\min (a_1,a_2,\mathrm{\dots },a_n)\le M(r)\le \max (a_1,a_2,\mathrm{\dots },a_n),$$

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Symbols:

$n$: nonnegative integer and $M(r)$: weighted mean

A&S Ref:

3.2.2

Permalink:

http://dlmf.nist.gov/1.7.E8

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Annotations for §1.7(iii), §1.7 and Ch.1

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1.7.9 $$M(r)\le M(s),$$ ,

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Symbols:

$M(r)$: weighted mean

A&S Ref:

3.2.4

Permalink:

http://dlmf.nist.gov/1.7.E9

Encodings:

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Annotations for §1.7(iii), §1.7 and Ch.1

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5: 15.17 Mathematical Applications

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

6: Bibliography Q

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On two problems concerning means. J. Hangzhou Inst. Elec. Engrg. 17, pp. 1–7 (Chinese).

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Notes:

In Chinese

Referenced by:

§19.9(i).

Encodings:

BibTeX

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7: 22.20 Methods of Computation

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§22.20(ii) Arithmetic-Geometric Mean

… ►Then as $n\to \mathrm{\infty }$ sequences $\{a_n\}$, $\{b_n\}$ converge to a common limit $M=M(a_0,b_0)$, 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 $\mathrm{am}(x,k)$. …

8: Need Help?

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• How can I view ‘VRML’? What is it? See Viewing DLMF Interactive 3D Graphics.

• What do the colors mean? See About Color Map.

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9: 19.22 Quadratic Transformations

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

►The AGM, $M(a_0,g_0)$, of two positive numbers $a_0$ and $g_0$ is defined in §19.8(i). … ►

19.22.8 $$\frac{2}{\pi }{R}_F(0,a_0^2,g_0^2)=\frac{1}{M(a_0,g_0)},$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $M(a,g)$: 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

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Annotations for §19.22(ii), §19.22 and Ch.19

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19.22.9 $$\frac{4}{\pi }{R}_G(0,a_0^2,g_0^2)=\frac{1}{M(a_0,g_0)}\left(a_0^2-\sum _{n=0}^{\mathrm{\infty }}{2}^{n-1}{c}_n^2\right)=\frac{1}{M(a_0,g_0)}\left(a_1^2-\sum _{n=2}^{\mathrm{\infty }}{2}^{n-1}{c}_n^2\right),$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $M(a,g)$: 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:

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See also:

Annotations for §19.22(ii), §19.22 and Ch.19

… ►As $n\to \mathrm{\infty }$, $p_n$ and ${\epsilon }_n$ converge quadratically to $M(a_0,g_0)$ and 0, respectively, and $Q_n$ converges to 0 faster than quadratically. …

10: 8.25 Methods of Computation

… ►See Allasia and Besenghi (1987b) for the numerical computation of $\mathrm{\Gamma }(a,z)$ from (8.6.4) by means of the trapezoidal rule. … ►The computation of $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5). …