# Cesàro means

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###### §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.
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: 1.15 Summability Methods
###### Cesàro (or (C,1)) Means
If $f(\theta)$ is periodic and integrable on $[0,2\pi]$, then as $n\to\infty$ the Abel means $A(r,\theta)$ and the (C,1) means $\sigma_{n}(\theta)$ converge to …
##### 3: 1.2 Elementary Algebra
###### §1.2(iv) Means
The arithmetic mean of $n$ numbers $a_{1},a_{2},\dots,a_{n}$ is … The geometric mean $G$ and harmonic mean $H$ of $n$ positive numbers $a_{1},a_{2},\dots,a_{n}$ are given by … If $r$ is a nonzero real number, then the weighted mean $M(r)$ of $n$ nonnegative numbers $a_{1},a_{2},\dots,a_{n}$, and $n$ positive numbers $p_{1},p_{2},\dots,p_{n}$ with …
$M(1)=A,$
##### 4: 1.7 Inequalities
###### §1.7(iii) Means
1.7.7 $H\leq G\leq A,$
1.7.8 $\min(a_{1},a_{2},\dots,a_{n})\leq M(r)\leq\max(a_{1},a_{2},\dots,a_{n}),$
1.7.9 $M(r)\leq M(s),$ $r,
##### 5: 15.17 Mathematical Applications
Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean19.22(ii)). … …
##### 6: 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 $\operatorname{am}\left(x,k\right)$. …
##### 7: Bibliography Q
• S.-L. Qiu and J.-M. Shen (1997) On two problems concerning means. J. Hangzhou Inst. Elec. Engrg. 17, pp. 1–7 (Chinese).
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• ##### 9: 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 $\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{M\left(a_{0},g_{% 0}\right)},$
19.22.9 $\frac{4}{\pi}R_{G}\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{M\left(a_{0},g_{% 0}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=\frac{1}{% M\left(a_{0},g_{0}\right)}\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}% \right),$
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. …
##### 10: 8.25 Methods of Computation
See Allasia and Besenghi (1987b) for the numerical computation of $\Gamma\left(a,z\right)$ from (8.6.4) by means of the trapezoidal rule. … The computation of $\gamma\left(a,z\right)$ and $\Gamma\left(a,z\right)$ by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5). …