# Abel 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
###### AbelMeans
$A(r,\theta)$ is a harmonic function in polar coordinates ((1.9.27)), and … 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: 22.18 Mathematical Applications
###### §22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem
For any two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ on this curve, their sum $(x_{3},y_{3})$, always a third point on the curve, is defined by the Jacobi–Abel addition law …a construction due to Abel; see Whittaker and Watson (1927, pp. 442, 496–497). …
##### 5: 2.10 Sums and Sequences
Another version is the Abel–Plana formula: …
• (c)

The first infinite integral in (2.10.2) converges.

• These problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula …
##### 6: 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,
##### 7: 15.17 Mathematical Applications
Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean19.22(ii)). … …
##### 8: 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 $\operatorname{am}\left(x,k\right)$. …
##### 9: 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).
• ##### 10: Bibliography B
• H. F. Baker (1995) Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.
• A. I. Bobenko (1991) Constant mean curvature surfaces and integrable equations. Uspekhi Mat. Nauk 46 (4(280)), pp. 3–42, 192 (Russian).