# arithmetic-geometric mean

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###### §19.8(i) Gauss’s Arithmetic-GeometricMean (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.
19.8.10 $p_{0}^{2}=1-(k^{2}/\alpha^{2}).$
##### 2: 15.17 Mathematical Applications
Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean19.22(ii)). … …
##### 3: 22.20 Methods of Computation
###### §22.20(ii) Arithmetic-GeometricMean
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 $\operatorname{am}\left(x,k\right)$. …
###### §19.22(ii) Gauss’s Arithmetic-GeometricMean (AGM)
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 $\frac{4}{\pi}R_{G}\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{M\left(a_{0},g_{% 0}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=\frac{1}{% M\left(a_{0},g_{0}\right)}\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}% \right),$
19.22.10 $R_{D}\left(0,g_{0}^{2},a_{0}^{2}\right)=\frac{3\pi}{4M\left(a_{0},g_{0}\right)% a_{0}^{2}}\sum_{n=0}^{\infty}Q_{n},$
19.22.12 $R_{J}\left(0,g_{0}^{2},a_{0}^{2},p_{0}^{2}\right)=\frac{3\pi}{4M\left(a_{0},g_% {0}\right)p_{0}^{2}}\sum_{n=0}^{\infty}Q_{n},$
##### 5: 23.22 Methods of Computation
• (a)

In the general case, given by $cd\neq 0$, we compute the roots $\alpha$, $\beta$, $\gamma$, say, of the cubic equation $4t^{3}-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent $e_{1}$, $e_{2}$, $e_{3}$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^{3}-27d^{2}>0$, then $e_{1}$, $e_{2}$, $e_{3}$ can be identified via (23.5.1), and $k^{2}$, ${k^{\prime}}^{2}$ obtained from (23.6.16).

If $c^{3}-27d^{2}<0$, or $c$ and $d$ are not both real, then we label $\alpha$, $\beta$, $\gamma$ so that the triangle with vertices $\alpha$, $\beta$, $\gamma$ is positively oriented and $[\alpha,\gamma]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$, $\beta$, $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha,\gamma)$. In consequence, $k^{2}=(\beta-\gamma)/(\alpha-\gamma)$, ${k^{\prime}}^{2}=(\alpha-\beta)/(\alpha-\gamma)$ satisfy $\Im k^{2}\geq 0\geq\Im{k^{\prime}}^{2}$ (with strict inequality unless $\alpha$, $\beta$, $\gamma$ are collinear); also $|k^{2}|$, $|{k^{\prime}}^{2}|\leq 1$.

Finally, on taking the principal square roots of $k^{2}$ and ${k^{\prime}}^{2}$ we obtain values for $k$ and $k^{\prime}$ that lie in the 1st and 4th quadrants, respectively, and $2\omega_{1}$, $2\omega_{3}$ are given by

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2\omega_{1}$, $2\omega_{3}$), corresponding to the 2 possible choices of the square root.

• ##### 6: Bibliography C
• D. A. Cox (1984) The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.
• D. A. Cox (1985) Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.
• ##### 7: Bibliography
• G. Almkvist and B. Berndt (1988) Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\pi$, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.
• ##### 8: Bibliography S
• K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
• ##### 9: 15.8 Transformations of Variable
This is used in a cubic analog of the arithmetic-geometric mean. …