# geometric

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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: 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
###### §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 $G=(a_{1}a_{2}\cdots a_{n})^{1/n},$
1.2.26 $\lim_{r\to 0}M(r)=G.$
##### 7: Alexander I. Bobenko
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. …
1.7.7 $H\leq G\leq A,$
The approximations converge geometrically3.8(i)) to the eigenvalues and coefficients of Lamé functions as $n\to\infty$. …