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ID: INPUT{id.yaml}

Title: >
  Values of the arithmetic-geometric mean

Definition: >
  The arithmetic-geometric mean $\text{agm}(a,b)$ of two
  non-negative real numbers $a$ and $b$ is defined
  as the simultaneous limit $\lim a_n = \lim g_n$ 
  of the sequences $(a_n)$ and $(g_n)$ given by
  $a_0 = a$, $g_0 = b$, 
  $a_{n+1} = (a_n + g_n)/2$, 
  $g_{n+1} = \sqrt{a_n g_n}$. 
  This table lists values of $\text{agm}(a,b)$ 
  for certain pairs of integers $a$ and $b$.
  
Parameters:
  a:
    display: $a$
    type: R
    constraints: $a \geq 0$

b:
    display: $b$
    type: R
    constraints: $b \geq 0$
   
Comments:
  comment-breaking-symmetry: >
    Due to the symmetry CITE{formula-symmetric} 
    and trivial cases CITE{formula-diagonal}, CITE{formula-edge}, 
    we only list values 
    for $a < b$.
  
Formulas: 
  formula-elliptic-integral: >
    $\text{agm}(a,b) = \frac{(a+b)\pi}{4K((a-b)/(a+b))}$, where
    $K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}$
    is the complete elliptic integral of the first kind CITE{WikiEllInt}.
  formula-symmetric: >
    $\text{agm}(a,b) = \text{agm}(b,a)$
  formula-diagonal: >
    $\text{agm}(a,a) = a$
  formula-edge: >
    $\text{agm}(0,a) = 0$

Programs:

References:

Links:
  MathWorld:
    title: "MathWorld: Arithmetic-Geometric Mean"
    url: https://mathworld.wolfram.com/Arithmetic-GeometricMean.html
  Wiki:
    title: "Wikipedia: Arithmetic-geometric mean"
    url: https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean
  WikiEllInt:
    title: "Wikipedia: Complete elliptic integral of the first kind"
    url: https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind

Similar tables:
  
Keywords:
  
Tags:
- number theory
- elliptic curves

Data properties:
  type: R
  complete: no

Display properties:
  number-header: $\text{agm}(a,b)$

Numbers: INPUT{numbers.yaml}

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Values of the arithmetic-geometric mean

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Numbers

$a$

$b$

$\text{agm}(a,b)$

INPUT{numbers.yaml} (not shown in preview)

Definition

The arithmetic-geometric mean $\text{agm}(a,b)$ of two non-negative real numbers $a$ and $b$ is defined as the simultaneous limit $\lim a_n = \lim g_n$ of the sequences $(a_n)$ and $(g_n)$ given by $a_0 = a$, $g_0 = b$, $a_{n+1} = (a_n + g_n)/2$, $g_{n+1} = \sqrt{a_n g_n}$. This table lists values of $\text{agm}(a,b)$ for certain pairs of integers $a$ and $b$.

Parameters

$a$

— real number ($a \geq 0$)

$b$

— real number ($b \geq 0$)

Formulas

(1)

$\text{agm}(a,b) = \frac{(a+b)\pi}{4K((a-b)/(a+b))}$, where $K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}$ is the complete elliptic integral of the first kind [3].

(2)

$\text{agm}(a,b) = \text{agm}(b,a)$

(3)

$\text{agm}(a,a) = a$

(4)

$\text{agm}(0,a) = 0$

Comments

(5)

Due to the symmetry (2) and trivial cases (3), (4), we only list values for $a < b$.

Links

[1]

MathWorld: Arithmetic-Geometric Mean