# 算术-几何平均数

${\displaystyle a_{1}={\frac {x+y}{2}}}$
${\displaystyle g_{1}={\sqrt {xy}}.}$

${\displaystyle a_{n+1}={\frac {a_{n}+g_{n}}{2}}}$
${\displaystyle g_{n+1}={\sqrt {a_{n}g_{n}}}.}$

## 例子

${\displaystyle a_{1}={\frac {24+6}{2}}=15,}$
${\displaystyle g_{1}={\sqrt {24\times 6}}=\,}$${\displaystyle 12}$

${\displaystyle a_{2}={\frac {15+12}{2}}=13.5,}$
${\displaystyle g_{2}={\sqrt {15\times 12}}=\,}$${\displaystyle 13.416407864999}$ etc.

n an gn
0 24 6
1 15 12
2 13.5 13.416407864999...
3 13.458203932499... 13.458139030991...
4 13.458171481745... 13.458171481706...

24和6的算术-几何平均数是两个数列的公共极限，大约为13.45817148173。

## 性质

${\displaystyle M(x,y)}$是一个介于${\displaystyle x}$${\displaystyle y}$的算术平均数和几何平均数之间的数。

${\displaystyle M(x,y)}$还可以写为如下形式：

${\displaystyle \mathrm {M} (x,y)={\frac {\pi }{4}}\cdot {\frac {x+y}{K\left({\frac {x-y}{x+y}}\right)}}}$

1和${\displaystyle {\sqrt {2}}}$的算术-几何平均数的倒数，称为高斯常数

${\displaystyle {\frac {1}{\mathrm {M} (1,{\sqrt {2}})}}=G=0.8346268\dots }$

## 存在性的证明

${\displaystyle g_{n}\leqslant a_{n}}$

${\displaystyle g_{n+1}={\sqrt {g_{n}\cdot a_{n}}}\geqslant {\sqrt {g_{n}\cdot g_{n}}}=g_{n}}$

${\displaystyle \lim _{n\to \infty }g_{n}=g}$

${\displaystyle a_{n}={\frac {g_{n+1}^{2}}{g_{n}}}}$

${\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }{\frac {g_{n+1}^{2}}{g_{n}}}={\frac {g^{2}}{g}}=g}$

## 关于积分表达式的证明

${\displaystyle I(x,y)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\sqrt {x^{2}\cos ^{2}\theta +y^{2}\sin ^{2}\theta }}},}$

${\displaystyle \sin \theta ={\frac {2x\sin \theta '}{(x+y)+(x-y)\sin ^{2}\theta '}},}$

{\displaystyle {\begin{aligned}I(x,y)&=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta '}{\sqrt {{\bigl (}{\frac {1}{2}}(x+y){\bigr )}^{2}\cos ^{2}\theta '+{\bigl (}{\sqrt {xy}}{\bigr )}^{2}\sin ^{2}\theta '}}}\\&=I{\bigl (}{\frac {1}{2}}(x+y),{\sqrt {xy}}{\bigr )}.\end{aligned}}}

{\displaystyle {\begin{aligned}I(x,y)&=I(a_{1},g_{1})=I(a_{2},g_{2})=\cdots \\&=I{\bigl (}M(x,y),M(x,y){\bigr )}={\frac {\pi }{2M(x,y)}}.\end{aligned}}}

${\displaystyle M(x,y)={\frac {\pi }{2I(x,y)}}.}$

## 参考文献

### 引用

1. ^ David A. Cox. The Arithmetic-Geometric Mean of Gauss. J.L. Berggren, Jonathan M. Borwein, Peter Borwein (编). Pi: A Source Book. Springer. 2004: 481 [2014-08-12]. ISBN 978-0-387-20571-7. （原始内容存档于2020-06-14）. first published in L'Enseignement Mathématique, t. 30 (1984), p. 275-330

### 来源

• Jonathan Borwein, Peter Borwein, Pi and the AGM. A study in analytic number theory and computational complexity. Reprint of the 1987 original. Canadian Mathematical Society Series of Monographs and Advanced Texts, 4. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1998. xvi+414 pp. ISBN 0-471-31515-X MR1641658