Arithmetic-geometric mean
The arithmetic-geometric mean M(x, y) of two positive real numbers x and y is defined as follows:
- First, you get the arithmetic average x e and by name a1, i.e. a1 =x+and) / 2.
- Then the geometric mean of x e and by name g1, i.e. g1 is the square root of xy.
- Then it was this operation with a1 instead of x and g1 instead of and. In this way, two successions are defined (an) and (gn):
an+1=an+gn2andgn+1=angn{displaystyle a_{n+1}={frac {a_{n}+g_{n}{n}}}{2}}{quad {mbox{y}}}quad g_{n+1}={sqrt {a_{n}{n}}}}}}
Both sequences converge to the same number, called the arithmetic-geometric mean M(x, y) of x and and.
Algebraic Origin of the Arithmetic - Geometric Mean
https://web.archive.org/web/20160818061731/http://www.cerano.com.mx/cerano_v2/media-aritmetico-geometrica/
Properties
It can also be shown that:
M(x,and)=π π 4⋅ ⋅ x+andK(x− − andx+and){displaystyle M(x,y)={frac {pi }{4}}}{cdot {frac {x+y}{Kleft({frac {x-y}{x+y}}}}}}}}}}}}}
where K(x) is the complete elliptic integral of the first kind. Another interesting identity involving the geometric arithmetic mean is the following:
1M(a,b)=2π π ∫ ∫ 0π π /2dθ θ a2#2 θ θ +b2without2 θ θ =1π π ∫ ∫ − − ∞ ∞ ∞ ∞ dt(a2+t2)(b2+t2){displaystyle {frac {1}{M(a,b)}}{frac {2{pi}}}{int _{0}{pi /2}{frac {dtheta}{sqrt} {a^{2}{2⁄2}{fs}{fs}{fs}{fs}{fs}{f}{fs}{fs}}{fs)}}{fs)}{fs)}{fs)}{fs}{fnx1⁄2}}{fs}}{fs}}{fs}{fs)}{fs}}{fs}{fs}}{fs}{fs}{fnx1⁄2}{f}{fs}{fs}{fs}{fs}}{fs}{fs}{f}{f}{
