Arithmetic-geometric mean

format_list_bulleted Contenido keyboard_arrow_down

ImprimirCitar

Comparison of different stockings. The arithmetic-geometric media is represented by the discontinuous line.

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 (a_n) and (g_n):

${displaystyle a_{n+1}={frac {a_{n}+g_{n}}{2}}quad {mbox{y}}quad g_{n+1}={sqrt {a_{n}g_{n}}}}$

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:

${displaystyle M(x,y)={frac {pi }{4}}cdot {frac {x+y}{Kleft({frac {x-y}{x+y}}right)}}}$

where K(x) is the complete elliptic integral of the first kind. Another interesting identity involving the geometric arithmetic mean is the following:

${displaystyle {frac {1}{M(a,b)}}={frac {2}{pi }}int _{0}^{pi /2}{frac {dtheta }{sqrt {a^{2}cos ^{2}theta +b^{2}sin ^{2}theta }}}={frac {1}{pi }}int _{-infty }^{infty }{frac {dt}{sqrt {(a^{2}+t^{2})(b^{2}+t^{2})}}}}$

Contenido relacionado

Más resultados...