Editing Arithmetic–geometric mean (section)

{{Short description|Mathematical function of two positive real arguments}}
{{about|the particular type of mean|the similarly named inequality|Inequality of arithmetic and geometric means}}

[[File:Generalized means + agm.png|400px|thumb|right|Plot of the arithmetic–geometric mean <math>\operatorname{agm}(1,x)</math> among several [[generalized mean]]s.]]
In [[mathematics]], the '''arithmetic–geometric mean''' (AGM or agM<ref name="Cox" />) of two [[positive real numbers]] {{math|''x''}} and {{math|''y''}} is the mutual limit of a sequence of [[arithmetic mean]]s and a sequence of [[geometric mean]]s. The arithmetic–geometric mean is used in fast [[algorithm]]s for [[exponential function|exponential]], [[trigonometric functions]], and other [[special functions]], as well as some [[mathematical constant]]s, in particular, [[computing π|computing {{mvar|&pi;}}]].

The AGM is defined as the limit of the interdependent [[sequence]]s <math>a_i</math> and <math>g_i</math>.  Assuming <math> x \geq y \geq 0</math>, we write:<math display=block>\begin{align}
 a_0 &= x,\\
 g_0 &= y\\
 a_{n+1} &= \tfrac12(a_n + g_n),\\
 g_{n+1} &= \sqrt{a_n g_n}\, .
\end{align}</math>These two sequences [[limit of a sequence|converge]] to the same number, the arithmetic–geometric mean of {{math|''x''}} and {{math|''y''}}; it is denoted by {{math|''M''(''x'', ''y'')}}, or sometimes by {{math|agm(''x'', ''y'')}} or {{math|AGM(''x'', ''y'')}}.

The arithmetic–geometric mean can be extended to [[Complex number|complex numbers]] and, when the [[Branch point#Branch cuts|branches]] of the square root are allowed to be taken inconsistently, it is a [[multivalued function]].<ref name="Cox">{{cite journal |last=Cox |first=David |date=January 1984 |title=The Arithmetic-Geometric Mean of Gauss|url=https://www.researchgate.net/publication/248675540 |journal=L'Enseignement Mathématique|volume=30|issue=2|pages=275–330}}</ref>