Introductions to ArithmeticGeometricMean

Introduction to the Arithmetic-Geometric Mean

General

The arithmetic-geometric mean appeared in the works of J. Landen (1771, 1775) and J.-L. Lagrange (1784-1785) who defined it through the following quite-natural limit procedure:

C. F. Gauss (1791–1799, 1800, 1876) continued to research this limit and in 1800 derived its representation through the hypergeometric function  (Subscript[, 2])(Subscript[F, 1])(a,b;c;z).

Definition of arithmetic-geometric mean

The arithmetic-geometric mean agm(a,b) is defined through the reciprocal value of the complete elliptic integral K(z) by the formula:

A quick look at the arithmetic-geometric mean

Here is a quick look at the graphic for the arithmetic-geometric mean over the real a–b-plane.

Connections within the arithmetic-geometric mean group and with other function groups

Representations through more general functions

The arithmetic-geometric mean agm(a,b) can be represented through the reciprocal function of the particular cases of hypergeometric and Meijer G functions:

Representations through related equivalent functions

The definition of the arithmetic-geometric mean agm(a,b) can be interpreted as a representation of agm(a,b) through related equivalent functions—the reciprocal of the complete elliptic integral K(z) with z==(((((a-b)/(a+b))))^2):

The best-known properties and formulas for the arithmetic-geometric mean

Values in points

The arithmetic-geometric mean agm(a,b) can be exactly evaluated in some points, for example:

Real values for real arguments

For real values of arguments a, b (with a b>0), the values of the arithmetic-geometric mean agm(a,b) are real.

Analyticity

The arithmetic-geometric mean agm(a,b) is an analytical function of a and b that is defined over (ℂ^2).

Poles and essential singularities

Branch points and branch cuts

Periodicity

The arithmetic-geometric mean agm(a,b) does not have periodicity.

Parity and symmetries

The arithmetic-geometric mean agm(a,b) is an odd function and has mirror and permutation symmetry:

The arithmetic-geometric mean agm(a,b) is the homogenous function:

Series representations

The arithmetic-geometric mean agm(a,b) has the following series representations at the points (b/a)->0, (b/a)->1, and |(b/a)|->∞:

Product representation

The arithmetic-geometric mean agm(1,b) has the following infinite product representation:

Integral representation

The arithmetic-geometric mean agm(a,b) has the following integral representation:

Limit representation

The arithmetic-geometric mean agm(a,b) has the following limit representation, which is often used for the definition of agm(a,b):

Transformations

The homogeneity property of the arithmetic-geometric mean agm(a,b) leads to the following transformations:

Another group of transformations is based on the first of the following properties:

Representations of derivatives

The first derivatives of the arithmetic-geometric mean agm(a,b) have rather simple representations:

The n(^th)-order symbolic derivatives are much more complicated. Here is an example:

Differential equations

The arithmetic-geometric mean agm(a,b) satisfies the following second-order ordinary nonlinear differential equation:

It can also be represented as partial solutions of the following partial differential equation:

Inequalities

The arithmetic-geometric mean agm(a,b) lies between the middle geometric mean and middle arithmetic mean, which is shown in the following famous inequality:

Applications of the arithmetic-geometric mean

Applications of the arithmetic-geometric mean include fast high-precision computation of π,log(z),(e^z),sin(z),cos(z), and so on.

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