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 .

Definition of arithmetic-geometric mean

The arithmetic-geometric mean is defined through the reciprocal value of the complete elliptic integral 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 –‐plane.

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

Representations through more general functions

The arithmetic‐geometric mean 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 can be interpreted as a representation of through related equivalent functions—the reciprocal of the complete elliptic integral with :

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

Values in points

The arithmetic‐geometric mean can be exactly evaluated in some points, for example:

Real values for real arguments

For real values of arguments , (with ), the values of the arithmetic‐geometric mean are real.

Analyticity

The arithmetic‐geometric mean is an analytical function of and that is defined over .

Poles and essential singularities

The arithmetic‐geometric mean does not have poles and essential singularities.

Branch points and branch cuts

The arithmetic‐geometric mean on the ‐plane has two branch points: . It is a single‐valued function on the ‐plane cut along the interval , where it is continuous from above:

Periodicity

The arithmetic‐geometric mean does not have periodicity.

Parity and symmetries

The arithmetic‐geometric mean is an odd function and has mirror and permutation symmetry:

The arithmetic‐geometric mean is the homogenous function:

Series representations

The arithmetic‐geometric mean has the following series representations at the points , , and :

Product representation

The arithmetic‐geometric mean has the following infinite product representation:

Integral representation

The arithmetic‐geometric mean has the following integral representation:

Limit representation

The arithmetic‐geometric mean has the following limit representation, which is often used for the definition of :

Transformations

The homogeneity property of the arithmetic‐geometric mean 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 have rather simple representations:

The -order symbolic derivatives are much more complicated. Here is an example:

Differential equations

The arithmetic‐geometric mean 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 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 , and so on.