Introduction to the Arithmetic‐Geometric Mean
General
The arithmeticgeometric mean appeared in the works of J. Landen (1771, 1775) and J.‐L. Lagrange (17841785) 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 arithmeticgeometric mean
The arithmeticgeometric mean is defined through the reciprocal value of the complete elliptic integral by the formula:
A quick look at the arithmeticgeometric mean Here is a quick look at the graphic for the arithmetic‐geometric mean over the real –‐plane.
Connections within the arithmeticgeometric 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 bestknown properties and formulas for the arithmeticgeometric 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 secondorder 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 arithmeticgeometric mean
Applications of the arithmetic‐geometric mean include fast high‐precision computation of , and so on.
