Introduction to the Arithmetic‐Geometric Mean
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.
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:
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:
The arithmetic‐geometric mean has the following series representations at the points , , and :
The arithmetic‐geometric mean has the following infinite product representation:
The arithmetic‐geometric mean has the following integral representation:
The arithmetic‐geometric mean has the following limit representation, which is often used for the definition of :
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:
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:
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.