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: