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: