|
The arithmetic‐geometric mean  can be exactly evaluated in some points, for example:
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  .
The arithmetic‐geometric mean  does not have poles and essential singularities.
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.
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:
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:
|
