Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











ArithmeticGeometricMean






Mathematica Notation

Traditional Notation









Elliptic Functions > ArithmeticGeometricMean[a,b] > Introduction to the Arithmetic‚ÄźGeometric Mean





The best-known properties and formulas for the arithmetic-geometric mean


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:





© 1998- Wolfram Research, Inc.