ArithmeticGeometricMean

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed a

For fixed b

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

agm(a,b) is an analytical function of a and b which is defined over (ℂ^2).

Symmetries and periodicities

Parity

agm(a,b) is an odd function.

Mirror symmetry

Permutation symmetry

Periodicity

No periodicity

Homogeneity

Poles and essential singularities

With respect to a

The function agm(a,b) does not have poles and essential singularities with respect to a.

With respect to b

The function agm(a,b) does not have poles and essential singularities with respect to b.

Branch points

Branch cuts

Series representations

Generalized power series

Expansions at b==0 for a==1

Expansions at b==1 for a==1

Expansions at b==∞ for a==1

Integral representations

On the real axis

Of the direct function

Product representations

Limit representations

Differential equations

Ordinary nonlinear differential equations

Partial differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Identities

Functional identities

Differentiation

Low-order differentiation

With respect to a

With respect to b

Symbolic differentiation

With respect to a

With respect to b

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1

Through Meijer G

Classical cases for the direct function itself

Through other functions

Involving some hypergeometric-type functions

Inequalities

Theorems

Representation of π

π==(lim_(n->∞))((2(Subscript[a, n+1]^2))/(1-(∑_(k=0)^n)(2^k)(Subscript[c, k]^2)))/; (Subscript[c, n+1])=(((Subscript[a, n])-(Subscript[b, n]))/2)∧ (Subscript[c, 0])=(Sqrt[(Subscript[a, 0]^2)-(Subscript[b, 0]^2)])

History

– J. Landen (1771,1775)
– J.-L. Lagrange (1784-85)
– C .F. Gauss (1791–1799, 1800, 1876); Gauss (1800) derived the relation to  2(Subscript[F, 1])(a,b;c;z)

Applications include fast high-precision computation of π,log(z),(e^z),sin(z),cos(z) , etc.

Copyright