ArithmeticGeometricMean

ArithmeticGeometricMean[a,b]

gives the arithmetic‐geometric mean of a and b.

Details

For certain special arguments, ArithmeticGeometricMean automatically evaluates to exact values.
ArithmeticGeometricMean can be evaluated to arbitrary numerical precision.
ArithmeticGeometricMean is a homogeneous function of a and b, and has a branch cut discontinuity in the complex plane, with a branch cut running from to 0.
ArithmeticGeometricMean automatically threads over lists.
ArithmeticGeometricMean can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope (25)

Numerical Evaluation (5)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Specific Values (4)

Values at fixed points:

Value at zero:

Evaluate symbolically:

Find a value of x for which ArithmeticGeometricMean[3,x]=1.5:

Visualization (2)

Plot the ArithmeticGeometricMean function for various orders:

Plot the real part of :

Plot the imaginary part of :

Function Properties (10)

Real domain of ArithmeticGeometricMean:

Complex domain:

ArithmeticGeometricMean achieves all real values:

ArithmeticGeometricMean threads elementwise over lists:

ArithmeticGeometricMean is not an analytic function:

It has both singularities and discontinuities:

is nondecreasing on its real domain:

is injective:

is not surjective:

is non-negative on its real domain:

is non-positive on its real domain:

is concave on its real domain:

TraditionalForm formatting:

Differentiation (2)

First derivative with respect to b:

Higher derivatives with respect to b:

Plot the higher derivatives with respect to b when a=3:

Series Expansions (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Applications (4)

Explicit form of the iterations yielding the arithmetic‐geometric mean:

Compare with ArithmeticGeometricMean:

Functional implementation of the previous iterative procedure:

Closed form of the iteration steps for calculating the arithmetic‐geometric mean expressed through ArithmeticGeometricMean:

Show convergence speed using arbitrary‐precision arithmetic:

Compute a thousand digits of :

Compute Gauss's constant:

Compare to its expression in terms of beta function:

Plot the absolute value in the parameter plane:

Properties & Relations (3)

Derivatives of ArithmeticGeometricMean:

Use FunctionExpand to expand ArithmeticGeometricMean to other functions:

Show that ArithmeticGeometricMean obeys a hypergeometric‐type differential equation:

Proof that iterations lie between the arithmetic and the geometric means:

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