ArithmeticGeometricMean

ArithmeticGeometricMean[a,b]

gives the arithmetic‐geometric mean of a and b.

Details

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 TemplateBox[{2, z}, ArithmeticGeometricMean]:

Plot the imaginary part of TemplateBox[{2, z}, ArithmeticGeometricMean]:

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:

TemplateBox[{1, x}, ArithmeticGeometricMean] is nondecreasing on its real domain:

TemplateBox[{1, x}, ArithmeticGeometricMean] is injective:

TemplateBox[{1, x}, ArithmeticGeometricMean] is not surjective:

TemplateBox[{1, x}, ArithmeticGeometricMean] is non-negative on its real domain:

TemplateBox[{{-, 1}, x}, ArithmeticGeometricMean] is non-positive on its real domain:

TemplateBox[{1, x}, ArithmeticGeometricMean] 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:

Wolfram Research (1988), ArithmeticGeometricMean, Wolfram Language function, https://reference.wolfram.com/language/ref/ArithmeticGeometricMean.html (updated 2022).

Text

Wolfram Research (1988), ArithmeticGeometricMean, Wolfram Language function, https://reference.wolfram.com/language/ref/ArithmeticGeometricMean.html (updated 2022).

CMS

Wolfram Language. 1988. "ArithmeticGeometricMean." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/ArithmeticGeometricMean.html.

APA

Wolfram Language. (1988). ArithmeticGeometricMean. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArithmeticGeometricMean.html

BibTeX

@misc{reference.wolfram_2025_arithmeticgeometricmean, author="Wolfram Research", title="{ArithmeticGeometricMean}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/ArithmeticGeometricMean.html}", note=[Accessed: 26-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_arithmeticgeometricmean, organization={Wolfram Research}, title={ArithmeticGeometricMean}, year={2022}, url={https://reference.wolfram.com/language/ref/ArithmeticGeometricMean.html}, note=[Accessed: 26-July-2025 ]}