gives the arithmeticgeometric mean of a and b.



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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  (18)

Numerical Evaluation  (4)

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:

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, {x, +, {i,  , y}}}, ArithmeticGeometricMean]:

Plot the imaginary part of TemplateBox[{2, {x, +, {i,  , y}}}, ArithmeticGeometricMean]:

Function Properties  (4)

Real domain of ArithmeticGeometricMean:

Complex domain:

Approximate function range of ArithmeticGeometricMean:

ArithmeticGeometricMean threads elementwise over lists:

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 arithmeticgeometric mean:

Compare with ArithmeticGeometricMean:

Functional implementation of the previous iterative procedure:

Closed form of the iteration steps for calculating the arithmeticgeometric mean expressed through ArithmeticGeometricMean:

Show convergence speed using arbitraryprecision 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 hypergeometrictype differential equation:

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

Wolfram Research (1988), ArithmeticGeometricMean, Wolfram Language function,


Wolfram Research (1988), ArithmeticGeometricMean, Wolfram Language function,


@misc{reference.wolfram_2021_arithmeticgeometricmean, author="Wolfram Research", title="{ArithmeticGeometricMean}", year="1988", howpublished="\url{}", note=[Accessed: 28-September-2021 ]}


@online{reference.wolfram_2021_arithmeticgeometricmean, organization={Wolfram Research}, title={ArithmeticGeometricMean}, year={1988}, url={}, note=[Accessed: 28-September-2021 ]}


Wolfram Language. 1988. "ArithmeticGeometricMean." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (1988). ArithmeticGeometricMean. Wolfram Language & System Documentation Center. Retrieved from