ArithmeticGeometricMean
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.
Examples
open allclose allBasic Examples (4)
Scope (18)
Numerical Evaluation (4)
Specific Values (4)
Find a value of x for which ArithmeticGeometricMean[3,x]=1.5:
Visualization (2)
Plot the ArithmeticGeometricMean function for various orders:
Function Properties (4)
Real domain of ArithmeticGeometricMean:
Approximate function range of ArithmeticGeometricMean:
ArithmeticGeometricMean threads elementwise over lists:
TraditionalForm formatting:
Differentiation (2)
Series Expansions (2)
Find the Taylor expansion using Series:
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 :
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:
Text
Wolfram Research (1988), ArithmeticGeometricMean, Wolfram Language function, https://reference.wolfram.com/language/ref/ArithmeticGeometricMean.html.
BibTeX
BibLaTeX
CMS
Wolfram Language. 1988. "ArithmeticGeometricMean." Wolfram Language & System Documentation Center. Wolfram Research. 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