gives the arithmetic‐geometric mean of a and b.
- 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. »
Examplesopen allclose all
Basic Examples (4)
Numerical Evaluation (5)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
ArithmeticGeometricMean can be used with Interval and CenteredInterval objects:
Specific Values (4)
Find a value of x for which ArithmeticGeometricMean[3,x]=1.5:
Plot the ArithmeticGeometricMean function for various orders:
Function Properties (10)
Real domain of ArithmeticGeometricMean:
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 non-negative on its real domain:
is non-positive on its real domain:
is concave on its real domain:
Series Expansions (2)
Find the Taylor expansion using Series:
Plots of the first three approximations around :
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 :
Compare to its expression in terms of beta function:
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).
Wolfram Language. 1988. "ArithmeticGeometricMean." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/ArithmeticGeometricMean.html.
Wolfram Language. (1988). ArithmeticGeometricMean. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArithmeticGeometricMean.html