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ArithmeticGeometricMean

ArithmeticGeometricMean
gives the arithmetic-geometric mean of a and b.
  • 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 from to 0.
Evaluate numerically:
Evaluate numerically:
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Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Simple exact values are generated automatically:
ArithmeticGeometricMean threads element-wise over lists:
TraditionalForm formatting:
Explicit form of the iterations yielding the arithmetic-geometric mean:
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 :
Plot the absolute value in the parameter plane:
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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