ArithmeticGeometricMean

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:

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 :

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: