If
and
Then

In 1799, Gauss observed this purely numerically and wrote
that this result

''will surely open a whole new field of
analysis.''

# Equivalent Modular Parameterization

This is equivalent to the identities

and
where
These are modular forms. So for example

# A Cubic Analogue of the AGM

Let

and
then the common limit is

The convergence is cubic.

## Proof

: The proof is opaque. It works because
satisfies
In the above notation.

where is, as above, with,
Taking **k+1** terms of the sum and limit gives a cubically convergent
algorithm. Twenty one terms gives three billion
digits.The underlying cubic transformation is beautiful:

# Equivalent Modular Parameterization

If
and
Then
and
Note that if then
This is a cubic modular equation for **R**.
See * Some cubic modular identities of Ramanujan* by J. Borwein, P. Borwein & F. Garvan
in Trans. A.M.S. ** 343** (1994) 35-47.

# Some Explanations

where
Let

Then

and
With we get

If we can compute iteratively we can compute and by a second
linked iteration.

There are four particularly interesting cases.

Inverting the ratios
gives elliptic modular functions. Which reduces much of
this to an algebraic theory.

Finding and proving these iterations can (at least in principal) be
effected entirely computationally.

# The Quadratic s=1/4 Iteration

Let and
Then the common limit is

# Caveat Emptor:

Let
and
Then the common limit is not differentially algebraic.