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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.