If and ThenIn 1799, Gauss observed this purely numerically and wrote that this result
This is equivalent to the identities
These are modular forms. So for example
and then the common limit is
The convergence is cubic.
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:
See Some cubic modular identities of Ramanujan by J. Borwein, P. Borwein & F. Garvan in Trans. A.M.S. 343 (1994) 35-47.
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.
Let and Then the common limit is