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Modular Equations and Pi

We wish to connect the modular equations of Theorem 1 to

. This we contrive via the function alpha defined by:

where

This allows one to rewrite Legendre's equation (5.3) in a one-sided form without the conjugate variable as

We have suppressed, and will continue to suppress, the k variable. With (5.6) and (5.7) at hand we can write a q-expansion for

, namely,

and we can see that as r tends to infinity

tends to zero and

tends to

. In fact

The key now is iteratively to calculate

. This is the content of the next theorem.

Theorem 3

Let , and as in (5.13). Then

where represents the full derivative of with respect to . In particular, is algebraic for rational arguments.

We know that is related via to and we know that is related via differentiation to K. (See (5.7) and (5.13).) Note that corresponds to . Thus from (7.17) some relation like that of the above theorem must exist. The actual derivation requires some careful algebraic manipulation. (See [11]), where it has also been made entirely explicit for and 7, and where numerous algebraic values are determined for Examples .) In the case we can specialize with some considerable knowledge of quintic modular equations to get:

Theorem 4

Let . Then

This couples with Ramanujan's quintic modular equation to provide a derivation of Algorithm 2.

Algorithm 2 results from specializing Theorem 3 with and coupling it with a quartic modular equation. The quartic equation in question is just two steps of the corresponding quadratic equation which is Legendre's form of the arithmetic geometric mean iteration , namely:

An algebraic p-th order algorithm for is derived from coupling Theorem 3 with a pth-order modular equation. The substantial details which are skirted here are available in [11].

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