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RamanujanModular Equations,
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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
**p**th-order modular equation. The substantial details which are skirted here are
available in [11].

Contents
** Next:** Ramanujan's sum
**Up:**
RamanujanModular Equations,
** Previous:** Ramanujan's Solvable Modular