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The function

Before we can define the alpha-function we need the following classical theta functions:

The alpha-function can be defined as

As r tends to infinity we see that q tends to zero so that we have

In [8, Theorem 3, p. 215,] Borwein, Borwein and Bailey are able to express

in terms of

and various theta functions. Utilizing p-th order modular equations for the theta functions, they then are able to construct p-th order iterations that converge to

. In the next section we show how to construct p-th order iterations in a different way. Instead of a single alpha-function we construct an infinite family of functions

Contents Next: The functions Up: Approximations to via the Previous: Introduction