Part 7: Four Singular Moduli and the Complete Elliptic Integral of the First Kind
by Tito Piezas III
Abstract: Formulas for four kinds of singular moduli will be given in terms of the Dedekind eta function η(τ) at arguments τ = 1/p√m. However, at τ = (1+√d)/2, these can also be used to compute the complete elliptic integral of the first kind K(k_{d}).
I. Introduction II. Four Singular Moduli III. Complete Elliptic Integral of the First Kind
I. Introduction
In his second letter to Hardy, Ramanujan solved the equation involving the hypergeometric function _{2}F_{1}(a,b;c;z),
using the amazing radical,
δ = (√15√14)^{2} (83√7)^{2} (2√3)^{2} (6√35)^{2} (1√2)^{4} (3√10)^{4} (4√15)^{4} (√7√6)^{4} ≈ 2.7066 x 10^{(19)}
Note that these involve powers of fundamental units. For example,
(√15√14)^{2} = 292√210
and {x, y} = {29, 2} is the fundamental solution to the Pell equation x^{2}210y^{2} = 1. There is a general formula for δ, and similar singular moduli, using the Dedekind eta function to be given here. (However, to express them as products of units as Ramanujan ingeniously did is another matter entirely.)
II. Singular moduli
Solutions to the following four equations,
are given by,
or equivalently,
where j(τ) is the jfunction, η(τ) is the Dedekind eta function, and,
for p = {1, 2, 3, 4}, respectively. (The variables {α, β, γ, δ} and the r_{p}(τ) are also discussed in the previous article, Part 6: General forms for Ramanujan’s pi formulas.) For example, the solution to the equations,
are given, respectively, by,
p = 1, τ = (1/1)√7 and r_{1}(τ) = 255^{3}.
p = 2, τ = (1/2)√58 and r_{2}(τ) = 396^{4}.
p = 3, τ = (1/3)√15 and r_{3}(τ) = 15^{3}.
p = 4, τ = (1/4)√28 and r_{4}(τ) = 2^{12}.
such that the moduli are,
Ramanujan’s example is then p = 4, τ = (1/4)√840, and its r_{4}(τ) is easily calculated using the Dedekind eta function, though to express it as radicals ab initio is not so easy. In general, for m a positive integer, then α, β, γ, δ, are algebraic numbers.
III. Complete Elliptic Integral of the First Kind
The complete elliptic integral of the first kind K(k) is defined for 0 < k < 1 by,
or equivalently,
But it can also be expressed in terms of α, β, δ (missing the third moduli, γ) as,
where τ is now restricted to the form,
for positive d above a bound, otherwise r_{p}(τ) may be zero. For example, for p = 1, then at least d > 3, since for d = 3, it is wellknown that r_{1}(τ) = j(τ) = 0. But, say, for d = 163, then,
where,
and x is the real root of the cubic, x^{3}6x^{2}+4x2 = 0. Similarly, for p = 2, and d = 37,
where,
 End 
© Tito Piezas III, Dec 2011 You can email author at tpiezas@gmail.com.
