0028: Part 7, Singular Moduli and Dedekind eta function

 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(kd).     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 2F1(a,b;c;z),     using the amazing radical,   δ = (√15-√14)2 (8-3√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 = 29-2√210   and {x, y} = {29, 2} is the fundamental solution to the Pell equation x2-210y2 = 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 j-function, η(τ) is the Dedekind eta function, and,     for p = {1, 2, 3, 4}, respectively.  (The variables {α, β, γ, δ} and the rp(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 r1(τ) = 2553.   p = 2, τ = (1/2)√-58 and r2(τ) = 3964.   p = 3, τ = (1/3)√-15 and r3(τ) = 153.   p = 4, τ = (1/4)√-28 and r4(τ) = 212.   such that the moduli are,     Ramanujan’s example is then p = 4, τ = (1/4)√-840, and its r4(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 rp(τ) may be zero. For example, for p = 1, then at least d > 3, since for d = 3, it is well-known that r1(τ) = j(τ) = 0.  But, say, for d = 163, then,     where,     and x is the real root of the cubic, x3-6x2+4x-2 = 0.  Similarly, for p = 2, and d = 37,     where,         -- End --     © Tito Piezas III, Dec 2011 You can email author at tpiezas@gmail.com.       ◄