0026: Part 5, Complete Elliptic Integral of the First Kind

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Part 5: Complete Elliptic Integral of the First Kind K(k) and the Moonshine Functions

By Tito Piezas III

Abstract: Infinite series using the reciprocals of the j-function and other moonshine functions can be used to find values of the complete elliptic integral of the first kind.

I. Introduction

II. Some Formulas for the Complete elliptic integral of the first kind K(k)

III. Gamma functions

IV. Elliptic Modulus

V. Moonshine Functions

VI. Conjectures

VII. Some Remarks

I. Introduction

In Part 4: Watson’s Triple Integrals and Ramanujan-Type Pi Formulas, it was discussed how the j-function and other moonshine functions can be used to express the three beautiful Watson triple integrals I_i. For example, given I₂,

it’s long known that it evaluates to,

where K(k_d) is the complete elliptic integral of the first kind. However, it can also be expressed, among others, by the j-function j(√-3) = 2∙30³ as,

One can easily calculate these values in Mathematica or http://www.wolframalpha.com/ using the commands,

K(k₃) := EllipticK[ModularLambda[√-3]]

j(√-3) := 1728KleinInvariantJ[√-3]

The fact that the Watson triple integrals are just special values (which used d = 1,3,6) of a family was the clue that the formulas using the j-function and others in Part 4 were not so much about the triple integrals, but in fact were more about the K(k_d). Thus. there should be similar formulas for other d. In fact there are, and some interesting ones are,

where φ is the golden ratio, T is the tribonacci constant, and P is the plastic constant.

II. Some Formulas for the Complete Elliptic Integral of the First Kind K(k)

A. Class number h(-d) = 1

Recall that,

where (a)_n is the Pochhammer symbol. Using the j-function j(τ), then,

where u(τ) is the Dedekind eta quotient,

with,

One can see the j(τ) in the denominators, for example,

j((1+√-163)/2) = 1728KleinInvariantJ[(1+√-163)/2] = -640320³

and easily discern a pattern which will be given in the Conjectures section. (Note: It is understood that all appearances of "tau", τ, in this article refer to the form given above.) For positive integer d, u(τ) is a real algebraic number. For odd d with class number h(-d) = 1, (but with the exception when d = 7), then u(τ) is the real root of simple cubics with small coefficients, namely,

(Table 1)

For the special case d = 11, then x = (T+1)/T where T is the tribonacci constant. (See also, A Tale of Four Constants by this author.) These cubic roots x have the beautiful continued fraction,

where q is the real number q = -exp(-π√d) = -1/e^π√d. In general, the cfrac is valid for u(τ) and τ as defined above. Furthermore, for d with h(-d) = 1, especially the three largest d, then the 24th power of x is a near-integer. In particular,

B. Class number h(-4d) = 2

Let,

The three formulas below, however, involve another moonshine function.

which now uses the cube of u(τ),

The values of the u(τ) are easily calculated as,

(Table 2)

Consistent with the results of Table 1, then it is also the case that,

III. Gamma Functions

It’s long been proven that K(k), if k has a singular value, then K(k) can be expressed in terms of the gamma function. Focusing only on the infinite series part given above, then for odd fundamental d > 3 with class number h(-d) = 1, this is simply,

where the exponent of the gamma function Γ(n) is the Kronecker symbol. This can be implemented in Mathematica as KroneckerSymbol[-d,m]. Since it yields only the values {-1,0,1}, then it determines whether Γ(n) is in the numerator, denominator, or if it vanishes. For example, for d = 7, then,

where the simplification results from the fact that,

and similarly for d = {11, 19, 43, 67, 163}. Of course, after taking the square root of the infinite series, one can then get the K(k_d),

where the x_d are given in Table 1 above, and the 81 numerators N for d = 163 are,

N₁₆₃ = {1, 4, 6, 9, 10, 14, 15, 16, 21, 22, 24, 25, 26, 33, 34, 35, 36, 38, 39, 40, 41, 43, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 64, 65, 69, 71, 74, 77, 81, 83, 84, 85, 87, 88, 90, 91, 93, 95, 96, 97, 100, 104, 111, 113, 115, 118, 119, 121, 126, 131, 132, 133, 134, 135, 136, 140, 143, 144, 145, 146, 150, 151, 152, 155, 156, 158, 160, 161}

IV. The Elliptic Modulus

The complete elliptic integral of the first kind K(k) is defined for 0 < k < 1 by,

or equivalently by the infinite series,

where k is the elliptic modulus, and ₂F₁(a,b;c;x) is the hypergeometric function. The complementary modulus is k’ = √(1-k²), where 0 < k’ < 1. Let d be a positive integer, then the equation,

defines a unique real number k_d called the singular modulus. In Mathematica, when k has a singular value, then k² can be given by the elliptic lambda function, λ(n),

k² = λ(n) = ModularLambda[n]

for n = √-d. Exact values of k_d for small values of d are given in the link above and labeled as λ*(d). For example, the first four k_d are,

k₁ = λ*(1) = 1/√2

k₂ = λ*(2) = -1+√2

k₃ = λ*(3) = ¼ (-1+√3)√2

k₄ = λ*(4) = 3-2√2

K(k) can then be calculated in terms of the parameter m = k² as: K(k) = EllipticK[m] = EllipticK[ModularLambda[n]].

Example: Let n = √-4, and we find that,

k² = λ(n) = ModularLambda[√-4] = (3-2√2)²

hence k₄ = 3-2√2 and,

K(k₄) = K(3-2√2) = EllipticK[(3-2√2)²] = (1+√2) Γ²[1/4] / (2^7/2√π) = 1.58255…

Exact values of other K(k_d) are given in the page, Elliptic integral singular value.

V. Moonshine Functions

Let q = e^2πiτ = exp(2πiτ) and, as before, define four of the moonshine functions as,

r_1A(τ) = j(τ) = 1/q + 744 + 196884q + 21493760q² + 864299970q³ + … (A007240)

r_2A(τ) = (f₂ + 64)² / f₂ = 1/q + 104 + 4372q + 96256q² + 1240002q³ + … (A101558)

r_3A(τ) = (f₃ + 27)² / f₃ = 1/q + 42 + 783q + 8672q² + 65367q³ + … (A030197)

r_4A(τ) = (f₄ + 16)² / f₄ = 1/q + 24 + 276q + 2048q² + 11202q³ + … (A097340)

with,

f₂ = (η(τ)/η(2τ))²⁴

f₃ = (η(τ)/η(3τ))¹²

f₄ = (η(τ)/η(4τ))⁸

Also define,

h₁ = (6n)! / ((3n)! n!³)

h₂ = (4n)! / (n!⁴)

h₃ = (2n)!(3n)! / (n!⁵)

h₄ = (2n)!³ / (n!⁶)

or, equivalently,

h₁ = 1728ⁿ (1/2)_n (1/6)_n (5/6)_n / (n!³)

h₂ = 256ⁿ (1/2)_n (1/4)_n (3/4)_n / (n!³)

h₃ = 108ⁿ (1/2)_n (1/3)_n (2/3)_n / (n!³)

h₄ = 64ⁿ (1/2)_n (1/2)_n (1/2)_n / (n!³)

VI. Conjectures

Given the moonshine functions defined above and still let,

with d a positive integer above a bound. Then, we conjecture that,

where,

For j(τ), r_2A(τ), r_4A(τ), then d ≥ {7, 4, 2} respectively, otherwise the series diverges. Unfortunately, I have not yet been able to find the correct formulation using r_3A. On the other hand, there is a much simpler form using only the 24th power of u(τ) and h₄. Given a 48th root of unity ζ₄₈ = exp(2π i/48) = exp(π i/24), then,

for d ≥ 1, nice examples of which for d = {5, 11, 23} were given in the Introduction. It’s been known since Watson (1908) that, given the complementary modulus, k’, then,

where 0 < k ≤ 1/√2. But this can also be expressed as,

By comparing the two formulas, it must be the case that,

where, as in the rest of this article,

Note: As pointed out by Michael Somos, the sequence A002897 generated by [(2n)! / n!²]³ = [C(2n,n)]³ = {1, 8, 216, 8000, 343000,...}, or the cube of the central binomial coefficients, can also be interpreted as the expansion of [K(k) / (π/2)]² in powers of x = (k k'/4)², or,

[K(k) / (π/2)]² = 1+ 8x + 216x² + 8000x³ + 343000x⁴…

VII. Some Remarks

Of the four moonshine functions mentioned, only r_3A(τ) remains unformulated in relation to K(k). It seems in the formula,

that x, like for the other three functions, is always an algebraic number for integer d above a bound, in this case, d ≥ 3. For example, for d = 3, then,

and similarly for other d, though a closed-form expression for general x has not yet been found. However, it turns out that r_3A(τ) may yet have a role in another function, one similar to K(k). But that’s another story.

-- End --

© Tito Piezas III, Aug 2011

You can email author at tpiezas@gmail.com.

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