Modular lambda function: Difference between revisions

Revision as of 19:56, 7 March 2022

Latest revision as of 15:53, 9 February 2025

(10 intermediate revisions by 6 users not shown)

Line 1:

{{short description|Symmetric holomorphic function}}

[[File:Modular lambda function in range -3 to 3.png|thumb|Modular lambda function in the complex plane.]]

In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[holomorphic function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup|congruence group]] &Gamma;(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [&minus;1] involution.

The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)|nome]], is given by:

Line 55 ⟶ 56:

which is the ''j''-invariant of the elliptic curve of [[Legendre form]] <math>y^2=x(x-1)(x-\lambda)</math>

Given <math>m\in\mathbb{C}\setminus\{0,1\}</math>, let

:<math>\tau=i\frac{K\{1-m\}}{K\{m\}}</math>

where <math>K</math> is the [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]] with parameter <math>m=k^2</math>.

Then

:<math>\lambda (\tau)=m.</math>

==Modular equations==

Line 68 ⟶ 75:

:<math>\lambda (ni)=\prod_{k=1}^{n/2} \operatorname{sl}^8\frac{(2k-1)\varpi}{2n}\quad (n\,\text{even})</math>

:<math>\lambda (ni)=\frac{1}{2^n}\prod_{k=1}^{n-1} \left(1-\operatorname{sl}^2\frac{k\varpi}{n}\right)^2\quad (n\,\text{odd})</math>

where <math>\operatorname{sl}</math> is the [[Lemniscate elliptic functions|lemniscate sine]] and <math>\varpi</math> is the [[Lemniscate elliptic functions#Lemniscate constant|lemniscate constant]].

==Lambda-star==

Line 74 ⟶ 81:

===Definition and computation of lambda-star===

The function <math>\lambda^*(x)</math><ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 152</ref> (forwhere <math>x\in\mathbb{R}^+</math>) gives the value of the elliptic modulus <math>k</math>, for which the [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]] <math>K(k)</math> and its complementary counterpart <math>K(\sqrt{1-k^2})</math> are related by following expression:

:<math>\frac{K\left[\sqrt{1-\lambda^*(x)^2}\right]}{K[\lambda^*(x)]} = \sqrt{x}</math>

Line 96 ⟶ 103:

:<math>\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+.</math>

<math>K(\lambda^*(x))</math> and <math>E(\lambda^*(x))</math> (the [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integral of the second kind]]) can be expressed in closed form in terms of the [[gamma function]] for any <math>x\in\mathbb{Q}^+</math>, as Selberg and Chowla proved in 1949.<ref>{{Cite web|url=https://www.semanticscholar.org/paper/On-Epstein's-Zeta-Function-(I).-Chowla-Selberg/87dc02200853b431bfa900e297cd6c2f80a5a4b1journal|title=On Epstein's Zeta Function (I).|last1=Chowla|first1=S.|last2=Selberg|first2=A.|websitejournal=Proceedings of the National Academy of Sciences |date=1949 |volume=Semantic35 |issue=7 Scholar|page=373|doi=10.1073/PNAS.35.7.371 |s2cid=45071481 |doi-access=free|pmc=1063041}}</ref><ref>{{Cite web|url=https://eudml.org/doc/150803|title=On Epstein's Zeta-Function|last1=Chowla|first1=S.|last2=Selberg|first2=A.|website=EuDML|pagepages=86–110}}</ref>

The following expression is valid for all <math>n \in \mathbb{N}</math>:

Line 120 ⟶ 127:

:<math>\lambda^*(x) - \lambda^*(9x) = 2[\lambda^*(x)\lambda^*(9x)]^{1/4} - 2[\lambda^*(x)\lambda^*(9x)]^{3/4}</math>

<math display=block>\begin{align}

:<math>\left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/2} - \left[\frac{2\lambda^*(25x)}{1-\lambda^*(25x)^2}\right]^{1/2} = 2\left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\left[\frac{2\lambda^*(25x)}{1-\lambda^*(25x)^2}\right]^{1/12} + 2\left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{5/12}\left[\frac{2\lambda^*(25x)}{1-\lambda^*(25x)^2}\right]^{5/12} </math>

& a^{6}-f^{6} = 2af +2a^5f^5\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(f = \left[\frac{2\lambda^*(25x)}{1-\lambda^*(25x)^2}\right]^{1/12}\right) \\

:<math> &a^{8}+b^{8}-7a^4b^4 = 2\sqrt{2}ab+2\sqrt{2}a^7b^7\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(b = \left[\frac{2\lambda^*(49x)}{1-\lambda^*(49x)^2}\right]^{1/12}\right) </math>\\

:<math>& a^{12}-c^{12} = 2\sqrt{2}(ac+a^3c^3)(1+3a^2c^2+a^4c^4)(2+3a^2c^2+2a^4c^4)\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(c = \left[\frac{2\lambda^*(121x)}{1-\lambda^*(121x)^2}\right]^{1/12}\right) </math>\\

:<math> & (a^2-d^2)(a^4+d^4-7a^2d^2)[(a^2-d^2)^4-a^2d^2(a^2+d^2)^2] = 8ad+8a^{13}d^{13}\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(d = \left[\frac{2\lambda^*(169x)}{1-\lambda^*(169x)^2}\right]^{1/12}\right) </math>

\end{align}

</math>

{{Collapse top|title=Special values}}