Modular lambda function: Difference between revisions

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

In [[mathematics]], the '''elliptic modular lambda''' function &lambda;(&tau;) 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 &tau;, 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:

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

The function λ*(x) gives the value of the elliptic modulus k, for which the complete [[elliptic integral]] of the first kind <math>K(k)</math> and its complementary counterpart <math>K\left(\sqrt{1-k^2}\right)</math> are related by following expression:

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

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:<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>

===Ramanujan's class invariants===

These are the relations between lambda-star and the Ramanujan-G-function:

Ramanujan's class invariants are defined as<ref>Zhang, Liang-Cheng "Ramanujan’s class invariants, Kronecker’s limit formula and modular equations (III)"</ref>

:<math>G(x)=2^{-1/4}e^{\pi\sqrt{x}/24}\prod_{k=0}^\infty (1+e^{-(2k+1)\pi\sqrt{x}}),</math>

:<math>g(x)=2^{-1/4}e^{\pi\sqrt{x}/24}\prod_{k=0}^\infty (1-e^{-(2k+1)\pi\sqrt{x}}).</math>

These are the relations between lambda-star and the Ramanujan-G-function's class invariants:

:<math>G(x) = \sin\{2\arcsin[\lambda^*(x)]\}^{-1/12} = 1/\left[\sqrt[12]{2\lambda^*(x)}\sqrt[24]{1-\lambda^*(x)^2}\right] </math>