Modular lambda function: Difference between revisions

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Revision as of 08:24, 27 October 2021

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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 '''modular lambda''' function &lambda;λ(&tau;τ) 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;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:

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They all can be expressed by polynomials of the [[gamma function]], as Selberg and Chowla proved in 1967.

Following expression is valid for all n ∈ ℕ<math>\mathbb{N}</math>:

:<math>\sqrt{n} = \sum_{a = 1}^{n} \operatorname{dn}\left[\frac{2a}{n}K\left[\lambda^*\left(\frac{1}{n}\right)\right];\lambda^*\left(\frac{1}{n}\right)\right] </math>

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===Ramanujan's class invariants===

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

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

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