Modular lambda function: Difference between revisions

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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’s class invariants, Kronecker’s limit formula and modular equations (III)"</ref>

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

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

where <math>n\in\mathbb{Q}^+</math>.

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

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

:<math>g(x)g_n = \tan\{2\arctan[\lambda^*(xn)]\}^{-1/12} = \sqrt[12]{[1-\lambda^*(xn)^2]/[2\lambda^*(xn)]} </math>

:<math>\lambda^*(xn) = \tan\left\{ \frac{1}{2}\arctan[g(x)g_n^{-12}]\right\} = \sqrt{g(x)g_n^{24}+1}-g(x)g_n^{12} </math>

===Special Values===