Modular lambda function: Difference between revisions

Revision as of 09:15, 18 April 2021

Revision as of 09:51, 18 April 2021

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

:<math>\tan\{2\arctan[\lambda^*(x)]\}^{1/2} - \tan\{2\arctan[\lambda^*(25x)]\}^{1/2} = </math>

:<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>

:<math>= 2\tan\{2\arctan[\lambda^*(x)]\}^{1/12}\tan\{2\arctan[\lambda^*(25x)]\}^{1/12} + 2\tan\{2\arctan[\lambda^*(x)]\}^{5/12}\tan\{2\arctan[\lambda^*(25x)]\}^{5/12} </math>

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