Modular lambda function: Difference between revisions

Revision as of 10:58, 3 May 2021

Revision as of 09:26, 4 May 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\frac{2\arctan[\lambda^*(x)]}{1-\lambda^*(x)^2}]^{1/2} - [\tan\frac{2\arctan[lambda^*(25x)}{1-\lambda^*(25x)^2}]^{1/2} = 2[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}]^{1/12}[\frac{2\lambda^*(25x)}{1-\lambda^*(25x)^2}]^{1/12} =+ 2[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}]^{5/12}[\frac{2\lambda^*(25x)}{1-\lambda^*(25x)^2}]^{5/12} </math>

:<math>= 2\tan\{2\arctan[\lambda^*(x)]\}^{1/12}\tan\{2\arctan[\lambda^*(25x49x)]\}^{1/124} + 2\tan\{2\arctan[1-\lambda^*(x)^2]\}^{51/128}\tan\{2\arctan[1-\lambda^*(25x49x)^2]\}^{51/128} = 1 </math>

:<math>a^{12}-b^{12} = 4\sqrt{2}ab+22\sqrt{2}a^3b^3+44\sqrt{2}a^5b^5+44\sqrt{2}a^7b^7+22\sqrt{2}a^9b^9+4\sqrt{2}a^{11}b^{11}\, (a = [\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}]^{1/12}) (b = [\frac{2\lambda^*(121x)}{1-\lambda^*(121x)^2}]^{1/12}) </math>

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

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