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:<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>
These are the relations between lambda-star and the Ramanujan-G-function:
:<math>G(x) = \sin[2\arcsin[\lambda^*(x)]]^{-1/12} = 1/[\sqrt[12]{2\lambda^*(x)}\sqrt[24]{1-\lambda^*(x)^2}] </math>
:<math>g(x) = \tan[2\arctan[\lambda^*(x)]]^{-1/12} = \sqrt[12]{[1-\lambda^*(x)^2]/[2\lambda^*(x)]} </math>
:<math>\lambda^*(x) = \tan[\frac{1}{2}\arctan[g(x)^{-12}]] = \sqrt{g(x)^{24}+1}-g(x)^{12} </math>
===Special Values===
Lambda-star-values of integer numbers:
:<math>\lambda^*(1) = \frac{1}{\sqrt{2}}</math>
:<math>\lambda^*(2) = \sqrt{2}-1</math>
:<math>\lambda^*(3) = \frac{1}{2\sqrt{2}}(\sqrt{3}-1)</math>
:<math>\lambda^*(4) = (\sqrt{2}-1)^2</math>
:<math>\lambda^*(5) = \frac{1}{2\sqrt{2}}(\sqrt{2\sqrt{5}-2}-\sqrt{5}+1)</math>
:<math>\lambda^*(6) = (2-\sqrt{3})(\sqrt{3}-\sqrt{2})</math>
:<math>\lambda^*(7) = \frac{1}{4\sqrt{2}}(3-\sqrt{7})</math>
:<math>\lambda^*(8) = (\sqrt{2}+1)(\sqrt{\sqrt{2}+1}-\sqrt{2})^2</math>
:<math>\lambda^*(9) = \frac{1}{2}(\sqrt{3}-1)(\sqrt{2}-\sqrt[4]{3})</math>
:<math>\lambda^*(10) = (\sqrt{10}-3)(\sqrt{2}-1)^2</math>
:<math>\lambda^*(11) = \frac{1}{8\sqrt{2}}(\sqrt{11}+3)(\frac{1}{3}\sqrt[3]{6\sqrt{3}+2\sqrt{11}}-\frac{1}{3}\sqrt[3]{6\sqrt{3}-2\sqrt{11}}+\frac{1}{3}\sqrt{11}-1)^4</math>
:<math>\lambda^*(12) = (\sqrt{3}-\sqrt{2})^2(\sqrt{2}-1)^2</math>
:<math>\lambda^*(13) = \frac{1}{2\sqrt{2}}[(5+\sqrt{13})\sqrt{5\sqrt{13}-18}-5+\sqrt{13}]</math>
:<math>\lambda^*(14) = [2\sqrt{2}+2-(\sqrt{2}+1)^2\sqrt{4\sqrt{2}-5}][(\sqrt{2}+1)^2\sqrt{4\sqrt{2}-5}-\sqrt{8\sqrt{2}+10}]</math>
:<math>\lambda^*(15) = \frac{1}{8\sqrt{2}}(3-\sqrt{5})(\sqrt{5}-\sqrt{3})(2-\sqrt{3})</math>
:<math>\lambda^*(16) = (\sqrt{2}+1)^2(\sqrt[4]{2}-1)^4</math>
:<math>\lambda^*(17) = \frac{1}{8\sqrt{2}}(\sqrt{3\sqrt{17}+11}-\sqrt{5+\sqrt{17}}-\sqrt{2\sqrt{2\sqrt{17}+2}-4})^2</math>
:<math>\lambda^*(18) = (\sqrt{2}-1)^3(2-\sqrt{3})^2</math>
:<math>\lambda^*(19) = \frac{1}{8\sqrt{2}}(3\sqrt{19}+13)[\frac{1}{6}(\sqrt{19}-2+\sqrt{3})\sqrt[3]{3\sqrt{3}-\sqrt{19}}-\frac{1}{6}(\sqrt{19}-2-\sqrt{3})\sqrt[3]{3\sqrt{3}+\sqrt{19}}-\frac{1}{3}(5-\sqrt{19})]^4</math>
:<math>\lambda^*(20) = (\sqrt{10}-3)(\sqrt{5}+2)(\sqrt{2}-1)(\sqrt{\sqrt{5}-1}-1)^2</math>
:<math>\lambda^*(21) = \frac{1}{4\sqrt{2}}(\sqrt{7}-\sqrt{3})[(\sqrt{3}+1)\sqrt{2\sqrt{7}-4}-4+\sqrt{7}-\sqrt{3}]</math>
:<math>\lambda^*(22) = (10-3\sqrt{11})(3\sqrt{11}-7\sqrt{2})</math>
:<math>\lambda^*(23) = \frac{1}{16\sqrt{2}}(5+\sqrt{23})[\frac{1}{6}(\sqrt{3}+1)\sqrt[3]{100-12\sqrt{69}}-\frac{1}{6}(\sqrt{3}-1)\sqrt[3]{100+12\sqrt{69}}+\frac{2}{3}]^4</math>
:<math>\lambda^*(24) = (2+\sqrt{3})^2(\sqrt{3}+\sqrt{2})[\sqrt{\sqrt{3}+\sqrt{2}}-(\sqrt{3}-1)(\sqrt{2}+1)]^2</math>
:<math>\lambda^*(25) = \frac{1}{\sqrt{2}}(\sqrt{5}-2)(3-2\sqrt[4]{5})</math>
Lambda-star-values of rational fractions:
:<math>\lambda^*(\frac{1}{2}) = \sqrt{2\sqrt{2}-2}</math>
:<math>\lambda^*(\frac{1}{3}) = \frac{1}{2\sqrt{2}}(\sqrt{3}+1)</math>
:<math>\lambda^*(\frac{2}{3}) = (2-\sqrt{3})(\sqrt{3}+\sqrt{2})</math>
:<math>\lambda^*(\frac{1}{4}) = 2\sqrt[4]{2}(\sqrt{2}-1)</math>
:<math>\lambda^*(\frac{3}{4}) = \sqrt[4]{8}(\sqrt{3}-\sqrt{2})(\sqrt{2}+1)\sqrt{(\sqrt{3}-1)^3}</math>
:<math>\lambda^*(\frac{1}{5}) = \frac{1}{2\sqrt{2}}(\sqrt{2\sqrt{5}-2}+\sqrt{5}-1)</math>
:<math>\lambda^*(\frac{2}{5}) = (\sqrt{10}-3)(\sqrt{2}+1)^2</math>
:<math>\lambda^*(\frac{3}{5}) = \frac{1}{8\sqrt{2}}(3+\sqrt{5})(\sqrt{5}-\sqrt{3})(2+\sqrt{3})</math>
:<math>\lambda^*(\frac{4}{5}) = (\sqrt{10}+3)(\sqrt{5}+2)(\sqrt{2}+1)(\sqrt{\sqrt{5}-1}-1)^2</math>
== Other appearances ==
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