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===Definition and computation of lambda-star===
The function λ*(x) gives the value of the elliptic modulus k, for which the complete [[elliptic integral]] of the first kind K(k) and its complementary counterpart <math>K
:<math>\frac{K\left[\sqrt{1-\lambda^*(x)^2}\right]
The values of λ*(x) can be computed as follows:
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By knowing one λ*-value, this formula can be used to computate related λ*-values:
:<math>\lambda^*(n^2x) = \lambda^*(x)^n\prod_{a=1}^{n}\operatorname{sn}\left\{\frac{2a-1}{n}K[\lambda^*(x)];\lambda^*(x)\right\}^2 </math>
In that formula, sn is the Jacobi elliptic function sinus amplitudinis.
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:<math>[\lambda^*(x)+1][\lambda^*(4/x)+1] = 2 </math>
:<math>\lambda^*(4x) = \frac{1-\sqrt{1-\lambda^*(x)^2}}{1+\sqrt{1-\lambda^*(x)^2}} = \tan\left\{\frac{1}{2}\arcsin[\lambda^*(x)]\right\}^2 </math>
:<math>\lambda^*(x) - \lambda^*(9x) = 2[\lambda^*(x)\lambda^*(9x)]^{1/4} - 2[\lambda^*(x)\lambda^*(9x)]^{3/4}</math>
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These are the relations between lambda-star and the Ramanujan-G-function:
:<math>G(x) = \sin\{2\arcsin[\lambda^*(x)]\}^{-1/12} = 1/\left[\sqrt[12]{2\lambda^*(x)}\sqrt[24]{1-\lambda^*(x)^2}\right] </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\left\{ \frac{1}{2}\arctan[g(x)^{-12}]\right\} = \sqrt{g(x)^{24}+1}-g(x)^{12} </math>
===Special Values===
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:<math>\lambda^*(1) = \frac{1}{\sqrt{2}}</math>
:<math>\lambda^*(5) = \sin\left[\frac{1}{2}\arcsin\left(\sqrt{5}-2\right)\right]</math>
:<math>\lambda^*(9) = \frac{1}{2}(\sqrt{3}-1)(\sqrt{2}-\sqrt[4]{3})</math>
:<math>\lambda^*(13) = \sin\left[\frac{1}{2}\arcsin(5\sqrt{13}-18)\right]</math>
:<math>\lambda^*(17) = \sin\left\{\frac{1}{2}\arcsin\left[\frac{1}{64}\left(5+\sqrt{17}-\sqrt{10\sqrt{17}+26}\right)^3\right]\right\}</math>
:<math>\lambda^*(21) = \sin\left\{\frac{1}{2}\arcsin[(8-3\sqrt{7})(2\sqrt{7}-3\sqrt{3})]\right\}</math>
:<math>\lambda^*(25) = \frac{1}{\sqrt{2}}(\sqrt{5}-2)(3-2\sqrt[4]{5})</math>
:<math>\lambda^*(33) = \sin\left\{\frac{1}{2}\arcsin[(10-3\sqrt{11})(2-\sqrt{3})^3]\right\}</math>
:<math>\lambda^*(37) = \sin\left\{\frac{1}{2}\arcsin[(\sqrt{37}-6)^3]\right\}</math>
:<math>\lambda^*(45) = \sin\left\{\frac{1}{2}\arcsin[(4-\sqrt{15})^2(\sqrt{5}-2)^3]\right\}</math>
:<math>\lambda^*(49) = \frac{1}{4}(8+3\sqrt{7})(5-\sqrt{7}-\sqrt[4]{28})\left(\sqrt{14}-\sqrt{2}-\sqrt[8]{28}\sqrt{5-\sqrt{7}}\right)</math>
:<math>\lambda^*(57) = \sin\left\{\frac{1}{2}\arcsin[(170-39\sqrt{19})(2-\sqrt{3})^3]\right\}</math>
:<math>\lambda^*(73) = \sin\left\{\frac{1}{2}\arcsin\left[\frac{1}{64}\left(45+5\sqrt{73}-3\sqrt{50\sqrt{73}+426}\right)^3\right]\right\}</math>
Lambda-star-values of integer numbers of 4n-2-type:
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:<math>\lambda^*(10) = (\sqrt{10}-3)(\sqrt{2}-1)^2</math>
:<math>\lambda^*(14) = \tan\left\{\frac{1}{2}\arctan\left[\frac{1}{8}\left(2\sqrt{2}+1-\sqrt{4\sqrt{2}+5}\right)^3\right]\right\}</math>
:<math>\lambda^*(18) = (\sqrt{2}-1)^3(2-\sqrt{3})^2</math>
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:<math>\lambda^*(22) = (10-3\sqrt{11})(3\sqrt{11}-7\sqrt{2})</math>
:<math>\lambda^*(30) = \tan\left\{\frac{1}{2}\arctan[(\sqrt{10}-3)^2(\sqrt{5}-2)^2]\right\}</math>
:<math>\lambda^*(34) = \tan\left\{\frac{1}{4}\arcsin\left[\frac{1}{9}(\sqrt{17}-4)^2\right]\right\}</math>
:<math>\lambda^*(42) = \tan\left\{\frac{1}{2}\arctan[(2\sqrt{7}-3\sqrt{3})^2(2\sqrt{2}-\sqrt{7})^2]\right\}</math>
:<math>\lambda^*(46) = \tan\left\{\frac{1}{2}\arctan\left[\frac{1}{64}\left(3+\sqrt{2}-\sqrt{6\sqrt{2}+7}\right)^6\right]\right\}</math>
:<math>\lambda^*(58) = (13\sqrt{58}-99)(\sqrt{2}-1)^6</math>
:<math>\lambda^*(70) = \tan\left\{\frac{1}{2}\arctan[(\sqrt{5}-2)^4(\sqrt{2}-1)^6]\right\}</math>
:<math>\lambda^*(78) = \tan\left\{\frac{1}{2}\arctan[(5\sqrt{13}-18)^2(\sqrt{26}-5)^2]\right\}</math>
:<math>\lambda^*(82) = \tan\left\{\frac{1}{4}\arcsin\left[\frac{1}{4761}(8\sqrt{41}-51)^2\right]\right\}</math>
Lambda-star-values of integer numbers of 4n-1-type:
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:<math>\lambda^*(7) = \frac{1}{4\sqrt{2}}(3-\sqrt{7})</math>
:<math>\lambda^*(11) = \frac{1}{8\sqrt{2}}(\sqrt{11}+3)\left(\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\right)^4</math>
:<math>\lambda^*(15) = \frac{1}{8\sqrt{2}}(3-\sqrt{5})(\sqrt{5}-\sqrt{3})(2-\sqrt{3})</math>
:<math>\lambda^*(19) = \frac{1}{8\sqrt{2}}(3\sqrt{19}+13)\left[\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})\right]^4</math>
:<math>\lambda^*(23) = \frac{1}{16\sqrt{2}}(5+\sqrt{23})\left[\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}\right]^4</math>
:<math>\lambda^*(27) = \frac{1}{16\sqrt{2}}(\sqrt{3}-1)^3\left[\frac{1}{3}\sqrt{3}(\sqrt[3]{4}-\sqrt[3]{2}+1)-\sqrt[3]{2}+1\right]^4</math>
:<math>\lambda^*(39) = \sin\left\{\frac{1}{2}\arcsin\left[\frac{1}{16}\left(6-\sqrt{13}-3\sqrt{6\sqrt{13}-21}\right)\right]\right\}</math>
:<math>\lambda^*(55) = \sin\left\{\frac{1}{2}\arcsin\left[\frac{1}{512}\left(3\sqrt{5}-3-\sqrt{6\sqrt{5}-2}\right)^3\right]\right\}</math>
Lambda-star-values of integer numbers of 4n-type:
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:<math>\lambda^*(4) = (\sqrt{2}-1)^2</math>
:<math>\lambda^*(8) = \left(\sqrt{2}+1-\sqrt{2\sqrt{2}+2}\right)^2</math>
:<math>\lambda^*(12) = (\sqrt{3}-\sqrt{2})^2(\sqrt{2}-1)^2</math>
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:<math>\lambda^*(16) = (\sqrt{2}+1)^2(\sqrt[4]{2}-1)^4</math>
:<math>\lambda^*(20) = \tan\left[\frac{1}{4}\arcsin(\sqrt{5}-2)\right]^2</math>
:<math>\lambda^*(24) = \tan\left\{\frac{1}{2}\arcsin[(2-\sqrt{3})(\sqrt{3}-\sqrt{2})]\right\}^2</math>
:<math>\lambda^*(28) = (2\sqrt{2}-\sqrt{7})^2(\sqrt{2}-1)^4</math>
:<math>\lambda^*(32) = \tan\left\{\frac{1}{2}\arcsin\left[\left(\sqrt{2}+1-\sqrt{2\sqrt{2}+2}\right)^2\right]\right\}^2</math>
Lambda-star-values of rational fractions:
:<math>\lambda^*\left(\frac{1}{2}\right) = \sqrt{2\sqrt{2}-2}</math>
:<math>\lambda^*\left(\frac{1}{3}\right) = \frac{1}{2\sqrt{2}}(\sqrt{3}+1)</math>
:<math>\lambda^*\left(\frac{2}{3}\right) = (2-\sqrt{3})(\sqrt{3}+\sqrt{2})</math>
:<math>\lambda^*\left(\frac{1}{4}\right) = 2\sqrt[4]{2}(\sqrt{2}-1)</math>
:<math>\lambda^*\left(\frac{3}{4}\right) = \sqrt[4]{8}(\sqrt{3}-\sqrt{2})(\sqrt{2}+1)\sqrt{(\sqrt{3}-1)^3}</math>
:<math>\lambda^*\left(\frac{1}{5}\right) = \frac{1}{2\sqrt{2}}\left(\sqrt{2\sqrt{5}-2}+\sqrt{5}-1\right)</math>
:<math>\lambda^*\left(\frac{2}{5}\right) = (\sqrt{10}-3)(\sqrt{2}+1)^2</math>
:<math>\lambda^*\left(\frac{3}{5}\right) = \frac{1}{8\sqrt{2}}(3+\sqrt{5})(\sqrt{5}-\sqrt{3})(2+\sqrt{3})</math>
:<math>\lambda^*\left(\frac{4}{5}\right) = \tan\left[\frac{\pi}{4}-\frac{1}{4}\arcsin(\sqrt{5}-2)\right]^2</math>
== Other appearances ==
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===Moonshine===
The function <math>\tau\mapsto\frac{16}{\lambda(2\tau)} - 8</math> is the normalized [[Hauptmodul]] for the group <math>\Gamma_0(4)</math>, and its ''q''-expansion <math>q^{-1} + 20q - 62q^3 + \dots</math>, {{oeis|id=A007248}} where <math>q=e^{2\pi i\tau }</math>, is the graded character of any element in conjugacy class 4C of the [[monster group]] acting on the [[monster vertex algebra]].
==Footnotes==
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