Modular lambda function: Difference between revisions

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{{short description|Symmetric holomorphic function}}

In [[mathematics]], the '''elliptic modular lambda''' function &lambda;(&tau;) is a highly symmetric holomorphic function on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup|congruence group]] &Gamma;(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point &tau;, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [&minus;1] involution.

[[File:Modular lambda function in range -3 to 3.png|thumb|Modular lambda function in the complex plane.]]

In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[holomorphic function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup|congruence group]] &Gamma;(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [&minus;1] involution.

The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)|nome]], is given by:

Line 6 ⟶ 8:

By symmetrizing the lambda function under the canonical action of the symmetric group ''S''₃ on ''X''(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group <math>\operatorname{SL}_2(\mathbb{Z})</math>, and it is in fact Klein's modular [[j-invariant]].

[[File:Lambda function.svg|thumb|A plot of x→ λ(ix)]]

==Modular properties==

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:<math> \left\lbrace { \lambda, \frac{1}{1-\lambda}, \frac{\lambda-1}{\lambda}, \frac{1}{\lambda}, \frac{\lambda}{\lambda-1}, 1-\lambda } \right\rbrace \ .</math>

==Relations to other elliptic functions==

It is the [[Square (algebra)|square]] of the [[Jacobielliptic modulus]],<ref name=C108>Chandrasekharan (1985) p.108</ref> that is, <math>\lambda(\tau)=k^2(\tau)</math>. In terms of the [[Dedekind eta function]] <math>\eta(\tau)</math> and [[theta function]]s,<ref name=C108/>

:<math> \lambda(\tau) = \Bigg(\frac{\sqrt{2}\,\eta(\tfrac{\tau}{2})\eta^2(2\tau)}{\eta^3(\tau)}\Bigg)^8 = \frac{16}{\left(\frac{\eta(\tau/2)}{\eta(2\tau)}\right)^8 + 16} =\frac{\theta_2^4(0,\tau)}{\theta_3^4(0,\tau)} </math>

and,

:<math> \frac{1}{\big(\lambda(\tau)\big)^{1/4}}-\big(\lambda(\tau)\big)^{1/4} = \frac{1}{2}\left(\frac{\eta(\tfrac{\tau}{4})}{\eta(\tau)}\right)^4 = 2\,\frac{\theta_4^2(0,\tfrac{\tau}{2})}{\theta_2^2(0,\tfrac{\tau}{2})}</math>

where<ref name=C63>Chandrasekharan (1985) p.63</ref> for the [[Nome (mathematics)|nome]] <math>q = e^{\pi i \tau}</math>,

:<math>\theta_2(0,\tau) = \sum_{n=-\infty}^\infty qe^{\leftpi i\tau ({n+\frac12}\right1/2)^2}</math>

:<math>\theta_3(0,\tau) = \sum_{n=-\infty}^\infty qe^{\pi i\tau n^2} </math>

:<math>\theta_4(0,\tau) = \sum_{n=-\infty}^\infty (-1)^n qe^{\pi i\tau n^2} </math>

In terms of the half-periods of [[Weierstrass's elliptic functions]], let <math>[\omega_1,\omega_2]</math> be a [[fundamental pair of periods]] with <math>\tau=\frac{\omega_2}{\omega_1}</math>.

:<math> e_1 = \wp\left(\frac{\omega_1}{2}\right), \quad e_2 = \wp\left(\frac{\omega_2}{2}\right),\quad e_3 = \wp\left(\frac{\omega_1+\omega_2}{2}\right) </math>

we have<ref name=C108/>

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:<math> \lambda = \frac{e_3-e_2}{e_1-e_2} \, . </math>

Since the three half-period values are distinct, this shows that λ<math>\lambda</math> does not take the value 0 or 1.<ref name=C108/>

The relation to the [[j-invariant]] is<ref name=C117>Chandrasekharan (1985) p.117</ref><ref>Rankin (1977) pp.226–228</ref>

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which is the ''j''-invariant of the elliptic curve of [[Legendre form]] <math>y^2=x(x-1)(x-\lambda)</math>

Given <math>m\in\mathbb{C}\setminus\{0,1\}</math>, let

==Elliptic modulus==

:<math>\tau=i\frac{K\{1-m\}}{K\{m\}}</math>

where <math>K</math> is the [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]] with parameter <math>m=k^2</math>.

Then

:<math>\lambda (\tau)=m.</math>

==Modular equations==

The ''modular equation of degree'' <math>p</math> (where <math>p</math> is a prime number) is an algebraic equation in <math>\lambda (p\tau)</math> and <math>\lambda (\tau)</math>. If <math>\lambda (p\tau)=u^8</math> and <math>\lambda (\tau)=v^8</math>, the modular equations of degrees <math>p=2,3,5,7</math> are, respectively,<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 103–109, 134</ref>

:<math>(1+u^4)^2v^8-4u^4=0,</math>

:<math>u^4-v^4+2uv(1-u^2v^2)=0,</math>

:<math>u^6-v^6+5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0,</math>

:<math>(1-u^8)(1-v^8)-(1-uv)^8=0.</math>

The quantity <math>v</math> (and hence <math>u</math>) can be thought of as a [[holomorphic function]] on the upper half-plane <math>\operatorname{Im}\tau>0</math>:

:<math>\begin{align}v&=\prod_{k=1}^\infty \tanh\frac{(k-1/2)\pi i}{\tau}=\sqrt{2}e^{\pi i\tau/8}\frac{\sum_{k\in\mathbb{Z}}e^{(2k^2+k)\pi i\tau}}{\sum_{k\in\mathbb{Z}}e^{k^2\pi i\tau}}\\

&=\cfrac{\sqrt{2}e^{\pi i\tau/8}}{1+\cfrac{e^{\pi i\tau}}{1+e^{\pi i\tau}+\cfrac{e^{2\pi i\tau}}{1+e^{2\pi i\tau}+\cfrac{e^{3\pi i\tau}}{1+e^{3\pi i\tau}+\ddots}}}}\end{align}</math>

Since <math>\lambda(i)=1/2</math>, the modular equations can be used to give [[Algebraic number|algebraic values]] of <math>\lambda(pi)</math> for any prime <math>p</math>.<ref group="note">For any [[prime power]], we can iterate the modular equation of degree <math>p</math>. This process can be used to give algebraic values of <math>\lambda (ni)</math> for any <math>n\in\mathbb{N}.</math></ref> The algebraic values of <math>\lambda(ni)</math> are also given by<ref name="Jacobi">{{Cite book |last1=Jacobi |first1=Carl Gustav Jacob |author-link=Carl Gustav Jacob Jacobi|title=Fundamenta nova theoriae functionum ellipticarum|language=Latin|year=1829}} p. 42</ref><ref group="note"><math>\operatorname{sl}a\varpi</math> is algebraic for every <math>a\in\mathbb{Q}.</math></ref>

:<math>\lambda (ni)=\prod_{k=1}^{n/2} \operatorname{sl}^8\frac{(2k-1)\varpi}{2n}\quad (n\,\text{even})</math>

:<math>\lambda (ni)=\frac{1}{2^n}\prod_{k=1}^{n-1} \left(1-\operatorname{sl}^2\frac{k\varpi}{n}\right)^2\quad (n\,\text{odd})</math>

where <math>\operatorname{sl}</math> is the [[Lemniscate elliptic functions|lemniscate sine]] and <math>\varpi</math> is the [[lemniscate constant]].

==Lambda-star==

===Definition and computation of lambda-star===

The function λ<math>\lambda^*(x)</math><ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 152</ref> (where <math>x\in\mathbb{R}^+</math>) gives the value of the elliptic modulus <math>k</math>, for which the complete [[Elliptic integral#Complete elliptic integral]] of the first kind|complete elliptic integral of the first kind]] <math>K(k)</math> and its complementary counterpart <math>K[sqrt(\sqrt{1-k^2})]</math> are related by following expression:

:<math>\frac{K\left[\sqrt{1-\lambda^*(x)^2}\right]/}{K[\lambda^*(x)]} = \sqrt{x}</math>

The values of λ<math>\lambda^*(x)</math> can be computed as follows:

:<math>\lambda^*(x) = \frac{\varthetatheta^2_2[0;\exp(-\pii\sqrt{x})]}{\varthetatheta^2_3[0;\exp(-\pii\sqrt{x})]} </math>

:<math>\lambda^*(x) = \left[\sum_{a=-\infty}^\infty\exp[-(a+1/2)^2\pi\sqrt{x}]\right]^2\left[\sum_{a=-\infty}^\infty\exp(-a^2\pi\sqrt{x})\right]^{-2} </math>

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:<math>\lambda^*(x) = \left[\sum_{a=-\infty}^\infty\operatorname{sech}[(a+1/2)\pi\sqrt{x}]\right]\left[\sum_{a=-\infty}^\infty\operatorname{sech}(a\pi\sqrt{x})\right]^{-1} </math>

The functions λ<math>\lambda^*</math> and λ<math>\lambda</math> are related to each other in this way:

:<math>\lambda^*(x) = \sqrt{\lambda(i\sqrt{x})}</math>

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===Properties of lambda-star===

Every λ<math>\lambda^*-</math> value of a positive [[rational number]] is a positive [[algebraic number]]:

:<math>\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+ .</math>

<math>K(\lambda^*(x))</math> and <math>E(\lambda^*(x))</math> (the [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integral of the second kind]]) can be expressed in closed form in terms of the [[gamma function]] for any <math>x\in\mathbb{Q}^+</math>, as Selberg and Chowla proved in 1949.<ref>{{Cite journal|title=On Epstein's Zeta Function (I).|last1=Chowla|first1=S.|last2=Selberg|first2=A.|journal=Proceedings of the National Academy of Sciences |date=1949 |volume=35 |issue=7 |page=373|doi=10.1073/PNAS.35.7.371 |s2cid=45071481 |doi-access=free|pmc=1063041}}</ref><ref>{{Cite web|url=https://eudml.org/doc/150803|title=On Epstein's Zeta-Function|last1=Chowla|first1=S.|last2=Selberg|first2=A.|website=EuDML|pages=86–110}}</ref>

Elliptic integrals of the first and second kind of these special λ*-values are called elliptic integral singular values.

They all can be expressed by polynomials of the [[gamma function]], as Selberg and Chowla proved in 1967.

FollowingThe following expression is valid for all <math>n ∈\in ℕ\mathbb{N}</math>:

:<math>\sqrt{n} = \sum_{a = 1}^{n} \operatorname{dn}\left[\frac{2a}{n}K\left[\lambda^*\left(\frac{1}{n}\right)\right];\lambda^*\left(\frac{1}{n}\right)\right] </math>

In this formula,where <math>\operatorname{dn}</math> is the [[Jacobi elliptic function]] delta amplitudinis with modulus <math>k</math>.

By knowing one λ<math>\lambda^*-</math> value, this formula can be used to computatecompute related λ<math>\lambda^*-</math> values:<ref name="Jacobi"/>

:<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>

Inwhere that<math>n\in\mathbb{N}</math> formula,and <math>\operatorname{sn}</math> is the Jacobi elliptic function sinus amplitudinis with modulus <math>k</math>.

That formula works for all natural numbers.

Further relations:

Line 102 ⟶ 123:

:<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>

<math display=block>\begin{align}

:<math>[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}]^{1/2} - [\frac{2\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>

& a^{6}-f^{6} = 2af +2a^5f^5\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(f = \left[\frac{2\lambda^*(25x)}{1-\lambda^*(25x)^2}\right]^{1/12}\right) \\

&a^{8}+b^{8}-7a^4b^4 = 2\sqrt{2}ab+2\sqrt{2}a^7b^7\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(b = \left[\frac{2\lambda^*(49x)}{1-\lambda^*(49x)^2}\right]^{1/12}\right) \\

:<math>& a^{12}-c^{12} = 2\sqrt{2}(ac+a^3c^3)(1+3a^2c^2+a^4c^4)(2+3a^2c^2+2a^4c^4)\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(49xx)^2}\right]^{1/412}\right) +&\left(c = \left[1-\frac{2\lambda^*(x121x)^2]^}{1/8}[1-\lambda^*(49x121x)^2}\right]^{1/812}\right) = 1 </math>\\

:<math> & (a^{12}2-bd^{12} = 4\sqrt{2}ab+22\sqrt{2})(a^3b^34+44\sqrt{d^4-7a^2d^2})[(a^5b2-d^5+44\sqrt{2})^4-a^7b2d^7+22\sqrt{2}(a^9b^92+4\sqrt{d^2}a)^2] = 8ad+8a^{1113}bd^{1113}\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(bd = \left[\frac{2\lambda^*(121x169x)}{1-\lambda^*(121x169x)^2}\right]^{1/12}\right) </math>

\end{align}

</math>

{{Collapse top|title=Special values}}

:<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>

Lambda-star values of integer numbers of 4n-3-type:

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 of 4n-3-type:

:<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) = \sin\{\frac{1}{24}(8+3\arcsin[sqrt{7})(5-\fracsqrt{17}-\sqrt[4]{828})\left(4\sqrt{714}+9-3\sqrt{2}-\sqrt[8]{28}\sqrt{5-\sqrt{7}+21}\right)^3]\}</math>

:<math>\lambda^*(57) = \sin\left\{\frac{1}{2}\arcsin[(170-39\sqrt{19})(2-\sqrt{3})^3]\right\}</math>

Lambda-star-values of integer numbers of 4n-2-type:

:<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:

:<math>\lambda^*(2) = \sqrt{2}-1</math>

Line 156 ⟶ 175:

:<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>

Line 162 ⟶ 181:

:<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}{24}\arctanarcsin\left[\frac{1}{649}(\sqrt{17}+-4)^3(15-32\sqrt{17}-right]\sqrt{38\sqrt{17}-150})^3]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>

Lambda-star-values of integer numbers of 4n-1-type:

:<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:

:<math>\lambda^*(3) = \frac{1}{2\sqrt{2}}(\sqrt{3}-1)</math>

Line 180 ⟶ 203:

:<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:

:<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>

Line 204 ⟶ 227:

:<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>

{{Collapse bottom}}

Lambda-star-values of rational fractions:

===Ramanujan's class invariants===

:<math>\lambda^*(\frac{1}{2}) = \sqrt{2\sqrt{2}-2}</math>

[[Srinivasa Ramanujan|Ramanujan's]] class invariants <math>G_n</math> and <math>g_n</math> are defined as<ref>{{cite journal |last1=Berndt |first1=Bruce C. |last2=Chan |first2=Heng Huat|last3=Zhang|first3=Liang-Cheng |date=6 June 1997 |title=Ramanujan's class invariants, Kronecker's limit formula, and modular equations|url=https://www.ams.org/journals/tran/1997-349-06/ |journal=Transactions of the American Mathematical Society|volume=349|issue=6|pages=2125–2173}}</ref>

:<math>\lambda^*(\frac{1}{3}) = \frac{1}{2\sqrt{2}}(\sqrt{3}+1)</math>

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

:<math>g_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1-e^{-(2k+1)\pi\sqrt{n}}\right),</math>

where <math>n\in\mathbb{Q}^+</math>. For such <math>n</math>, the class invariants are algebraic numbers. For example

:<math>g_{58}=\lambda^*(sqrt{\frac{25+\sqrt{29}}{32})}, =\quad (2-g_{190}=\sqrt{3})(\sqrt{35}+2)(\sqrt{210}+3)}.</math>

Identities with the class invariants include<ref>{{Cite book |last1=Eymard |first1=Pierre |last2=Lafon| first2=Jean-Pierre |title=Autour du nombre Pi |language=French|publisher=HERMANN |year=1999 |isbn=2705614435}} p. 240</ref>

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

:<math>\lambda^*(\fracG_n=G_{31/n},\quad g_{4n}) = \sqrt[4]frac{81}(\sqrt{3g_{4/n}}-,\sqrtquad g_{24n})(\sqrt{=2}+1)\sqrt^{(\sqrt{3}-1)^3/4}g_nG_n.</math>

The class invariants are very closely related to the [[Weber modular function|Weber modular functions]] <math>\mathfrak{f}</math> and <math>\mathfrak{f}_1</math>. These are the relations between lambda-star and the class invariants:

:<math>\lambda^*(\frac{1}{5}) = \frac{1}{2\sqrt{2}}(\sqrt{2\sqrt{5}-2}+\sqrt{5}-1)</math>

:<math>G_n = \sin\{2\arcsin[\lambda^*(n)]\frac{2}^{5-1/12}) = (1\Big /\left[\sqrt[12]{10}-3)2\lambda^*(n)}\sqrt[24]{2}+1-\lambda^*(n)^2}\right] </math>

:<math>g_n = \tan\{2\arctan[\lambda^*(n)]\frac{3}^{5-1/12}) = \fracsqrt[12]{[1}{8-\sqrt{lambda^*(n)^2]/[2}}(3+\sqrt{5})lambda^*(\sqrt{5}-\sqrt{3}n)(2+\sqrt{3]}) </math>

:<math>\lambda^*(\frac{4}{5}n) = \tan[\frac{left\pi}{4}- \frac{1}{42}\arcsin(arctan[g_n^{-12}]\right\} = \sqrt{5g_n^{24}+1}-2)]g_n^2{12} </math>

== Other appearances ==

Line 238 ⟶ 284:

===Moonshine===

The function <math>\frac{tau\mapsto 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==

Line 244 ⟶ 290:

==References==

===Notes===

{{reflist|group=note}}

===Other===

* {{Citation | editor1-last=Abramowitz | editor1-first=Milton | editor1-link=Milton Abramowitz | editor2-last=Stegun | editor2-first=Irene A. | editor2-link=Irene Stegun | title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | publisher=[[Dover Publications]] | ___location=New York | isbn=978-0-486-61272-0 | year=1972 | zbl=0543.33001 | url-access=registration | url=https://archive.org/details/handbookofmathe000abra }}

* {{citation | last=Chandrasekharan | first=K. | authorlink=K. S. Chandrasekharan | title=Elliptic Functions | series=Grundlehren der mathematischen Wissenschaften | volume=281 | publisher=[[Springer-Verlag]] | year=1985 | isbn=3-540-15295-4 | zbl=0575.33001 | pages=108–121 }}

Line 256 ⟶ 305:

* Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J. reine angew. Math. 227, 86-110, 1967.

==External links==

* [https://fungrim.org/topic/Modular_lambda_function/ Modular lambda function] at [https://fungrim.org/ Fungrim]

{{DEFAULTSORT:Modular Lambda Function}}