{{short description|Symmetric holomorphic function}} 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 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 group Γ(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 [−1] involution.
The q-expansion, where <math>q = e^{\pi i \tau}</math> is the nome, is given by:
: <math> \lambda(\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \dots</math>. {{oeis|id=A115977 }}
By symmetrizing the lambda function under the canonical action of the symmetric group ''S''<sub>3</sub> 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. thumb|A plot of x→ λ(ix)
==Modular properties== The function <math> \lambda(\tau) </math> is invariant under the group generated by<ref name=C115>Chandrasekharan (1985) p.115</ref>
:<math> \tau \mapsto \tau+2 \ ;\ \tau \mapsto \frac{\tau}{1-2\tau} \ . </math>
The generators of the modular group act by<ref name=C109>Chandrasekharan (1985) p.109</ref>
:<math> \tau \mapsto \tau+1 \ :\ \lambda \mapsto \frac{\lambda}{\lambda-1} \, ;</math> :<math> \tau \mapsto -\frac{1}{\tau} \ :\ \lambda \mapsto 1 - \lambda \ . </math>
Consequently, the action of the modular group on <math> \lambda(\tau) </math> is that of the anharmonic group, giving the six values of the cross-ratio:<ref name=C110>Chandrasekharan (1985) p.110</ref>
:<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 functions== It is the square of the elliptic 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 functions,<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(\tau)}{\theta_3^4(\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(\tfrac{\tau}{2})}{\theta_2^2(\tfrac{\tau}{2})}</math>
where<ref name=C63>Chandrasekharan (1985) p.63</ref>
:<math>\theta_2(\tau) =\sum_{n=-\infty}^\infty e^{\pi i\tau (n+1/2)^2}</math>
:<math>\theta_3(\tau) = \sum_{n=-\infty}^\infty e^{\pi i\tau n^2}</math>
:<math>\theta_4(\tau) = \sum_{n=-\infty}^\infty (-1)^n e^{\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/>
:<math> \lambda = \frac{e_3-e_2}{e_1-e_2} \, . </math>
Since the three half-period values are distinct, this shows that <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>
:<math> j(\tau) = \frac{256(1-\lambda(1-\lambda))^3}{(\lambda(1-\lambda))^2} = \frac{256(1-\lambda+\lambda^2)^3}{\lambda^2 (1-\lambda)^2} \ . </math>
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 :<math>\tau=i\frac{K\{1-m\}}{K\{m\}}</math> where <math>K</math> is the 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 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 sine and <math>\varpi</math> is the lemniscate constant.
==Lambda-star==
===Definition and computation of lambda-star===
The function <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 of the first kind <math>K(k)</math> and its complementary counterpart <math>K(\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 <math>\lambda^*(x)</math> can be computed as follows:
:<math>\lambda^*(x) = \frac{\theta^2_2(i\sqrt{x})}{\theta^2_3(i\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>
:<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 <math>\lambda^*</math> and <math>\lambda</math> are related to each other in this way:
:<math>\lambda^*(x) = \sqrt{\lambda(i\sqrt{x})}</math>
===Properties of lambda-star===
Every <math>\lambda^*</math> value of a positive rational number is a positive algebraic number:
:<math>\lambda^*(x) \in \overline{\mathbb{Q}}_+ \quad \forall x\in\mathbb{Q}^+.</math>
<math>K(\lambda^*(x))</math> and <math>E(\lambda^*(x))</math> (the 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>
The following expression is valid for all <math>n \in \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>
where <math>\operatorname{dn}</math> is the Jacobi elliptic function delta amplitudinis with modulus <math>k</math>.
By knowing one <math>\lambda^*</math> value, this formula can be used to compute related <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>
where <math>n\in\mathbb{N}</math> and <math>\operatorname{sn}</math> is the Jacobi elliptic function sinus amplitudinis with modulus <math>k</math>.
Further relations:
:<math>\lambda^*(x)^2 + \lambda^*(1/x)^2 = 1 </math>
:<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} & 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) \\
& 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^*(x)^2}\right]^{1/12}\right) &\left(c = \left[\frac{2\lambda^*(121x)}{1-\lambda^*(121x)^2}\right]^{1/12}\right) \\
& (a^2-d^2)(a^4+d^4-7a^2d^2)[(a^2-d^2)^4-a^2d^2(a^2+d^2)^2] = 8ad+8a^{13}d^{13}\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(d = \left[\frac{2\lambda^*(169x)}{1-\lambda^*(169x)^2}\right]^{1/12}\right) \end{align} </math>
{{Collapse top|title=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) = \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:
:<math>\lambda^*(2) = \sqrt{2}-1</math>
:<math>\lambda^*(6) = (2-\sqrt{3})(\sqrt{3}-\sqrt{2})</math>
:<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>
:<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:
:<math>\lambda^*(3) = \frac{1}{2\sqrt{2}}(\sqrt{3}-1)</math>
:<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>
:<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}}
===Ramanujan's class invariants===
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>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}=\sqrt{\frac{5+\sqrt{29}}{2}}, \quad g_{190}=\sqrt{(\sqrt{5}+2)(\sqrt{10}+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>G_n=G_{1/n},\quad g_{n}=\frac{1}{g_{4/n}},\quad g_{4n}=2^{1/4}g_nG_n.</math>
The class invariants are very closely related to the 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>G_n = \sin\{2\arcsin[\lambda^*(n)]\}^{-1/12} = 1\Big /\left[\sqrt[12]{2\lambda^*(n)}\sqrt[24]{1-\lambda^*(n)^2}\right] </math>
:<math>g_n = \tan\{2\arctan[\lambda^*(n)]\}^{-1/12} = \sqrt[12]{[1-\lambda^*(n)^2]/[2\lambda^*(n)]} </math>
:<math>\lambda^*(n) = \tan\left\{ \frac{1}{2}\arctan[g_n^{-12}]\right\} = \sqrt{g_n^{24}+1}-g_n^{12} </math>
== Other appearances ==
===Little Picard theorem=== The lambda function is used in the original proof of the Little Picard theorem, that an entire non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard in 1879.<ref>Chandrasekharan (1985) p.121</ref> Suppose if possible that ''f'' is entire and does not take the values 0 and 1. Since λ is holomorphic, it has a local holomorphic inverse ω defined away from 0,1,∞. Consider the function ''z'' → ω(''f''(''z'')). By the Monodromy theorem this is holomorphic and maps the complex plane '''C''' to the upper half plane. From this it is easy to construct a holomorphic function from '''C''' to the unit disc, which by Liouville's theorem must be constant.<ref>Chandrasekharan (1985) p.118</ref>
===Moonshine=== The function <math>\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== {{reflist}}
==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 }} * {{citation|first1=John Horton|last1=Conway|author1-link=John Horton Conway|first2=Simon|last2=Norton|author2-link=Simon P. Norton|title=Monstrous moonshine|journal=Bulletin of the London Mathematical Society|volume=11|issue=3|year=1979|pages=308–339|mr=0554399|zbl=0424.20010 |doi=10.1112/blms/11.3.308}} * {{citation | last=Rankin | first=Robert A. | authorlink=Robert Alexander Rankin | title=Modular Forms and Functions | publisher=Cambridge University Press | year=1977 | isbn=0-521-21212-X | zbl=0376.10020 }} * {{dlmf|id=23.15.E6|title=Elliptic Modular Function|first= W. P. |last=Reinhardt|first2=P. L.|last2= Walker}}
* Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.
* Conway, J. H. and Norton, S. P. "Monstrous Moonshine." Bull. London Math. Soc. 11, 308-339, 1979.
* 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}} Category:Modular forms Category:Elliptic functions