modular lambda function

File:Modular lambda function in range -3 to 3.png

In mathematics, the modular lambda function λ(τ) 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 $\mathbb{C}/\langle 1, \tau \rangle$ , where the map is defined as the quotient by the [−1] involution.

The q-expansion, where $q = e^{\pi i \tau}$ is the nome, is given by:

: $\lambda(\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \dots$ . {{oeis|id=A115977 }}

By symmetrizing the lambda function under the canonical action of the symmetric group S₃ on X(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group $\operatorname{SL}_2(\mathbb{Z})$ , and it is in fact Klein's modular j-invariant.

File:Lambda function.svg

Modular properties

The function $\lambda(\tau)$ is invariant under the group generated byChandrasekharan (1985) p.115

: $\tau \mapsto \tau+2 \ ;\ \tau \mapsto \frac{\tau}{1-2\tau} \ .$

The generators of the modular group act byChandrasekharan (1985) p.109

: $\tau \mapsto \tau+1 \ :\ \lambda \mapsto \frac{\lambda}{\lambda-1} \, ;$

: $\tau \mapsto -\frac{1}{\tau} \ :\ \lambda \mapsto 1 - \lambda \ .$

Consequently, the action of the modular group on $\lambda(\tau)$ is that of the anharmonic group, giving the six values of the cross-ratio:Chandrasekharan (1985) p.110

: $\left\lbrace { \lambda, \frac{1}{1-\lambda}, \frac{\lambda-1}{\lambda}, \frac{1}{\lambda}, \frac{\lambda}{\lambda-1}, 1-\lambda } \right\rbrace \ .$

Relations to other functions

It is the square of the elliptic modulus,Chandrasekharan (1985) p.108 that is, $\lambda(\tau)=k^2(\tau)$ . In terms of the Dedekind eta function $\eta(\tau)$ and theta functions,

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

and,

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

whereChandrasekharan (1985) p.63

: $\theta_2(\tau) =\sum_{n=-\infty}^\infty e^{\pi i\tau (n+1/2)^2}$

: $\theta_3(\tau) = \sum_{n=-\infty}^\infty e^{\pi i\tau n^2}$

: $\theta_4(\tau) = \sum_{n=-\infty}^\infty (-1)^n e^{\pi i\tau n^2}$

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

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

we have

: $\lambda = \frac{e_3-e_2}{e_1-e_2} \, .$

Since the three half-period values are distinct, this shows that $\lambda$ does not take the value 0 or 1.

The relation to the j-invariant isChandrasekharan (1985) p.117Rankin (1977) pp.226–228

: $j(\tau) = \frac{256(1-\lambda(1-\lambda))^3}{(\lambda(1-\lambda))^2} = \frac{256(1-\lambda+\lambda^2)^3}{\lambda^2 (1-\lambda)^2} \ .$

which is the j-invariant of the elliptic curve of Legendre form $y^2=x(x-1)(x-\lambda)$

Given $m\in\mathbb{C}\setminus\{0,1\}$ , let

: $\tau=i\frac{K\{1-m\}}{K\{m\}}$

where $K$ is the complete elliptic integral of the first kind with parameter $m=k^2$ .

Then

: $\lambda (\tau)=m.$

Modular equations

The modular equation of degree $p$ (where $p$ is a prime number) is an algebraic equation in $\lambda (p\tau)$ and $\lambda (\tau)$ . If $\lambda (p\tau)=u^8$ and $\lambda (\tau)=v^8$ , the modular equations of degrees $p=2,3,5,7$ are, respectively,{{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

: $(1+u^4)^2v^8-4u^4=0,$

: $u^4-v^4+2uv(1-u^2v^2)=0,$

: $u^6-v^6+5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0,$

: $(1-u^8)(1-v^8)-(1-uv)^8=0.$

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

: $\begin{align}v&=\prod_{k=1}^\infty \tanh\frac{(k-1/2)\pi i}{\tau}=\sqrt{2}e^{\pi i\tau/8}\frac{\sum_{k\in\mathbb{Z}}e^{(2k^2+k)\pi i\tau}}{\sum_{k\in\mathbb{Z}}e^{k^2\pi i\tau}}\\
&=\cfrac{\sqrt{2}e^{\pi i\tau/8}}{1+\cfrac{e^{\pi i\tau}}{1+e^{\pi i\tau}+\cfrac{e^{2\pi i\tau}}{1+e^{2\pi i\tau}+\cfrac{e^{3\pi i\tau}}{1+e^{3\pi i\tau}+\ddots}}}}\end{align}$

Since $\lambda(i)=1/2$ , the modular equations can be used to give algebraic values of $\lambda(pi)$ for any prime $p$ .For any prime power, we can iterate the modular equation of degree $p$ . This process can be used to give algebraic values of $\lambda (ni)$ for any $n\in\mathbb{N}.$ The algebraic values of $\lambda(ni)$ are also given by{{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 $\operatorname{sl}a\varpi$ is algebraic for every $a\in\mathbb{Q}.$

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

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

where $\operatorname{sl}$ is the lemniscate sine and $\varpi$ is the lemniscate constant.

Lambda-star

=Definition and computation of lambda-star=

The function $\lambda^*(x)$ {{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 (where $x\in\mathbb{R}^+$ ) gives the value of the elliptic modulus $k$ , for which the complete elliptic integral of the first kind $K(k)$ and its complementary counterpart $K(\sqrt{1-k^2})$ are related by following expression:

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

The values of $\lambda^*(x)$ can be computed as follows:

: $\lambda^*(x) = \frac{\theta^2_2(i\sqrt{x})}{\theta^2_3(i\sqrt{x})}$

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

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

The functions $\lambda^*$ and $\lambda$ are related to each other in this way:

: $\lambda^*(x) = \sqrt{\lambda(i\sqrt{x})}$

=Properties of lambda-star=

Every $\lambda^*$ value of a positive rational number is a positive algebraic number:

: $\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+.$

$K(\lambda^*(x))$ and $E(\lambda^*(x))$ (the complete elliptic integral of the second kind) can be expressed in closed form in terms of the gamma function for any $x\in\mathbb{Q}^+$ , as Selberg and Chowla proved in 1949.{{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}}{{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}}

The following expression is valid for all $n \in \mathbb{N}$ :

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

where $\operatorname{dn}$ is the Jacobi elliptic function delta amplitudinis with modulus $k$ .

By knowing one $\lambda^*$ value, this formula can be used to compute related $\lambda^*$ values:

: $\lambda^*(n^2x) = \lambda^*(x)^n\prod_{a=1}^{n}\operatorname{sn}\left\{\frac{2a-1}{n}K[\lambda^*(x)];\lambda^*(x)\right\}^2$

where $n\in\mathbb{N}$ and $\operatorname{sn}$ is the Jacobi elliptic function sinus amplitudinis with modulus $k$ .

Further relations:

: $\lambda^*(x)^2 + \lambda^*(1/x)^2 = 1$

: $[\lambda^*(x)+1][\lambda^*(4/x)+1] = 2$

: $\lambda^*(4x) = \frac{1-\sqrt{1-\lambda^*(x)^2}}{1+\sqrt{1-\lambda^*(x)^2}} = \tan\left\{\frac{1}{2}\arcsin[\lambda^*(x)]\right\}^2$

: $\lambda^*(x) - \lambda^*(9x) = 2[\lambda^*(x)\lambda^*(9x)]^{1/4} - 2[\lambda^*(x)\lambda^*(9x)]^{3/4}$

$\begin{align}
& a^{6}-f^{6} = 2af +2a^5f^5\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(f = \left[\frac{2\lambda^*(25x)}{1-\lambda^*(25x)^2}\right]^{1/12}\right) \\
&a^{8}+b^{8}-7a^4b^4 = 2\sqrt{2}ab+2\sqrt{2}a^7b^7\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(b = \left[\frac{2\lambda^*(49x)}{1-\lambda^*(49x)^2}\right]^{1/12}\right) \\
& a^{12}-c^{12} = 2\sqrt{2}(ac+a^3c^3)(1+3a^2c^2+a^4c^4)(2+3a^2c^2+2a^4c^4)\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(c = \left[\frac{2\lambda^*(121x)}{1-\lambda^*(121x)^2}\right]^{1/12}\right) \\
& (a^2-d^2)(a^4+d^4-7a^2d^2)[(a^2-d^2)^4-a^2d^2(a^2+d^2)^2] = 8ad+8a^{13}d^{13}\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(d = \left[\frac{2\lambda^*(169x)}{1-\lambda^*(169x)^2}\right]^{1/12}\right)
\end{align}$

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

: $\lambda^*(1) = \frac{1}{\sqrt{2}}$

: $\lambda^*(5) = \sin\left[\frac{1}{2}\arcsin\left(\sqrt{5}-2\right)\right]$

: $\lambda^*(9) = \frac{1}{2}(\sqrt{3}-1)(\sqrt{2}-\sqrt[4]{3})$

: $\lambda^*(13) = \sin\left[\frac{1}{2}\arcsin(5\sqrt{13}-18)\right]$

: $\lambda^*(17) = \sin\left\{\frac{1}{2}\arcsin\left[\frac{1}{64}\left(5+\sqrt{17}-\sqrt{10\sqrt{17}+26}\right)^3\right]\right\}$

: $\lambda^*(21) = \sin\left\{\frac{1}{2}\arcsin[(8-3\sqrt{7})(2\sqrt{7}-3\sqrt{3})]\right\}$

: $\lambda^*(25) = \frac{1}{\sqrt{2}}(\sqrt{5}-2)(3-2\sqrt[4]{5})$

: $\lambda^*(33) = \sin\left\{\frac{1}{2}\arcsin[(10-3\sqrt{11})(2-\sqrt{3})^3]\right\}$

: $\lambda^*(37) = \sin\left\{\frac{1}{2}\arcsin[(\sqrt{37}-6)^3]\right\}$

: $\lambda^*(45) = \sin\left\{\frac{1}{2}\arcsin[(4-\sqrt{15})^2(\sqrt{5}-2)^3]\right\}$

: $\lambda^*(49) = \frac{1}{4}(8+3\sqrt{7})(5-\sqrt{7}-\sqrt[4]{28})\left(\sqrt{14}-\sqrt{2}-\sqrt[8]{28}\sqrt{5-\sqrt{7}}\right)$

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

: $\lambda^*(73) = \sin\left\{\frac{1}{2}\arcsin\left[\frac{1}{64}\left(45+5\sqrt{73}-3\sqrt{50\sqrt{73}+426}\right)^3\right]\right\}$

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

: $\lambda^*(2) = \sqrt{2}-1$

: $\lambda^*(6) = (2-\sqrt{3})(\sqrt{3}-\sqrt{2})$

: $\lambda^*(10) = (\sqrt{10}-3)(\sqrt{2}-1)^2$

: $\lambda^*(14) = \tan\left\{\frac{1}{2}\arctan\left[\frac{1}{8}\left(2\sqrt{2}+1-\sqrt{4\sqrt{2}+5}\right)^3\right]\right\}$

: $\lambda^*(18) = (\sqrt{2}-1)^3(2-\sqrt{3})^2$

: $\lambda^*(22) = (10-3\sqrt{11})(3\sqrt{11}-7\sqrt{2})$

: $\lambda^*(30) = \tan\left\{\frac{1}{2}\arctan[(\sqrt{10}-3)^2(\sqrt{5}-2)^2]\right\}$

: $\lambda^*(34) = \tan\left\{\frac{1}{4}\arcsin\left[\frac{1}{9}(\sqrt{17}-4)^2\right]\right\}$

: $\lambda^*(42) = \tan\left\{\frac{1}{2}\arctan[(2\sqrt{7}-3\sqrt{3})^2(2\sqrt{2}-\sqrt{7})^2]\right\}$

: $\lambda^*(46) = \tan\left\{\frac{1}{2}\arctan\left[\frac{1}{64}\left(3+\sqrt{2}-\sqrt{6\sqrt{2}+7}\right)^6\right]\right\}$

: $\lambda^*(58) = (13\sqrt{58}-99)(\sqrt{2}-1)^6$

: $\lambda^*(70) = \tan\left\{\frac{1}{2}\arctan[(\sqrt{5}-2)^4(\sqrt{2}-1)^6]\right\}$

: $\lambda^*(78) = \tan\left\{\frac{1}{2}\arctan[(5\sqrt{13}-18)^2(\sqrt{26}-5)^2]\right\}$

: $\lambda^*(82) = \tan\left\{\frac{1}{4}\arcsin\left[\frac{1}{4761}(8\sqrt{41}-51)^2\right]\right\}$

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

: $\lambda^*(3) = \frac{1}{2\sqrt{2}}(\sqrt{3}-1)$

: $\lambda^*(7) = \frac{1}{4\sqrt{2}}(3-\sqrt{7})$

: $\lambda^*(11) = \frac{1}{8\sqrt{2}}(\sqrt{11}+3)\left(\frac{1}{3}\sqrt[3]{6\sqrt{3}+2\sqrt{11}}-\frac{1}{3}\sqrt[3]{6\sqrt{3}-2\sqrt{11}}+\frac{1}{3}\sqrt{11}-1\right)^4$

: $\lambda^*(15) = \frac{1}{8\sqrt{2}}(3-\sqrt{5})(\sqrt{5}-\sqrt{3})(2-\sqrt{3})$

: $\lambda^*(19) = \frac{1}{8\sqrt{2}}(3\sqrt{19}+13)\left[\frac{1}{6}(\sqrt{19}-2+\sqrt{3})\sqrt[3]{3\sqrt{3}-\sqrt{19}}-\frac{1}{6}(\sqrt{19}-2-\sqrt{3})\sqrt[3]{3\sqrt{3}+\sqrt{19}}-\frac{1}{3}(5-\sqrt{19})\right]^4$

: $\lambda^*(23) = \frac{1}{16\sqrt{2}}(5+\sqrt{23})\left[\frac{1}{6}(\sqrt{3}+1)\sqrt[3]{100-12\sqrt{69}}-\frac{1}{6}(\sqrt{3}-1)\sqrt[3]{100+12\sqrt{69}}+\frac{2}{3}\right]^4$

: $\lambda^*(27) = \frac{1}{16\sqrt{2}}(\sqrt{3}-1)^3\left[\frac{1}{3}\sqrt{3}(\sqrt[3]{4}-\sqrt[3]{2}+1)-\sqrt[3]{2}+1\right]^4$

: $\lambda^*(39) = \sin\left\{\frac{1}{2}\arcsin\left[\frac{1}{16}\left(6-\sqrt{13}-3\sqrt{6\sqrt{13}-21}\right)\right]\right\}$

: $\lambda^*(55) = \sin\left\{\frac{1}{2}\arcsin\left[\frac{1}{512}\left(3\sqrt{5}-3-\sqrt{6\sqrt{5}-2}\right)^3\right]\right\}$

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

: $\lambda^*(4) = (\sqrt{2}-1)^2$

: $\lambda^*(8) = \left(\sqrt{2}+1-\sqrt{2\sqrt{2}+2}\right)^2$

: $\lambda^*(12) = (\sqrt{3}-\sqrt{2})^2(\sqrt{2}-1)^2$

: $\lambda^*(16) = (\sqrt{2}+1)^2(\sqrt[4]{2}-1)^4$

: $\lambda^*(20) = \tan\left[\frac{1}{4}\arcsin(\sqrt{5}-2)\right]^2$

: $\lambda^*(24) = \tan\left\{\frac{1}{2}\arcsin[(2-\sqrt{3})(\sqrt{3}-\sqrt{2})]\right\}^2$

: $\lambda^*(28) = (2\sqrt{2}-\sqrt{7})^2(\sqrt{2}-1)^4$

: $\lambda^*(32) = \tan\left\{\frac{1}{2}\arcsin\left[\left(\sqrt{2}+1-\sqrt{2\sqrt{2}+2}\right)^2\right]\right\}^2$

Lambda-star values of rational fractions:

: $\lambda^*\left(\frac{1}{2}\right) = \sqrt{2\sqrt{2}-2}$

: $\lambda^*\left(\frac{1}{3}\right) = \frac{1}{2\sqrt{2}}(\sqrt{3}+1)$

: $\lambda^*\left(\frac{2}{3}\right) = (2-\sqrt{3})(\sqrt{3}+\sqrt{2})$

: $\lambda^*\left(\frac{1}{4}\right) = 2\sqrt[4]{2}(\sqrt{2}-1)$

: $\lambda^*\left(\frac{3}{4}\right) = \sqrt[4]{8}(\sqrt{3}-\sqrt{2})(\sqrt{2}+1)\sqrt{(\sqrt{3}-1)^3}$

: $\lambda^*\left(\frac{1}{5}\right) = \frac{1}{2\sqrt{2}}\left(\sqrt{2\sqrt{5}-2}+\sqrt{5}-1\right)$

: $\lambda^*\left(\frac{2}{5}\right) = (\sqrt{10}-3)(\sqrt{2}+1)^2$

: $\lambda^*\left(\frac{3}{5}\right) = \frac{1}{8\sqrt{2}}(3+\sqrt{5})(\sqrt{5}-\sqrt{3})(2+\sqrt{3})$

: $\lambda^*\left(\frac{4}{5}\right) = \tan\left[\frac{\pi}{4}-\frac{1}{4}\arcsin(\sqrt{5}-2)\right]^2$

=Ramanujan's class invariants=

Ramanujan's class invariants $G_n$ and $g_n$ are defined as{{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}}

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

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

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

: $g_{58}=\sqrt{\frac{5+\sqrt{29}}{2}}, \quad g_{190}=\sqrt{(\sqrt{5}+2)(\sqrt{10}+3)}.$

Identities with the class invariants include{{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

: $G_n=G_{1/n},\quad g_{n}=\frac{1}{g_{4/n}},\quad g_{4n}=2^{1/4}g_nG_n.$

The class invariants are very closely related to the Weber modular functions $\mathfrak{f}$ and $\mathfrak{f}_1$ . These are the relations between lambda-star and the class invariants:

: $G_n = \sin\{2\arcsin[\lambda^*(n)]\}^{-1/12} = 1\Big /\left[\sqrt[12]{2\lambda^*(n)}\sqrt[24]{1-\lambda^*(n)^2}\right]$

: $g_n = \tan\{2\arctan[\lambda^*(n)]\}^{-1/12} = \sqrt[12]{[1-\lambda^*(n)^2]/[2\lambda^*(n)]}$

: $\lambda^*(n) = \tan\left\{ \frac{1}{2}\arctan[g_n^{-12}]\right\} = \sqrt{g_n^{24}+1}-g_n^{12}$

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.Chandrasekharan (1985) p.121 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.Chandrasekharan (1985) p.118

=Moonshine=

The function $\tau\mapsto 16/\lambda(2\tau) - 8$ is the normalized Hauptmodul for the group $\Gamma_0(4)$ , and its q-expansion $q^{-1} + 20q - 62q^3 + \dots$ , {{oeis|id=A007248}} where $q=e^{2\pi i\tau }$ , is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.

Footnotes

References

=Notes=

=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]

Category:Modular forms

Category:Elliptic functions