Modular lambda function

Modular lambda function in the complex plane.

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:

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

A plot of x→ λ(ix)

Modular properties

The function $\lambda (\tau )$ is invariant under the group generated by

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

The generators of the modular group act by

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

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

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

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

Relations to other functions

It is the square of the elliptic modulus, 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,

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

where

$\theta _{2}(\tau )=\sum _{n=-\infty }^{\infty }e^{\pi i\tau (n+1/2)^{2}}$

$\theta _{3}(\tau )=\sum _{n=-\infty }^{\infty }e^{\pi i\tau n^{2}}$

$\theta _{4}(\tau )=\sum _{n=-\infty }^{\infty }(-1)^{n}e^{\pi i\tau n^{2}}$

In terms of the half-periods of Weierstrass's elliptic functions, let $[\omega _{1},\omega _{2}]$ be a fundamental pair of periods with $\tau ={\frac {\omega _{2}}{\omega _{1}}}$.

$e_{1}=\wp \left({\frac {\omega _{1}}{2}}\right),e_{2}=\wp \left({\frac {\omega _{2}}{2}}\right),e_{3}=\wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)$

we have

$\lambda ={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}}\,.$

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

The relation to the j-invariant is

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

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

Modular equations

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

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

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

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

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

The quantities $u$ and $v$ have the following product representations which define them as holomorphic functions in the whole upper half-plane:

$u={\sqrt {2}}e^{p\pi i\tau /8}\prod _{k=1}^{\infty }{\frac {1+e^{2kp\pi i\tau }}{1+e^{(2k-1)p\pi i\tau }}},\quad v={\sqrt {2}}e^{\pi i\tau /8}\prod _{k=1}^{\infty }{\frac {1+e^{2k\pi i\tau }}{1+e^{(2k-1)\pi i\tau }}}.$

Since $\lambda (i)=1/2$, the modular equations can be used to give algebraic values of $\lambda (pi)$ for any prime $p$. For any odd $n$, the algebraic values of $\lambda (ni)$ are given by

$\lambda (ni)={\frac {1}{2^{n}}}\prod _{k=1}^{n-1}\left(1-\operatorname {sl} ^{2}{\frac {k\varpi }{n}}\right)^{2}$

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

Lambda-star

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 $K\left({\sqrt {1-k^{2}}}\right)$ are related by following expression:

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

The values of λ*(x) can be computed as follows:

$\lambda ^{*}(x)={\frac {\theta _{2}^{2}(i{\sqrt {x}})}{\theta _{3}^{2}(i{\sqrt {x}})}}$

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

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

The functions λ* and λ are related to each other in this way:

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

Properties of lambda-star

Every λ*-value of a positive rational number is a positive algebraic number:

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

Elliptic integrals of the first and second kind of these special λ*-values can be expressed in terms of the gamma function, as Selberg and Chowla proved in 1949.

Following expression is valid for all n ∈ $\mathbb {N}$:

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

In this formula, dn is the Jacobi elliptic function delta amplitudinis.

By knowing one λ*-value, this formula can be used to compute related λ*-values:

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

In that formula, sn is the Jacobi elliptic function sinus amplitudinis. That formula works for all natural numbers.

Further relations:

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

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

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

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

$\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/2}-\left[{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}\right]^{1/2}=2\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/12}\left[{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}\right]^{1/12}+2\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{5/12}\left[{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}\right]^{5/12}$

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

$a^{12}-c^{12}=2{\sqrt {2}}(ac+a^{3}c^{3})(1+3a^{2}c^{2}+a^{4}c^{4})(2+3a^{2}c^{2}+2a^{4}c^{4})\,\left(a=\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/12}\right)\left(c=\left[{\frac {2\lambda ^{*}(121x)}{1-\lambda ^{*}(121x)^{2}}}\right]^{1/12}\right)$

$(a^{2}-d^{2})(a^{4}+d^{4}-7a^{2}d^{2})[(a^{2}-d^{2})^{4}-a^{2}d^{2}(a^{2}+d^{2})^{2}]=8ad+8a^{13}d^{13}\,\left(a=\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/12}\right)\left(d=\left[{\frac {2\lambda ^{*}(169x)}{1-\lambda ^{*}(169x)^{2}}}\right]^{1/12}\right)$

Ramanujan's class invariants

Ramanujan's class invariants $G_{n}$ and $g_{n}$ are defined as

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

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

where $n\in \mathbb {Q} ^{+}$.

These are the relations between lambda-star and Ramanujan's class invariants:

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

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

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

Special Values

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:

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

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

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

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

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

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

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

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

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

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. 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.

Moonshine

The function $\tau \mapsto {\frac {16}{\lambda (2\tau )}}-8$ is the normalized Hauptmodul for the group $\Gamma _{0}(4)$, and its q-expansion $q^{-1}+20q-62q^{3}+\dots$, OEIS: 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.

References

Other

• Abramowitz, Milton; Stegun, Irene A., eds. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, ISBN 978-0-486-61272-0, Zbl 0543.33001
• Chandrasekharan, K. (1985), Elliptic Functions, Grundlehren der mathematischen Wissenschaften, vol. 281, Springer-Verlag, pp. 108–121, ISBN 3-540-15295-4, Zbl 0575.33001
• Conway, John Horton; Norton, Simon (1979), "Monstrous moonshine", Bulletin of the London Mathematical Society, 11 (3): 308–339, doi:10.1112/blms/11.3.308, MR 0554399, Zbl 0424.20010
• Rankin, Robert A. (1977), Modular Forms and Functions, Cambridge University Press, ISBN 0-521-21212-X, Zbl 0376.10020
• Reinhardt, W. P.; Walker, P. L. (2010), "Elliptic Modular Function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.

• 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

• Modular lambda function at Fungrim