Modular lambda function

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

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

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

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

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

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

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

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

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

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

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.[8][9]

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

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

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:

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

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

Further relations:

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

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

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

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

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

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

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

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

Ramanujan's class invariants ${\displaystyle G_{n}}$ and ${\displaystyle g_{n}}$ are defined as[10]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Lambda-star-values of rational fractions:

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

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

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

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

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

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

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

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

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

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.[11] 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.[12]

The function ${\displaystyle \tau \mapsto {\frac {16}{\lambda (2\tau )}}-8}$ is the normalized Hauptmodul for the group ${\displaystyle \Gamma _{0}(4)}$, and its q-expansion ${\displaystyle q^{-1}+20q-62q^{3}+\dots }$, OEIS: A007248 where ${\displaystyle q=e^{2\pi i\tau }}$, is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.