Modular lambda function

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

Modular lambda function in the complex plane.

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

${\displaystyle \lambda (\tau )=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+\dots }$. 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 ${\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}$, and it is in fact Klein's modular j-invariant.

A plot of x→ λ(ix)

Modular properties

The function ${\displaystyle \lambda (\tau )}$ is invariant under the group generated by^[1]

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

The generators of the modular group act by^[2]

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

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

Consequently, the action of the modular group on ${\displaystyle \lambda (\tau )}$ is that of the anharmonic group, giving the six values of the cross-ratio:^[3]

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

Relations to other functions

It is the square of the elliptic modulus,^[4] that is, ${\displaystyle \lambda (\tau )=k^{2}(\tau )}$. In terms of the Dedekind eta function ${\displaystyle \eta (\tau )}$ and theta functions,^[4]

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

and,

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

where^[5]

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

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

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

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

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

we have^[4]

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

Since the three half-period values are distinct, this shows that ${\displaystyle \lambda }$ does not take the value 0 or 1.^[4]

The relation to the j-invariant is^[6][7]

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

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

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

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

where ${\displaystyle K}$ is the complete elliptic integral of the first kind with parameter ${\displaystyle m=k^{2}}$. Then

${\displaystyle \lambda (\tau )=m.}$

Modular equations

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

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

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

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

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

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

${\displaystyle {\begin{aligned}v&=\prod _{k=1}^{\infty }\tanh {\frac {(k-1/2)\pi i}{\tau }}={\sqrt {2}}e^{\pi i\tau /8}{\frac {\sum _{k\in \mathbb {Z} }e^{(2k^{2}+k)\pi i\tau }}{\sum _{k\in \mathbb {Z} }e^{k^{2}\pi i\tau }}}\\&={\cfrac {{\sqrt {2}}e^{\pi i\tau /8}}{1+{\cfrac {e^{\pi i\tau }}{1+e^{\pi i\tau }+{\cfrac {e^{2\pi i\tau }}{1+e^{2\pi i\tau }+{\cfrac {e^{3\pi i\tau }}{1+e^{3\pi i\tau }+\ddots }}}}}}}}\end{aligned}}}$

Since ${\displaystyle \lambda (i)=1/2}$, the modular equations can be used to give algebraic values of ${\displaystyle \lambda (pi)}$ for any prime ${\displaystyle p}$.^{[note 2]} The algebraic values of ${\displaystyle \lambda (ni)}$ are also given by^{[9][note 3]}

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

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

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

Lambda-star

Definition and computation of lambda-star

The function ${\displaystyle \lambda ^{*}(x)}$^[10] (where ${\displaystyle x\in \mathbb {R} ^{+}}$) gives the value of the elliptic modulus ${\displaystyle k}$, for which the complete elliptic integral of the first kind ${\displaystyle K(k)}$ and its complementary counterpart ${\displaystyle K({\sqrt {1-k^{2}}})}$ are related by following expression:

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

The values of ${\displaystyle \lambda ^{*}(x)}$ 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 ${\displaystyle \lambda ^{*}}$ and ${\displaystyle \lambda }$ are related to each other in this way:

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

Properties of lambda-star

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

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

${\displaystyle K(\lambda ^{*}(x))}$ and ${\displaystyle E(\lambda ^{*}(x))}$ (the complete elliptic integral of the second kind) can be expressed in closed form in terms of the gamma function for any ${\displaystyle x\in \mathbb {Q} ^{+}}$, as Selberg and Chowla proved in 1949.^[11][12]

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

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

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

By knowing one ${\displaystyle \lambda ^{*}}$ value, this formula can be used to compute related ${\displaystyle \lambda ^{*}}$ values:^[9]

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

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

Further relations:

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

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

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

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

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

Special values

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: ${\displaystyle \lambda ^{*}\left({\frac {1}{2}}\right)={\sqrt {2{\sqrt {2}}-2}}}$ ${\displaystyle \lambda ^{*}\left({\frac {1}{3}}\right)={\frac {1}{2{\sqrt {2}}}}({\sqrt {3}}+1)}$ ${\displaystyle \lambda ^{*}\left({\frac {2}{3}}\right)=(2-{\sqrt {3}})({\sqrt {3}}+{\sqrt {2}})}$ ${\displaystyle \lambda ^{*}\left({\frac {1}{4}}\right)=2{\sqrt[{4}]{2}}({\sqrt {2}}-1)}$ ${\displaystyle \lambda ^{*}\left({\frac {3}{4}}\right)={\sqrt[{4}]{8}}({\sqrt {3}}-{\sqrt {2}})({\sqrt {2}}+1){\sqrt {({\sqrt {3}}-1)^{3}}}}$ ${\displaystyle \lambda ^{*}\left({\frac {1}{5}}\right)={\frac {1}{2{\sqrt {2}}}}\left({\sqrt {2{\sqrt {5}}-2}}+{\sqrt {5}}-1\right)}$ ${\displaystyle \lambda ^{*}\left({\frac {2}{5}}\right)=({\sqrt {10}}-3)({\sqrt {2}}+1)^{2}}$ ${\displaystyle \lambda ^{*}\left({\frac {3}{5}}\right)={\frac {1}{8{\sqrt {2}}}}(3+{\sqrt {5}})({\sqrt {5}}-{\sqrt {3}})(2+{\sqrt {3}})}$ ${\displaystyle \lambda ^{*}\left({\frac {4}{5}}\right)=\tan \left[{\frac {\pi }{4}}-{\frac {1}{4}}\arcsin({\sqrt {5}}-2)\right]^{2}}$

Ramanujan's class invariants

Ramanujan's class invariants ${\displaystyle G_{n}}$ and ${\displaystyle g_{n}}$ are defined as^[13]

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

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

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

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

Identities with the class invariants include^[14]

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

The class invariants are very closely related to the Weber modular functions ${\displaystyle {\mathfrak {f}}}$ and ${\displaystyle {\mathfrak {f}}_{1}}$. These are the relations between lambda-star and the class invariants:

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

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

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

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.^[15] 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.^[16]

Moonshine

The function ${\displaystyle \tau \mapsto 16/\lambda (2\tau )-8}$ 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.