Modular lambda function (original) (raw)

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Symmetric holomorphic function

Modular lambda function in the complex plane.

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 C / ⟨ 1 , τ ⟩ {\displaystyle \mathbb {C} /\langle 1,\tau \rangle } $\mathbb {C} /\langle 1,\tau \rangle$ , where the map is defined as the quotient by the [−1] involution.

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

λ ( τ ) = 16 q − 128 q 2 + 704 q 3 − 3072 q 4 + 11488 q 5 − 38400 q 6 + … {\displaystyle \lambda (\tau )=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+\dots } $\lambda (\tau )=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+\dots$ . OEIS: A115977

By symmetrizing the lambda function under the canonical action of the symmetric group _S_3 on X(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group SL 2 ⁡ ( Z ) {\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )} $\operatorname {SL} _{2}(\mathbb {Z} )$ , and it is in fact Klein's modular j-invariant.

A plot of x→ λ(ix)

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

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

The generators of the modular group act by[2]

τ ↦ τ + 1 : λ ↦ λ λ − 1 ; {\displaystyle \tau \mapsto \tau +1\ :\ \lambda \mapsto {\frac {\lambda }{\lambda -1}}\,;} $\tau \mapsto \tau +1\ :\ \lambda \mapsto {\frac {\lambda }{\lambda -1}}\,;$

τ ↦ − 1 τ : λ ↦ 1 − λ . {\displaystyle \tau \mapsto -{\frac {1}{\tau }}\ :\ \lambda \mapsto 1-\lambda \ .} $\tau \mapsto -{\frac {1}{\tau }}\ :\ \lambda \mapsto 1-\lambda \ .$

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

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

Relations to other functions

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

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

and,

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

where[5]

θ 2 ( τ ) = ∑ n = − ∞ ∞ e π i τ ( n + 1 / 2 ) 2 {\displaystyle \theta _{2}(\tau )=\sum _{n=-\infty }^{\infty }e^{\pi i\tau (n+1/2)^{2}}} $\theta _{2}(\tau )=\sum _{n=-\infty }^{\infty }e^{\pi i\tau (n+1/2)^{2}}$

θ 3 ( τ ) = ∑ n = − ∞ ∞ e π i τ n 2 {\displaystyle \theta _{3}(\tau )=\sum _{n=-\infty }^{\infty }e^{\pi i\tau n^{2}}} $\theta _{3}(\tau )=\sum _{n=-\infty }^{\infty }e^{\pi i\tau n^{2}}$

θ 4 ( τ ) = ∑ n = − ∞ ∞ ( − 1 ) n e π i τ n 2 {\displaystyle \theta _{4}(\tau )=\sum _{n=-\infty }^{\infty }(-1)^{n}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 [ ω 1 , ω 2 ] {\displaystyle [\omega _{1},\omega _{2}]} $[\omega _{1},\omega _{2}]$ be a fundamental pair of periods with τ = ω 2 ω 1 {\displaystyle \tau ={\frac {\omega _{2}}{\omega _{1}}}} $\tau ={\frac {\omega _{2}}{\omega _{1}}}$ .

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

we have[4]

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

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

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

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

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

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

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

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

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

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

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

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

u 6 − v 6 + 5 u 2 v 2 ( u 2 − v 2 ) + 4 u v ( 1 − u 4 v 4 ) = 0 , {\displaystyle u^{6}-v^{6}+5u^{2}v^{2}(u^{2}-v^{2})+4uv(1-u^{4}v^{4})=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 − u v ) 8 = 0. {\displaystyle (1-u^{8})(1-v^{8})-(1-uv)^{8}=0.} $(1-u^{8})(1-v^{8})-(1-uv)^{8}=0.$

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

v = ∏ k = 1 ∞ tanh ⁡ ( k − 1 / 2 ) π i τ = 2 e π i τ / 8 ∑ k ∈ Z e ( 2 k 2 + k ) π i τ ∑ k ∈ Z e k 2 π i τ = 2 e π i τ / 8 1 + e π i τ 1 + e π i τ + e 2 π i τ 1 + e 2 π i τ + e 3 π i τ 1 + e 3 π i τ + ⋱ {\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}}} ${\begin{aligned}v&=\prod _{k=1}^{\infty }\tanh {\frac {(k-1/2)\pi i}{\tau }}={\sqrt {2}}e^{\pi i\tau /8}{\frac {\sum _{k\in \mathbb {Z} }e^{(2k^{2}+k)\pi i\tau }}{\sum _{k\in \mathbb {Z} }e^{k^{2}\pi i\tau }}}\\&={\cfrac {{\sqrt {2}}e^{\pi i\tau /8}}{1+{\cfrac {e^{\pi i\tau }}{1+e^{\pi i\tau }+{\cfrac {e^{2\pi i\tau }}{1+e^{2\pi i\tau }+{\cfrac {e^{3\pi i\tau }}{1+e^{3\pi i\tau }+\ddots }}}}}}}}\end{aligned}}$

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

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

λ ( n i ) = 1 2 n ∏ k = 1 n − 1 ( 1 − sl 2 ⁡ k ϖ n ) 2 ( n odd ) {\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}})} $\lambda (ni)={\frac {1}{2^{n}}}\prod _{k=1}^{n-1}\left(1-\operatorname {sl} ^{2}{\frac {k\varpi }{n}}\right)^{2}\quad (n\,{\text{odd}})$

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

Definition and computation of lambda-star

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

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

The values of λ ∗ ( x ) {\displaystyle \lambda ^{*}(x)} $\lambda ^{*}(x)$ can be computed as follows:

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

λ ∗ ( x ) = [ ∑ a = − ∞ ∞ exp ⁡ [ − ( a + 1 / 2 ) 2 π x ] ] 2 [ ∑ a = − ∞ ∞ exp ⁡ ( − a 2 π x ) ] − 2 {\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 }\exp[-(a+1/2)^{2}\pi {\sqrt {x}}]\right]^{2}\left[\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {x}})\right]^{-2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b8070bd15a4b2a9b9bd621ffb37eece379cb864)

λ ∗ ( x ) = [ ∑ a = − ∞ ∞ sech ⁡ [ ( a + 1 / 2 ) π x ] ] [ ∑ a = − ∞ ∞ sech ⁡ ( a π x ) ] − 1 {\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}} ![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d176424e648ac0676545cec2e165cf27d61f78)

The functions λ ∗ {\displaystyle \lambda ^{*}} $\lambda ^{*}$ and λ {\displaystyle \lambda } $\lambda$ are related to each other in this way:

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

Properties of lambda-star

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

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

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

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

n = ∑ a = 1 n dn ⁡ [ 2 a n K [ λ ∗ ( 1 n ) ] ; λ ∗ ( 1 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]} ![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe98aa85642729039559cf8044f68b293548acf6)

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

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

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

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

Further relations:

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

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

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

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

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

Ramanujan's class invariants

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

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

g n = 2 − 1 / 4 e π n / 24 ∏ k = 0 ∞ ( 1 − e − ( 2 k + 1 ) π n ) , {\displaystyle 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 ∈ Q + {\displaystyle n\in \mathbb {Q} ^{+}} $n\in \mathbb {Q} ^{+}$ . For such n {\displaystyle n} $n$ , the class invariants are algebraic numbers. For example

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

Identities with the class invariants include[14]

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

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

G n = sin ⁡ { 2 arcsin ⁡ [ λ ∗ ( n ) ] } − 1 / 12 = 1 / [ 2 λ ∗ ( n ) 12 1 − λ ∗ ( n ) 2 24 ] {\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}=\sin{2\arcsin[\lambda ^{}(n)]}^{-1/12}=1{\Big /}\left[{\sqrt[{12}]{2\lambda ^{}(n)}}{\sqrt[{24}]{1-\lambda ^{*}(n)^{2}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d7ba82805bad5b61422ef8ab1e6d8ff0653f93)

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

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

Little Picard theorem

[edit]

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]

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

^ Chandrasekharan (1985) p.115
^ Chandrasekharan (1985) p.109
^ Chandrasekharan (1985) p.110
^ a b c d Chandrasekharan (1985) p.108
^ Chandrasekharan (1985) p.63
^ Chandrasekharan (1985) p.117
^ Rankin (1977) pp.226–228
^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 103–109, 134
^ a b Jacobi, Carl Gustav Jacob (1829). Fundamenta nova theoriae functionum ellipticarum (in Latin). p. 42
^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 152
^ Chowla, S.; Selberg, A. (1949). "On Epstein's Zeta Function (I)". Proceedings of the National Academy of Sciences. 35 (7): 373. doi:10.1073/PNAS.35.7.371. PMC 1063041. S2CID 45071481.
^ Chowla, S.; Selberg, A. "On Epstein's Zeta-Function". EuDML. pp. 86–110.
^ Berndt, Bruce C.; Chan, Heng Huat; Zhang, Liang-Cheng (6 June 1997). "Ramanujan's class invariants, Kronecker's limit formula, and modular equations". Transactions of the American Mathematical Society. 349 (6): 2125–2173.
^ Eymard, Pierre; Lafon, Jean-Pierre (1999). Autour du nombre Pi (in French). HERMANN. ISBN 2705614435. p. 240
^ Chandrasekharan (1985) p.121
^ Chandrasekharan (1985) p.118

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Modular lambda function at Fungrim