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Modular lambda function

A highly symmetric holomorphic function on the complex upper half-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 C / ⟨ 1 , τ ⟩ {\displaystyle \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 }} 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 } . 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 SL 2 ⁡ ( Z ) {\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )} , and it is in fact Klein's modular j-invariant.

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Modular properties

The function λ ( τ ) {\displaystyle \lambda (\tau )} is invariant under the group generated by²

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

The generators of the modular group act by³

τ ↦ τ + 1 : λ ↦ λ λ − 1 ; {\displaystyle \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 \ .}

Consequently, the action of the modular group on λ ( τ ) {\displaystyle \lambda (\tau )} is that of the anharmonic group, giving the six values of the cross-ratio:⁴

{ 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 \ .}

Relations to other functions

It is the square of the elliptic modulus,⁵ that is, λ ( τ ) = k 2 ( τ ) {\displaystyle \lambda (\tau )=k^{2}(\tau )} . In terms of the Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )} and theta functions,⁶

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 )}}}

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}})}}}

where⁷

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}}} θ 3 ( τ ) = ∑ n = − ∞ ∞ e π i τ n 2 {\displaystyle \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}}}

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

we have⁸

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

Since the three half-period values are distinct, this shows that λ {\displaystyle \lambda } does not take the value 0 or 1.⁹

The relation to the j-invariant is¹⁰ ¹¹

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}}}\ .}

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 )}

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

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

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

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

Modular equations

The modular equation of degree p {\displaystyle p} (where p {\displaystyle p} is a prime number) is an algebraic equation in λ ( p τ ) {\displaystyle \lambda (p\tau )} and λ ( τ ) {\displaystyle \lambda (\tau )} . If λ ( p τ ) = u 8 {\displaystyle \lambda (p\tau )=u^{8}} and λ ( τ ) = v 8 {\displaystyle \lambda (\tau )=v^{8}} , the modular equations of degrees p = 2 , 3 , 5 , 7 {\displaystyle p=2,3,5,7} are, respectively,¹²

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

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

Since λ ( i ) = 1 / 2 {\displaystyle \lambda (i)=1/2} , the modular equations can be used to give algebraic values of λ ( p i ) {\displaystyle \lambda (pi)} for any prime p {\displaystyle p} .¹³ The algebraic values of λ ( n i ) {\displaystyle \lambda (ni)} are also given by¹⁴ ¹⁵

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}})} 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}})}

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

Lambda-star

Definition and computation of lambda-star

The function λ ∗ ( x ) {\displaystyle \lambda ^{*}(x)} ¹⁶ (where x ∈ R + {\displaystyle x\in \mathbb {R} ^{+}} ) gives the value of the elliptic modulus k {\displaystyle k} , for which the complete elliptic integral of the first kind K ( k ) {\displaystyle K(k)} and its complementary counterpart K ( 1 − k 2 ) {\displaystyle 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}}}

The values of λ ∗ ( x ) {\displaystyle \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}})}}} 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}} 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}}

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

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

Properties of lambda-star

Every λ ∗ {\displaystyle \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} ^{+}.}

K ( λ ∗ ( x ) ) {\displaystyle K(\lambda ^{*}(x))} and E ( λ ∗ ( x ) ) {\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 x ∈ Q + {\displaystyle x\in \mathbb {Q} ^{+}} , as Selberg and Chowla proved in 1949.¹⁷ ¹⁸

The following expression is valid for all n ∈ N {\displaystyle 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]}

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

By knowing one λ ∗ {\displaystyle \lambda ^{*}} value, this formula can be used to compute related λ ∗ {\displaystyle \lambda ^{*}} values:¹⁹

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}}

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

Further relations:

λ ∗ ( x ) 2 + λ ∗ ( 1 / x ) 2 = 1 {\displaystyle \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} 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}} 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}}

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}}}

Special values
Lambda-star values of integer numbers of 4n-3-type: λ ∗ ( 1 ) = 1 2 {\displaystyle \lambda ^{}(1)={\frac {1}{\sqrt {2}}}} λ ∗ ( 5 ) = sin ⁡ [ 1 2 arcsin ⁡ ( 5 − 2 ) ] {\displaystyle \lambda ^{}(5)=\sin \left[{\frac {1}{2}}\arcsin \left({\sqrt {5}}-2\right)\right]} λ ∗ ( 9 ) = 1 2 ( 3 − 1 ) ( 2 − 3 4 ) {\displaystyle \lambda ^{}(9)={\frac {1}{2}}({\sqrt {3}}-1)({\sqrt {2}}-{\sqrt[{4}]{3}})} λ ∗ ( 13 ) = sin ⁡ [ 1 2 arcsin ⁡ ( 5 13 − 18 ) ] {\displaystyle \lambda ^{}(13)=\sin \left[{\frac {1}{2}}\arcsin(5{\sqrt {13}}-18)\right]} λ ∗ ( 17 ) = sin ⁡ { 1 2 arcsin ⁡ [ 1 64 ( 5 + 17 − 10 17 + 26 ) 3 ] } {\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\}} λ ∗ ( 21 ) = sin ⁡ { 1 2 arcsin ⁡ [ ( 8 − 3 7 ) ( 2 7 − 3 3 ) ] } {\displaystyle \lambda ^{}(21)=\sin \left\{{\frac {1}{2}}\arcsin[(8-3{\sqrt {7}})(2{\sqrt {7}}-3{\sqrt {3}})]\right\}} λ ∗ ( 25 ) = 1 2 ( 5 − 2 ) ( 3 − 2 5 4 ) {\displaystyle \lambda ^{}(25)={\frac {1}{\sqrt {2}}}({\sqrt {5}}-2)(3-2{\sqrt[{4}]{5}})} λ ∗ ( 33 ) = sin ⁡ { 1 2 arcsin ⁡ [ ( 10 − 3 11 ) ( 2 − 3 ) 3 ] } {\displaystyle \lambda ^{}(33)=\sin \left\{{\frac {1}{2}}\arcsin[(10-3{\sqrt {11}})(2-{\sqrt {3}})^{3}]\right\}} λ ∗ ( 37 ) = sin ⁡ { 1 2 arcsin ⁡ [ ( 37 − 6 ) 3 ] } {\displaystyle \lambda ^{}(37)=\sin \left\{{\frac {1}{2}}\arcsin[({\sqrt {37}}-6)^{3}]\right\}} λ ∗ ( 45 ) = sin ⁡ { 1 2 arcsin ⁡ [ ( 4 − 15 ) 2 ( 5 − 2 ) 3 ] } {\displaystyle \lambda ^{}(45)=\sin \left\{{\frac {1}{2}}\arcsin[(4-{\sqrt {15}})^{2}({\sqrt {5}}-2)^{3}]\right\}} λ ∗ ( 49 ) = 1 4 ( 8 + 3 7 ) ( 5 − 7 − 28 4 ) ( 14 − 2 − 28 8 5 − 7 ) {\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)} λ ∗ ( 57 ) = sin ⁡ { 1 2 arcsin ⁡ [ ( 170 − 39 19 ) ( 2 − 3 ) 3 ] } {\displaystyle \lambda ^{}(57)=\sin \left\{{\frac {1}{2}}\arcsin[(170-39{\sqrt {19}})(2-{\sqrt {3}})^{3}]\right\}} λ ∗ ( 73 ) = sin ⁡ { 1 2 arcsin ⁡ [ 1 64 ( 45 + 5 73 − 3 50 73 + 426 ) 3 ] } {\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: λ ∗ ( 2 ) = 2 − 1 {\displaystyle \lambda ^{}(2)={\sqrt {2}}-1} λ ∗ ( 6 ) = ( 2 − 3 ) ( 3 − 2 ) {\displaystyle \lambda ^{}(6)=(2-{\sqrt {3}})({\sqrt {3}}-{\sqrt {2}})} λ ∗ ( 10 ) = ( 10 − 3 ) ( 2 − 1 ) 2 {\displaystyle \lambda ^{}(10)=({\sqrt {10}}-3)({\sqrt {2}}-1)^{2}} λ ∗ ( 14 ) = tan ⁡ { 1 2 arctan ⁡ [ 1 8 ( 2 2 + 1 − 4 2 + 5 ) 3 ] } {\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\}} λ ∗ ( 18 ) = ( 2 − 1 ) 3 ( 2 − 3 ) 2 {\displaystyle \lambda ^{}(18)=({\sqrt {2}}-1)^{3}(2-{\sqrt {3}})^{2}} λ ∗ ( 22 ) = ( 10 − 3 11 ) ( 3 11 − 7 2 ) {\displaystyle \lambda ^{}(22)=(10-3{\sqrt {11}})(3{\sqrt {11}}-7{\sqrt {2}})} λ ∗ ( 30 ) = tan ⁡ { 1 2 arctan ⁡ [ ( 10 − 3 ) 2 ( 5 − 2 ) 2 ] } {\displaystyle \lambda ^{}(30)=\tan \left\{{\frac {1}{2}}\arctan[({\sqrt {10}}-3)^{2}({\sqrt {5}}-2)^{2}]\right\}} λ ∗ ( 34 ) = tan ⁡ { 1 4 arcsin ⁡ [ 1 9 ( 17 − 4 ) 2 ] } {\displaystyle \lambda ^{}(34)=\tan \left\{{\frac {1}{4}}\arcsin \left[{\frac {1}{9}}({\sqrt {17}}-4)^{2}\right]\right\}} λ ∗ ( 42 ) = tan ⁡ { 1 2 arctan ⁡ [ ( 2 7 − 3 3 ) 2 ( 2 2 − 7 ) 2 ] } {\displaystyle \lambda ^{}(42)=\tan \left\{{\frac {1}{2}}\arctan[(2{\sqrt {7}}-3{\sqrt {3}})^{2}(2{\sqrt {2}}-{\sqrt {7}})^{2}]\right\}} λ ∗ ( 46 ) = tan ⁡ { 1 2 arctan ⁡ [ 1 64 ( 3 + 2 − 6 2 + 7 ) 6 ] } {\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\}} λ ∗ ( 58 ) = ( 13 58 − 99 ) ( 2 − 1 ) 6 {\displaystyle \lambda ^{}(58)=(13{\sqrt {58}}-99)({\sqrt {2}}-1)^{6}} λ ∗ ( 70 ) = tan ⁡ { 1 2 arctan ⁡ [ ( 5 − 2 ) 4 ( 2 − 1 ) 6 ] } {\displaystyle \lambda ^{}(70)=\tan \left\{{\frac {1}{2}}\arctan[({\sqrt {5}}-2)^{4}({\sqrt {2}}-1)^{6}]\right\}} λ ∗ ( 78 ) = tan ⁡ { 1 2 arctan ⁡ [ ( 5 13 − 18 ) 2 ( 26 − 5 ) 2 ] } {\displaystyle \lambda ^{}(78)=\tan \left\{{\frac {1}{2}}\arctan[(5{\sqrt {13}}-18)^{2}({\sqrt {26}}-5)^{2}]\right\}} λ ∗ ( 82 ) = tan ⁡ { 1 4 arcsin ⁡ [ 1 4761 ( 8 41 − 51 ) 2 ] } {\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: λ ∗ ( 3 ) = 1 2 2 ( 3 − 1 ) {\displaystyle \lambda ^{}(3)={\frac {1}{2{\sqrt {2}}}}({\sqrt {3}}-1)} λ ∗ ( 7 ) = 1 4 2 ( 3 − 7 ) {\displaystyle \lambda ^{}(7)={\frac {1}{4{\sqrt {2}}}}(3-{\sqrt {7}})} λ ∗ ( 11 ) = 1 8 2 ( 11 + 3 ) ( 1 3 6 3 + 2 11 3 − 1 3 6 3 − 2 11 3 + 1 3 11 − 1 ) 4 {\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}} λ ∗ ( 15 ) = 1 8 2 ( 3 − 5 ) ( 5 − 3 ) ( 2 − 3 ) {\displaystyle \lambda ^{}(15)={\frac {1}{8{\sqrt {2}}}}(3-{\sqrt {5}})({\sqrt {5}}-{\sqrt {3}})(2-{\sqrt {3}})} λ ∗ ( 19 ) = 1 8 2 ( 3 19 + 13 ) [ 1 6 ( 19 − 2 + 3 ) 3 3 − 19 3 − 1 6 ( 19 − 2 − 3 ) 3 3 + 19 3 − 1 3 ( 5 − 19 ) ] 4 {\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}} λ ∗ ( 23 ) = 1 16 2 ( 5 + 23 ) [ 1 6 ( 3 + 1 ) 100 − 12 69 3 − 1 6 ( 3 − 1 ) 100 + 12 69 3 + 2 3 ] 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}} λ ∗ ( 27 ) = 1 16 2 ( 3 − 1 ) 3 [ 1 3 3 ( 4 3 − 2 3 + 1 ) − 2 3 + 1 ] 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}} λ ∗ ( 39 ) = sin ⁡ { 1 2 arcsin ⁡ [ 1 16 ( 6 − 13 − 3 6 13 − 21 ) ] } {\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\}} λ ∗ ( 55 ) = sin ⁡ { 1 2 arcsin ⁡ [ 1 512 ( 3 5 − 3 − 6 5 − 2 ) 3 ] } {\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: λ ∗ ( 4 ) = ( 2 − 1 ) 2 {\displaystyle \lambda ^{}(4)=({\sqrt {2}}-1)^{2}} λ ∗ ( 8 ) = ( 2 + 1 − 2 2 + 2 ) 2 {\displaystyle \lambda ^{}(8)=\left({\sqrt {2}}+1-{\sqrt {2{\sqrt {2}}+2}}\right)^{2}} λ ∗ ( 12 ) = ( 3 − 2 ) 2 ( 2 − 1 ) 2 {\displaystyle \lambda ^{}(12)=({\sqrt {3}}-{\sqrt {2}})^{2}({\sqrt {2}}-1)^{2}} λ ∗ ( 16 ) = ( 2 + 1 ) 2 ( 2 4 − 1 ) 4 {\displaystyle \lambda ^{}(16)=({\sqrt {2}}+1)^{2}({\sqrt[{4}]{2}}-1)^{4}} λ ∗ ( 20 ) = tan ⁡ [ 1 4 arcsin ⁡ ( 5 − 2 ) ] 2 {\displaystyle \lambda ^{}(20)=\tan \left[{\frac {1}{4}}\arcsin({\sqrt {5}}-2)\right]^{2}} λ ∗ ( 24 ) = tan ⁡ { 1 2 arcsin ⁡ [ ( 2 − 3 ) ( 3 − 2 ) ] } 2 {\displaystyle \lambda ^{}(24)=\tan \left\{{\frac {1}{2}}\arcsin[(2-{\sqrt {3}})({\sqrt {3}}-{\sqrt {2}})]\right\}^{2}} λ ∗ ( 28 ) = ( 2 2 − 7 ) 2 ( 2 − 1 ) 4 {\displaystyle \lambda ^{}(28)=(2{\sqrt {2}}-{\sqrt {7}})^{2}({\sqrt {2}}-1)^{4}} λ ∗ ( 32 ) = tan ⁡ { 1 2 arcsin ⁡ [ ( 2 + 1 − 2 2 + 2 ) 2 ] } 2 {\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: λ ∗ ( 1 2 ) = 2 2 − 2 {\displaystyle \lambda ^{}\left({\frac {1}{2}}\right)={\sqrt {2{\sqrt {2}}-2}}} λ ∗ ( 1 3 ) = 1 2 2 ( 3 + 1 ) {\displaystyle \lambda ^{}\left({\frac {1}{3}}\right)={\frac {1}{2{\sqrt {2}}}}({\sqrt {3}}+1)} λ ∗ ( 2 3 ) = ( 2 − 3 ) ( 3 + 2 ) {\displaystyle \lambda ^{}\left({\frac {2}{3}}\right)=(2-{\sqrt {3}})({\sqrt {3}}+{\sqrt {2}})} λ ∗ ( 1 4 ) = 2 2 4 ( 2 − 1 ) {\displaystyle \lambda ^{}\left({\frac {1}{4}}\right)=2{\sqrt[{4}]{2}}({\sqrt {2}}-1)} λ ∗ ( 3 4 ) = 8 4 ( 3 − 2 ) ( 2 + 1 ) ( 3 − 1 ) 3 {\displaystyle \lambda ^{}\left({\frac {3}{4}}\right)={\sqrt[{4}]{8}}({\sqrt {3}}-{\sqrt {2}})({\sqrt {2}}+1){\sqrt {({\sqrt {3}}-1)^{3}}}} λ ∗ ( 1 5 ) = 1 2 2 ( 2 5 − 2 + 5 − 1 ) {\displaystyle \lambda ^{}\left({\frac {1}{5}}\right)={\frac {1}{2{\sqrt {2}}}}\left({\sqrt {2{\sqrt {5}}-2}}+{\sqrt {5}}-1\right)} λ ∗ ( 2 5 ) = ( 10 − 3 ) ( 2 + 1 ) 2 {\displaystyle \lambda ^{}\left({\frac {2}{5}}\right)=({\sqrt {10}}-3)({\sqrt {2}}+1)^{2}} λ ∗ ( 3 5 ) = 1 8 2 ( 3 + 5 ) ( 5 − 3 ) ( 2 + 3 ) {\displaystyle \lambda ^{}\left({\frac {3}{5}}\right)={\frac {1}{8{\sqrt {2}}}}(3+{\sqrt {5}})({\sqrt {5}}-{\sqrt {3}})(2+{\sqrt {3}})} λ ∗ ( 4 5 ) = tan ⁡ [ π 4 − 1 4 arcsin ⁡ ( 5 − 2 ) ] 2 {\displaystyle \lambda ^{*}\left({\frac {4}{5}}\right)=\tan \left[{\frac {\pi }{4}}-{\frac {1}{4}}\arcsin({\sqrt {5}}-2)\right]^{2}}

Special values

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

λ ∗ ( 1 ) = 1 2 {\displaystyle \lambda ^{*}(1)={\frac {1}{\sqrt {2}}}} λ ∗ ( 5 ) = sin ⁡ [ 1 2 arcsin ⁡ ( 5 − 2 ) ] {\displaystyle \lambda ^{*}(5)=\sin \left[{\frac {1}{2}}\arcsin \left({\sqrt {5}}-2\right)\right]} λ ∗ ( 9 ) = 1 2 ( 3 − 1 ) ( 2 − 3 4 ) {\displaystyle \lambda ^{*}(9)={\frac {1}{2}}({\sqrt {3}}-1)({\sqrt {2}}-{\sqrt[{4}]{3}})} λ ∗ ( 13 ) = sin ⁡ [ 1 2 arcsin ⁡ ( 5 13 − 18 ) ] {\displaystyle \lambda ^{*}(13)=\sin \left[{\frac {1}{2}}\arcsin(5{\sqrt {13}}-18)\right]} λ ∗ ( 17 ) = sin ⁡ { 1 2 arcsin ⁡ [ 1 64 ( 5 + 17 − 10 17 + 26 ) 3 ] } {\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\}} λ ∗ ( 21 ) = sin ⁡ { 1 2 arcsin ⁡ [ ( 8 − 3 7 ) ( 2 7 − 3 3 ) ] } {\displaystyle \lambda ^{*}(21)=\sin \left\{{\frac {1}{2}}\arcsin[(8-3{\sqrt {7}})(2{\sqrt {7}}-3{\sqrt {3}})]\right\}} λ ∗ ( 25 ) = 1 2 ( 5 − 2 ) ( 3 − 2 5 4 ) {\displaystyle \lambda ^{*}(25)={\frac {1}{\sqrt {2}}}({\sqrt {5}}-2)(3-2{\sqrt[{4}]{5}})} λ ∗ ( 33 ) = sin ⁡ { 1 2 arcsin ⁡ [ ( 10 − 3 11 ) ( 2 − 3 ) 3 ] } {\displaystyle \lambda ^{*}(33)=\sin \left\{{\frac {1}{2}}\arcsin[(10-3{\sqrt {11}})(2-{\sqrt {3}})^{3}]\right\}} λ ∗ ( 37 ) = sin ⁡ { 1 2 arcsin ⁡ [ ( 37 − 6 ) 3 ] } {\displaystyle \lambda ^{*}(37)=\sin \left\{{\frac {1}{2}}\arcsin[({\sqrt {37}}-6)^{3}]\right\}} λ ∗ ( 45 ) = sin ⁡ { 1 2 arcsin ⁡ [ ( 4 − 15 ) 2 ( 5 − 2 ) 3 ] } {\displaystyle \lambda ^{*}(45)=\sin \left\{{\frac {1}{2}}\arcsin[(4-{\sqrt {15}})^{2}({\sqrt {5}}-2)^{3}]\right\}} λ ∗ ( 49 ) = 1 4 ( 8 + 3 7 ) ( 5 − 7 − 28 4 ) ( 14 − 2 − 28 8 5 − 7 ) {\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)} λ ∗ ( 57 ) = sin ⁡ { 1 2 arcsin ⁡ [ ( 170 − 39 19 ) ( 2 − 3 ) 3 ] } {\displaystyle \lambda ^{*}(57)=\sin \left\{{\frac {1}{2}}\arcsin[(170-39{\sqrt {19}})(2-{\sqrt {3}})^{3}]\right\}} λ ∗ ( 73 ) = sin ⁡ { 1 2 arcsin ⁡ [ 1 64 ( 45 + 5 73 − 3 50 73 + 426 ) 3 ] } {\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:

λ ∗ ( 2 ) = 2 − 1 {\displaystyle \lambda ^{*}(2)={\sqrt {2}}-1} λ ∗ ( 6 ) = ( 2 − 3 ) ( 3 − 2 ) {\displaystyle \lambda ^{*}(6)=(2-{\sqrt {3}})({\sqrt {3}}-{\sqrt {2}})} λ ∗ ( 10 ) = ( 10 − 3 ) ( 2 − 1 ) 2 {\displaystyle \lambda ^{*}(10)=({\sqrt {10}}-3)({\sqrt {2}}-1)^{2}} λ ∗ ( 14 ) = tan ⁡ { 1 2 arctan ⁡ [ 1 8 ( 2 2 + 1 − 4 2 + 5 ) 3 ] } {\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\}} λ ∗ ( 18 ) = ( 2 − 1 ) 3 ( 2 − 3 ) 2 {\displaystyle \lambda ^{*}(18)=({\sqrt {2}}-1)^{3}(2-{\sqrt {3}})^{2}} λ ∗ ( 22 ) = ( 10 − 3 11 ) ( 3 11 − 7 2 ) {\displaystyle \lambda ^{*}(22)=(10-3{\sqrt {11}})(3{\sqrt {11}}-7{\sqrt {2}})} λ ∗ ( 30 ) = tan ⁡ { 1 2 arctan ⁡ [ ( 10 − 3 ) 2 ( 5 − 2 ) 2 ] } {\displaystyle \lambda ^{*}(30)=\tan \left\{{\frac {1}{2}}\arctan[({\sqrt {10}}-3)^{2}({\sqrt {5}}-2)^{2}]\right\}} λ ∗ ( 34 ) = tan ⁡ { 1 4 arcsin ⁡ [ 1 9 ( 17 − 4 ) 2 ] } {\displaystyle \lambda ^{*}(34)=\tan \left\{{\frac {1}{4}}\arcsin \left[{\frac {1}{9}}({\sqrt {17}}-4)^{2}\right]\right\}} λ ∗ ( 42 ) = tan ⁡ { 1 2 arctan ⁡ [ ( 2 7 − 3 3 ) 2 ( 2 2 − 7 ) 2 ] } {\displaystyle \lambda ^{*}(42)=\tan \left\{{\frac {1}{2}}\arctan[(2{\sqrt {7}}-3{\sqrt {3}})^{2}(2{\sqrt {2}}-{\sqrt {7}})^{2}]\right\}} λ ∗ ( 46 ) = tan ⁡ { 1 2 arctan ⁡ [ 1 64 ( 3 + 2 − 6 2 + 7 ) 6 ] } {\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\}} λ ∗ ( 58 ) = ( 13 58 − 99 ) ( 2 − 1 ) 6 {\displaystyle \lambda ^{*}(58)=(13{\sqrt {58}}-99)({\sqrt {2}}-1)^{6}} λ ∗ ( 70 ) = tan ⁡ { 1 2 arctan ⁡ [ ( 5 − 2 ) 4 ( 2 − 1 ) 6 ] } {\displaystyle \lambda ^{*}(70)=\tan \left\{{\frac {1}{2}}\arctan[({\sqrt {5}}-2)^{4}({\sqrt {2}}-1)^{6}]\right\}} λ ∗ ( 78 ) = tan ⁡ { 1 2 arctan ⁡ [ ( 5 13 − 18 ) 2 ( 26 − 5 ) 2 ] } {\displaystyle \lambda ^{*}(78)=\tan \left\{{\frac {1}{2}}\arctan[(5{\sqrt {13}}-18)^{2}({\sqrt {26}}-5)^{2}]\right\}} λ ∗ ( 82 ) = tan ⁡ { 1 4 arcsin ⁡ [ 1 4761 ( 8 41 − 51 ) 2 ] } {\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:

λ ∗ ( 3 ) = 1 2 2 ( 3 − 1 ) {\displaystyle \lambda ^{*}(3)={\frac {1}{2{\sqrt {2}}}}({\sqrt {3}}-1)} λ ∗ ( 7 ) = 1 4 2 ( 3 − 7 ) {\displaystyle \lambda ^{*}(7)={\frac {1}{4{\sqrt {2}}}}(3-{\sqrt {7}})} λ ∗ ( 11 ) = 1 8 2 ( 11 + 3 ) ( 1 3 6 3 + 2 11 3 − 1 3 6 3 − 2 11 3 + 1 3 11 − 1 ) 4 {\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}} λ ∗ ( 15 ) = 1 8 2 ( 3 − 5 ) ( 5 − 3 ) ( 2 − 3 ) {\displaystyle \lambda ^{*}(15)={\frac {1}{8{\sqrt {2}}}}(3-{\sqrt {5}})({\sqrt {5}}-{\sqrt {3}})(2-{\sqrt {3}})} λ ∗ ( 19 ) = 1 8 2 ( 3 19 + 13 ) [ 1 6 ( 19 − 2 + 3 ) 3 3 − 19 3 − 1 6 ( 19 − 2 − 3 ) 3 3 + 19 3 − 1 3 ( 5 − 19 ) ] 4 {\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}} λ ∗ ( 23 ) = 1 16 2 ( 5 + 23 ) [ 1 6 ( 3 + 1 ) 100 − 12 69 3 − 1 6 ( 3 − 1 ) 100 + 12 69 3 + 2 3 ] 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}} λ ∗ ( 27 ) = 1 16 2 ( 3 − 1 ) 3 [ 1 3 3 ( 4 3 − 2 3 + 1 ) − 2 3 + 1 ] 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}} λ ∗ ( 39 ) = sin ⁡ { 1 2 arcsin ⁡ [ 1 16 ( 6 − 13 − 3 6 13 − 21 ) ] } {\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\}} λ ∗ ( 55 ) = sin ⁡ { 1 2 arcsin ⁡ [ 1 512 ( 3 5 − 3 − 6 5 − 2 ) 3 ] } {\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:

λ ∗ ( 4 ) = ( 2 − 1 ) 2 {\displaystyle \lambda ^{*}(4)=({\sqrt {2}}-1)^{2}} λ ∗ ( 8 ) = ( 2 + 1 − 2 2 + 2 ) 2 {\displaystyle \lambda ^{*}(8)=\left({\sqrt {2}}+1-{\sqrt {2{\sqrt {2}}+2}}\right)^{2}} λ ∗ ( 12 ) = ( 3 − 2 ) 2 ( 2 − 1 ) 2 {\displaystyle \lambda ^{*}(12)=({\sqrt {3}}-{\sqrt {2}})^{2}({\sqrt {2}}-1)^{2}} λ ∗ ( 16 ) = ( 2 + 1 ) 2 ( 2 4 − 1 ) 4 {\displaystyle \lambda ^{*}(16)=({\sqrt {2}}+1)^{2}({\sqrt[{4}]{2}}-1)^{4}} λ ∗ ( 20 ) = tan ⁡ [ 1 4 arcsin ⁡ ( 5 − 2 ) ] 2 {\displaystyle \lambda ^{*}(20)=\tan \left[{\frac {1}{4}}\arcsin({\sqrt {5}}-2)\right]^{2}} λ ∗ ( 24 ) = tan ⁡ { 1 2 arcsin ⁡ [ ( 2 − 3 ) ( 3 − 2 ) ] } 2 {\displaystyle \lambda ^{*}(24)=\tan \left\{{\frac {1}{2}}\arcsin[(2-{\sqrt {3}})({\sqrt {3}}-{\sqrt {2}})]\right\}^{2}} λ ∗ ( 28 ) = ( 2 2 − 7 ) 2 ( 2 − 1 ) 4 {\displaystyle \lambda ^{*}(28)=(2{\sqrt {2}}-{\sqrt {7}})^{2}({\sqrt {2}}-1)^{4}} λ ∗ ( 32 ) = tan ⁡ { 1 2 arcsin ⁡ [ ( 2 + 1 − 2 2 + 2 ) 2 ] } 2 {\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:

λ ∗ ( 1 2 ) = 2 2 − 2 {\displaystyle \lambda ^{*}\left({\frac {1}{2}}\right)={\sqrt {2{\sqrt {2}}-2}}} λ ∗ ( 1 3 ) = 1 2 2 ( 3 + 1 ) {\displaystyle \lambda ^{*}\left({\frac {1}{3}}\right)={\frac {1}{2{\sqrt {2}}}}({\sqrt {3}}+1)} λ ∗ ( 2 3 ) = ( 2 − 3 ) ( 3 + 2 ) {\displaystyle \lambda ^{*}\left({\frac {2}{3}}\right)=(2-{\sqrt {3}})({\sqrt {3}}+{\sqrt {2}})} λ ∗ ( 1 4 ) = 2 2 4 ( 2 − 1 ) {\displaystyle \lambda ^{*}\left({\frac {1}{4}}\right)=2{\sqrt[{4}]{2}}({\sqrt {2}}-1)} λ ∗ ( 3 4 ) = 8 4 ( 3 − 2 ) ( 2 + 1 ) ( 3 − 1 ) 3 {\displaystyle \lambda ^{*}\left({\frac {3}{4}}\right)={\sqrt[{4}]{8}}({\sqrt {3}}-{\sqrt {2}})({\sqrt {2}}+1){\sqrt {({\sqrt {3}}-1)^{3}}}} λ ∗ ( 1 5 ) = 1 2 2 ( 2 5 − 2 + 5 − 1 ) {\displaystyle \lambda ^{*}\left({\frac {1}{5}}\right)={\frac {1}{2{\sqrt {2}}}}\left({\sqrt {2{\sqrt {5}}-2}}+{\sqrt {5}}-1\right)} λ ∗ ( 2 5 ) = ( 10 − 3 ) ( 2 + 1 ) 2 {\displaystyle \lambda ^{*}\left({\frac {2}{5}}\right)=({\sqrt {10}}-3)({\sqrt {2}}+1)^{2}} λ ∗ ( 3 5 ) = 1 8 2 ( 3 + 5 ) ( 5 − 3 ) ( 2 + 3 ) {\displaystyle \lambda ^{*}\left({\frac {3}{5}}\right)={\frac {1}{8{\sqrt {2}}}}(3+{\sqrt {5}})({\sqrt {5}}-{\sqrt {3}})(2+{\sqrt {3}})} λ ∗ ( 4 5 ) = tan ⁡ [ π 4 − 1 4 arcsin ⁡ ( 5 − 2 ) ] 2 {\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 G n {\displaystyle G_{n}} and g n {\displaystyle g_{n}} are defined as²⁰

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 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),}

where n ∈ Q + {\displaystyle n\in \mathbb {Q} ^{+}} . For such n {\displaystyle 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)}}.}

Identities with the class invariants include²¹

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}.}

The class invariants are very closely related to the Weber modular functions f {\displaystyle {\mathfrak {f}}} and f 1 {\displaystyle {\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]} 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)]}}} 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}}

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 1879.²² 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.²³

Moonshine

The function τ ↦ 16 / λ ( 2 τ ) − 8 {\displaystyle \tau \mapsto 16/\lambda (2\tau )-8} is the normalized Hauptmodul for the group Γ 0 ( 4 ) {\displaystyle \Gamma _{0}(4)} , and its q-expansion q − 1 + 20 q − 62 q 3 + … {\displaystyle q^{-1}+20q-62q^{3}+\dots } , OEIS: A007248 where q = e 2 π i τ {\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.

Footnotes

Notes

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

References

λ ( τ ) {\displaystyle \lambda (\tau )} is not a modular function (per the Wikipedia definition), but every modular function is a rational function in λ ( τ ) {\displaystyle \lambda (\tau )} . Some authors use a non-equivalent definition of "modular functions". /wiki/Modular_form#Modular_functions ↩
Chandrasekharan (1985) p.115 ↩
Chandrasekharan (1985) p.109 ↩
Chandrasekharan (1985) p.110 ↩
Chandrasekharan (1985) p.108 ↩
Chandrasekharan (1985) p.108 ↩
Chandrasekharan (1985) p.63 ↩
Chandrasekharan (1985) p.108 ↩
Chandrasekharan (1985) p.108 ↩
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 0-471-83138-7 ↩
For any prime power, we can iterate the modular equation of degree p {\displaystyle p} . This process can be used to give algebraic values of λ ( n i ) {\displaystyle \lambda (ni)} for any n ∈ N . {\displaystyle n\in \mathbb {N} .} /wiki/Prime_power ↩
Jacobi, Carl Gustav Jacob (1829). Fundamenta nova theoriae functionum ellipticarum (in Latin). p. 42 /wiki/Carl_Gustav_Jacob_Jacobi ↩
sl ⁡ a ϖ {\displaystyle \operatorname {sl} a\varpi } is algebraic for every a ∈ Q . {\displaystyle a\in \mathbb {Q} .} ↩
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 0-471-83138-7 ↩
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. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063041 ↩
Chowla, S.; Selberg, A. "On Epstein's Zeta-Function". EuDML. pp. 86–110. https://eudml.org/doc/150803 ↩
Jacobi, Carl Gustav Jacob (1829). Fundamenta nova theoriae functionum ellipticarum (in Latin). p. 42 /wiki/Carl_Gustav_Jacob_Jacobi ↩
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. https://www.ams.org/journals/tran/1997-349-06/ ↩
Eymard, Pierre; Lafon, Jean-Pierre (1999). Autour du nombre Pi (in French). HERMANN. ISBN 2705614435. p. 240 2705614435 ↩
Chandrasekharan (1985) p.121 ↩
Chandrasekharan (1985) p.118 ↩