Modular lambda function: Difference between revisions

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{{short description|Symmetric holomorphic function}}

[[File:Modular lambda function in range -3 to 3.png|thumb|Modular lambda function in the complex plane.]]

In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[holomorphic function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup|congruence group]] &Gamma;(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]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [&minus;1] involution.

The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)|nome]], is given by:

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In terms of the half-periods of [[Weierstrass's elliptic functions]], let <math>[\omega_1,\omega_2]</math> be a [[fundamental pair of periods]] with <math>\tau=\frac{\omega_2}{\omega_1}</math>.

:<math> 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) </math>

we have<ref name=C108/>

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:<math> \lambda = \frac{e_3-e_2}{e_1-e_2} \, . </math>

Since the three half-period values are distinct, this shows that λ<math>\lambda</math> does not take the value 0 or 1.<ref name=C108/>

The relation to the [[j-invariant]] is<ref name=C117>Chandrasekharan (1985) p.117</ref><ref>Rankin (1977) pp.226–228</ref>

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which is the ''j''-invariant of the elliptic curve of [[Legendre form]] <math>y^2=x(x-1)(x-\lambda)</math>

whereGiven <math>nm\in\mathbb{QC}\setminus\{0,1\}^+</math>., let

:<math>\tau=i\frac{K\{1-m\}}{K\{m\}}</math>

where <math>K</math> is the [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]] with parameter <math>m=k^2</math>.

Then

:<math>\lambda (\tau)=m.</math>

==Modular equations==

The ''modular equation of degree'' <math>p</math> (where <math>p</math> is a prime number) is an algebraic equation in <math>\alpha = \lambda (p\tau)</math> and <math>\beta =\lambda (\tau)</math>. If <math>\alphalambda (p\tau)=u^8</math> and <math>\betalambda (\tau)=v^8</math>, the modular equations of degrees <math>p=2,3,5,7</math> are, respectively,<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 103–109, 134</ref>

:<math>(1+u^4)^2v^8-4u^4=0,</math>

:<math>u^4-v^4+2uv(1-u^2v^2)=0,</math>

:<math>u^6-v^6+5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0,</math>

:<math>(1-u^8)(1-v^8)-(1-uv)^8=0.</math>

The quantitiesquantity <math>uv</math> (and hence <math>vu</math>) havecan the followingbe productthought representationsof whichas define them asa [[Holomorphicholomorphic function|holomorphic functions]] inon the whole upper half-plane <math>\operatorname{Im}\tau>0</math>:

:<math>u=\sqrtbegin{2}e^{p\pi i\tau/8align}v&=\prod_{k=1}^\infty \tanh\frac{1+e^{2kp\pi i\tau}}{1+e^{(2kk-1/2)p\pi i}{\tau}},\quad v=\sqrt{2}e^{\pi i\tau/8}\prod_frac{\sum_{k=1}^\infty in\fracmathbb{1+Z}}e^{(2k^2+k)\pi i\tau}}{1+\sum_{k\in\mathbb{Z}}e^{(2k-1)k^2\pi i\tau}}.</math>\\

&=\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{align}</math>

Since <math>\lambda(i)=1/2</math>, the modular equations can be used to give [[Algebraic number|algebraic values]] of <math>\lambda(pi)</math> for any prime <math>p</math>.<ref group="note">For any [[prime power]], we can iterate the modular equation of degree <math>p.</math>. This process can be used to give algebraic values of <math>\lambda (ni)</refmath> Forfor any odd <math>n\in\mathbb{N}.</math>,</ref> theThe algebraic values of <math>\lambda(ni)</math> are also given by<ref name="Jacobi">{{Cite book |last1=Jacobi |first1=Carl Gustav Jacob |author-link=Carl Gustav Jacob Jacobi|title=Fundamenta nova theoriae functionum ellipticarum|language=Latin|year=1829}} p. 42</ref><ref group="note"><math>\operatorname{sl}a\varpi</math> is algebraic for every <math>a\in\mathbb{Q}.</math></ref>

:<math>\lambda (ni)=\frac{1}{2^n}\prod_{k=1}^{n-1} \left(1-\operatorname{sl}^2\frac{k\varpi}{n}\right)^2</math>

:<math>\lambda (ni)=\prod_{k=1}^{n/2} \operatorname{sl}^8\frac{(2k-1)\varpi}{2n}\quad (n\,\text{even})</math>

where <math>\operatorname{sl}</math> is the [[Lemniscate elliptic functions|lemniscate sine]] and <math>\varpi</math> is the [[Lemniscate elliptic functions#Lemniscate constant|lemniscate constant]].

:<math>\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})</math>

where <math>\operatorname{sl}</math> is the [[Lemniscate elliptic functions|lemniscate sine]] and <math>\varpi</math> is the [[Lemniscate elliptic functions#Lemniscate constant|lemniscate constant]].

==Lambda-star==

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===Definition and computation of lambda-star===

The function λ<math>\lambda^*(x)</math><ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 152</ref> (where <math>x\in\mathbb{R}^+</math>) gives the value of the elliptic modulus <math>k</math>, for which the complete [[Elliptic integral#Complete elliptic integral]] of the first kind|complete elliptic integral of the first kind]] <math>K(k)</math> and its complementary counterpart <math>K\left(\sqrt{1-k^2}\right)</math> are related by following expression:

:<math>\frac{K\left[\sqrt{1-\lambda^*(x)^2}\right]}{K[\lambda^*(x)]} = \sqrt{x}</math>

The values of λ<math>\lambda^*(x)</math> can be computed as follows:

:<math>\lambda^*(x) = \frac{\theta^2_2(i\sqrt{x})}{\theta^2_3(i\sqrt{x})} </math>

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:<math>\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} </math>

The functions λ<math>\lambda^*</math> and λ<math>\lambda</math> are related to each other in this way:

:<math>\lambda^*(x) = \sqrt{\lambda(i\sqrt{x})}</math>

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===Properties of lambda-star===

Every λ<math>\lambda^*-</math> value of a positive [[rational number]] is a positive [[algebraic number]]:

:<math>\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+ .</math>

Elliptic<math>K(\lambda^*(x))</math> integralsand of<math>E(\lambda^*(x))</math> (the first[[Elliptic andintegral#Complete elliptic integral of the second kind|complete elliptic integral of thesethe specialsecond λ*-valueskind]]) can be expressed in closed form in terms of the [[gamma function]] for any <math>x\in\mathbb{Q}^+</math>, as Selberg and Chowla proved in 1949.<ref>{{Cite web|url=https://www.semanticscholar.org/paper/On-Epstein's-Zeta-Function-(I).-Chowla-Selberg/87dc02200853b431bfa900e297cd6c2f80a5a4b1journal|title=On Epstein's Zeta Function (I).|last1=Chowla|first1=S.|last2=Selberg|first2=A.|websitejournal=SemanticProceedings of the National Academy of Sciences |date=1949 |volume=35 |issue=7 Scholar|page=373|doi=10.1073/PNAS.35.7.371 |s2cid=45071481 |doi-access=free|pmc=1063041}}</ref><ref>{{Cite web|url=https://eudml.org/doc/150803|title=On Epstein's Zeta-Function|last1=Chowla|first1=S.|last2=Selberg|first2=A.|website=EuDML|pagepages=86–110}}</ref>

FollowingThe following expression is valid for all n ∈ <math>n \in \mathbb{N}</math>:

:<math>\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] </math>

In this formula,where <math>\operatorname{dn}</math> is the [[Jacobi elliptic function]] delta amplitudinis with modulus <math>k</math>.

By knowing one λ<math>\lambda^*-</math> value, this formula can be used to compute related λ<math>\lambda^*-</math> values:<ref name="Jacobi"/>

:<math>\lambda^*(n^2x) = \lambda^*(x)^n\prod_{a=1}^{n}\operatorname{sn}\left\{\frac{2a-1}{n}K[\lambda^*(x)];\lambda^*(x)\right\}^2 </math>

Inwhere that<math>n\in\mathbb{N}</math> formula,and <math>\operatorname{sn}</math> is the Jacobi elliptic function sinus amplitudinis with modulus <math>k</math>.

That formula works for all natural numbers.

Further relations:

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:<math>\lambda^*(x) - \lambda^*(9x) = 2[\lambda^*(x)\lambda^*(9x)]^{1/4} - 2[\lambda^*(x)\lambda^*(9x)]^{3/4}</math>

<math display=block>\begin{align}

:<math>\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} </math>

:<math>(& a^2{6}-df^2)(a^4+d^4-7a^2d^2)[(a^2-d^2)^4-a^2d^2(a^2+d^2)^2]{6} = 8ad2af +8a2a^{13}d5f^{13}5\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(df = \left[\frac{2\lambda^*(169x25x)}{1-\lambda^*(169x25x)^2}\right]^{1/12}\right) </math>\\

&a^{8}+b^{8}-7a^4b^4 = 2\sqrt{2}ab+2\sqrt{2}a^7b^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) \\

:<math>& a^{812}+b-c^{812}-7a^4b^4 = 2\sqrt{2}ab(ac+a^3c^3)(1+3a^2c^2\sqrt{2}+a^7b4c^74)(2+3a^2c^2+2a^4c^4)\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(bc = \left[\frac{2\lambda^*(49x121x)}{1-\lambda^*(49x121x)^2}\right]^{1/12}\right) </math>\\

:<math> & (a^{12}2-cd^{12} = 2\sqrt{2})(ac+a^3c4+d^34-7a^2d^2)[(1+3aa^2c2-d^2+)^4-a^4c2d^4)2(a^2+3ad^2c2)^2] = 8ad+2a8a^4c{13}d^4){13}\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(cd = \left[\frac{2\lambda^*(121x169x)}{1-\lambda^*(121x169x)^2}\right]^{1/12}\right) </math>

\end{align}

</math>

{{Collapse top|title=Special values}}

:<math>(a^2-d^2)(a^4+d^4-7a^2d^2)[(a^2-d^2)^4-a^2d^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) </math>

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

===Ramanujan's class invariants===

Ramanujan's class invariants <math>G_n</math> and <math>g_n</math> are defined as<ref>Zhang, Liang-Cheng "Ramanujan's class invariants, Kronecker's limit formula and modular equations (III)"</ref>

:<math>G_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1+e^{-(2k+1)\pi\sqrt{n}}\right),</math>

:<math>g_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1-e^{-(2k+1)\pi\sqrt{n}}\right),</math>

where <math>n\in\mathbb{Q}^+</math>.

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

:<math>G_n = \sin\{2\arcsin[\lambda^*(n)]\}^{-1/12} = 1\Big /\left[\sqrt[12]{2\lambda^*(n)}\sqrt[24]{1-\lambda^*(n)^2}\right] </math>

:<math>g_n = \tan\{2\arctan[\lambda^*(n)]\}^{-1/12} = \sqrt[12]{[1-\lambda^*(n)^2]/[2\lambda^*(n)]} </math>

:<math>\lambda^*(n) = \tan\left\{ \frac{1}{2}\arctan[g_n^{-12}]\right\} = \sqrt{g_n^{24}+1}-g_n^{12} </math>

===Special Values===

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

:<math>\lambda^*(1) = \frac{1}{\sqrt{2}}</math>

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:<math>\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\}</math>

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

:<math>\lambda^*(2) = \sqrt{2}-1</math>

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:<math>\lambda^*(82) = \tan\left\{\frac{1}{4}\arcsin\left[\frac{1}{4761}(8\sqrt{41}-51)^2\right]\right\}</math>

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

:<math>\lambda^*(3) = \frac{1}{2\sqrt{2}}(\sqrt{3}-1)</math>

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:<math>\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\}</math>

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

:<math>\lambda^*(4) = (\sqrt{2}-1)^2</math>

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:<math>\lambda^*(32) = \tan\left\{\frac{1}{2}\arcsin\left[\left(\sqrt{2}+1-\sqrt{2\sqrt{2}+2}\right)^2\right]\right\}^2</math>

Lambda-star- values of rational fractions:

:<math>\lambda^*\left(\frac{1}{2}\right) = \sqrt{2\sqrt{2}-2}</math>

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:<math>\lambda^*\left(\frac{4}{5}\right) = \tan\left[\frac{\pi}{4}-\frac{1}{4}\arcsin(\sqrt{5}-2)\right]^2</math>

{{Collapse bottom}}

===Ramanujan's class invariants===

[[Srinivasa Ramanujan|Ramanujan's]] class invariants <math>G_n</math> and <math>g_n</math> are defined as<ref>{{cite journal |last1=Berndt |first1=Bruce C. |last2=Chan |first2=Heng Huat|last3=Zhang|first3=Liang-Cheng |date=6 June 1997 |title=Ramanujan's class invariants, Kronecker's limit formula, and modular equations|url=https://www.ams.org/journals/tran/1997-349-06/ |journal=Transactions of the American Mathematical Society|volume=349|issue=6|pages=2125–2173}}</ref>

:<math>G_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1+e^{-(2k+1)\pi\sqrt{n}}\right),</math>

:<math>g_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1-e^{-(2k+1)\pi\sqrt{n}}\right),</math>

where <math>n\in\mathbb{Q}^+</math>. For such <math>n</math>, the class invariants are algebraic numbers. For example

:<math>g_{58}=\sqrt{\frac{5+\sqrt{29}}{2}}, \quad g_{190}=\sqrt{(\sqrt{5}+2)(\sqrt{10}+3)}.</math>

Identities with the class invariants include<ref>{{Cite book |last1=Eymard |first1=Pierre |last2=Lafon| first2=Jean-Pierre |title=Autour du nombre Pi |language=French|publisher=HERMANN |year=1999 |isbn=2705614435}} p. 240</ref>

:<math>G_n=G_{1/n},\quad g_{n}=\frac{1}{g_{4/n}},\quad g_{4n}=2^{1/4}g_nG_n.</math>

The class invariants are very closely related to the [[Weber modular function|Weber modular functions]] <math>\mathfrak{f}</math> and <math>\mathfrak{f}_1</math>. These are the relations between lambda-star and the class invariants:

:<math>G_n = \sin\{2\arcsin[\lambda^*(n)]\}^{-1/12} = 1\Big /\left[\sqrt[12]{2\lambda^*(n)}\sqrt[24]{1-\lambda^*(n)^2}\right] </math>

:<math>g_n = \tan\{2\arctan[\lambda^*(n)]\}^{-1/12} = \sqrt[12]{[1-\lambda^*(n)^2]/[2\lambda^*(n)]} </math>

:<math>\lambda^*(n) = \tan\left\{ \frac{1}{2}\arctan[g_n^{-12}]\right\} = \sqrt{g_n^{24}+1}-g_n^{12} </math>

== Other appearances ==

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===Moonshine===

The function <math>\tau\mapsto\frac{ 16}{/\lambda(2\tau)} - 8</math> is the normalized [[Hauptmodul]] for the group <math>\Gamma_0(4)</math>, and its ''q''-expansion <math>q^{-1} + 20q - 62q^3 + \dots</math>, {{oeis|id=A007248}} where <math>q=e^{2\pi i\tau }</math>, is the graded character of any element in conjugacy class 4C of the [[monster group]] acting on the [[monster vertex algebra]].

==Footnotes==