LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle \lambda(\tau) }

${\displaystyle \lambda (\tau )}$

... is translated to the CAS output ...

Semantic latex: \modularlambdatau@{\tau}

Confidence: 0.90733333333333

Mathematica

Translation: ModularLambda[\[Tau]]

Information

Sub Equations

ModularLambda[\[Tau]]

Free variables

\[Tau]

Symbol info

Elliptic modular function; Example: \modularlambdatau@{\tau}

Will be translated to: ModularLambda[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/23.15#E6 Mathematica: https://reference.wolfram.com/language/ref/ModularLambda.html

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \modularlambdatau [\modularlambdatau]

Tests

Symbolic

Numeric

Maple

Translation:

Information

Symbol info

(LaTeX -> Maple) No translation possible for given token: Cannot extract information from feature set: \modularlambdatau [\modularlambdatau]

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\tau$

$\lambda$

Is part of

$\lambda (\tau )=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+\dots$

$\lambda (\tau )=k^{2}(\tau )$

$\lambda (\tau )={\Bigg (}{\frac {{\sqrt {2}}\,\eta ({\tfrac {\tau }{2}})\eta ^{2}(2\tau )}{\eta ^{3}(\tau )}}{\Bigg )}^{8}={\frac {16}{\left({\frac {\eta (\tau /2)}{\eta (2\tau )}}\right)^{8}+16}}={\frac {\theta _{2}^{4}(0,\tau )}{\theta _{3}^{4}(0,\tau )}}$

${\frac {1}{{\big (}\lambda (\tau ){\big )}^{1/4}}}-{\big (}\lambda (\tau ){\big )}^{1/4}={\frac {1}{2}}\left({\frac {\eta ({\tfrac {\tau }{4}})}{\eta (\tau )}}\right)^{4}=2\,{\frac {\theta _{4}^{2}(0,{\tfrac {\tau }{2}})}{\theta _{2}^{2}(0,{\tfrac {\tau }{2}})}}$

Failed to parse (syntax error): {\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} \}

${\frac {16}{\lambda (2\tau )}}-8$

Description

function

elliptic modular lambda function

symmetric holomorphic function on the complex upper half-plane

mathematics

nome

q-expansion

square of the Jacobi modulus

term of the Dedekind eta function

term of the half-period

relation to the j-invariant

Hauptmodul for the group

group

action of the modular group

graded character of any element

fundamental pair of period

theta function

expansion

anharmonic group

Weierstrass 's elliptic function

conjugacy class 4c of the monster group

value

monster vertex algebra

Complete translation information:

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  "formula" : "\\lambda(\\tau)",
  "semanticFormula" : "\\modularlambdatau@{\\tau}",
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      "translation" : "ModularLambda[\\[Tau]]",
      "translationInformation" : {
        "subEquations" : [ "ModularLambda[\\[Tau]]" ],
        "freeVariables" : [ "\\[Tau]" ],
        "tokenTranslations" : {
          "\\modularlambdatau" : "Elliptic modular function; Example: \\modularlambdatau@{\\tau}\nWill be translated to: ModularLambda[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/23.15#E6\nMathematica:  https://reference.wolfram.com/language/ref/ModularLambda.html"
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    "Maple" : {
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  "includes" : [ "\\tau", "\\lambda" ],
  "isPartOf" : [ "\\lambda(\\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \\dots", "\\lambda(\\tau)=k^2(\\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(0,\\tau)}{\\theta_3^4(0,\\tau)}", "\\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(0,\\tfrac{\\tau}{2})}{\\theta_2^2(0,\\tfrac{\\tau}{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} \\", "\\frac{16}{\\lambda(2\\tau)} - 8" ],
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    "definition" : "elliptic modular lambda function",
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    "definition" : "symmetric holomorphic function on the complex upper half-plane",
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    "definition" : "q-expansion",
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    "definition" : "square of the Jacobi modulus",
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    "definition" : "term of the Dedekind eta function",
    "score" : 0.35175394239749164
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    "definition" : "term of the half-period",
    "score" : 0.35175394239749164
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    "definition" : "relation to the j-invariant",
    "score" : 0.3515992754597499
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    "definition" : "Hauptmodul for the group",
    "score" : 0.35159851049127155
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    "definition" : "action of the modular group",
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    "definition" : "graded character of any element",
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    "definition" : "fundamental pair of period",
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    "definition" : "theta function",
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    "definition" : "anharmonic group",
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  }, {
    "definition" : "Weierstrass 's elliptic function",
    "score" : 0.23797293143965273
  }, {
    "definition" : "conjugacy class 4c of the monster group",
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    "definition" : "monster vertex algebra",
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}

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Formula input
Wikitext context	In [[mathematics]], the '''elliptic 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 subgroup\|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]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution. The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)\|nome]], is given by: : <math> \lambda(\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \dots</math>. {{oeis\|id=A115977 }} By symmetrizing the lambda function under the canonical action of the symmetric group ''S''<sub>3</sub> on ''X''(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group <math>SL_2(\mathbb{Z})</math>, and it is in fact Klein's modular [[j-invariant]]. ==Modular properties== The function <math> \lambda(\tau) </math> is invariant under the group generated by<ref name=C115>Chandrasekharan (1985) p.115</ref> :<math> \tau \mapsto \tau+2 \ ;\ \tau \mapsto \frac{\tau}{1-2\tau} \ . </math> The generators of the modular group act by<ref name=C109>Chandrasekharan (1985) p.109</ref> :<math> \tau \mapsto \tau+1 \ :\ \lambda \mapsto \frac{\lambda}{\lambda-1} \, ;</math> :<math> \tau \mapsto -\frac{1}{\tau} \ :\ \lambda \mapsto 1 - \lambda \ . </math> Consequently, the action of the modular group on <math> \lambda(\tau) </math> is that of the [[anharmonic group]], giving the six values of the [[cross-ratio]]:<ref name=C110>Chandrasekharan (1985) p.110</ref> :<math> \left\lbrace { \lambda, \frac{1}{1-\lambda}, \frac{\lambda-1}{\lambda}, \frac{1}{\lambda}, \frac{\lambda}{\lambda-1}, 1-\lambda } \right\rbrace \ .</math> == Other appearances == ===Other elliptic functions=== It is the [[Square (algebra)\|square]] of the [[Jacobi modulus]],<ref name=C108>Chandrasekharan (1985) p.108</ref> that is, <math>\lambda(\tau)=k^2(\tau)</math>. In terms of the [[Dedekind eta function]] <math>\eta(\tau)</math> and [[theta function]]s,<ref name=C108/> :<math> \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(0,\tau)}{\theta_3^4(0,\tau)} </math> and, :<math> \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(0,\tfrac{\tau}{2})}{\theta_2^2(0,\tfrac{\tau}{2})}</math> where<ref name=C63>Chandrasekharan (1985) p.63</ref> for the [[Nome (mathematics)\|nome]] <math>q = e^{\pi i \tau}</math>, :<math>\theta_2(0,\tau) = \sum_{n=-\infty}^\infty q^{\left({n+\frac12}\right)^2}</math> :<math>\theta_3(0,\tau) = \sum_{n=-\infty}^\infty q^{n^2} </math> :<math>\theta_4(0,\tau) = \sum_{n=-\infty}^\infty (-1)^n q^{n^2} </math> 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), e_2 = \wp\left(\frac{\omega_2}{2}\right), e_3 = \wp\left(\frac{\omega_1+\omega_2}{2}\right) </math> we have<ref name=C108/> :<math> \lambda = \frac{e_3-e_2}{e_1-e_2} \, . </math> Since the three half-period values are distinct, this shows that λ 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> :<math> 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} \ . </math> which is the ''j''-invariant of the elliptic curve of [[Legendre form]] <math>y^2=x(x-1)(x-\lambda)</math> ===Little Picard theorem=== The lambda function is used in the original proof of the [[Little Picard theorem]], that an [[entire function\|entire]] non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard in 1879.<ref>Chandrasekharan (1985) p.121</ref> 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 (complex analysis)\|Liouville's theorem]] must be constant.<ref>Chandrasekharan (1985) p.118</ref> ===Moonshine=== The function <math>\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== {{reflist}} ==References== {{Citation \| editor1-last=Abramowitz \| editor1-first=Milton \| editor1-link=Milton Abramowitz \| editor2-last=Stegun \| editor2-first=Irene A. \| editor2-link=Irene Stegun \| title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables \| publisher=[[Dover Publications]] \| location=New York \| isbn=978-0-486-61272-0 \| year=1972 \| zbl=0543.33001 \| url-access=registration \| url=https://archive.org/details/handbookofmathe000abra }} {{citation \| last=Chandrasekharan \| first=K. \| authorlink=K. S. Chandrasekharan \| title=Elliptic Functions \| series=Grundlehren der mathematischen Wissenschaften \| volume=281 \| publisher=[[Springer-Verlag]] \| year=1985 \| isbn=3-540-15295-4 \| zbl=0575.33001 \| pages=108–121 }} {{citation\|first1=John Horton\|last1=Conway\|author1-link=John Horton Conway\|first2=Simon\|last2=Norton\|author2-link=Simon P. Norton\|title=Monstrous moonshine\|journal=Bulletin of the London Mathematical Society\|volume=11\|issue=3\|year=1979\|pages=308–339\|mr=0554399\|zbl=0424.20010 \|doi=10.1112/blms/11.3.308}} {{citation \| last=Rankin \| first=Robert A. \| authorlink=Robert Alexander Rankin \| title=Modular Forms and Functions \| publisher=[[Cambridge University Press]] \| year=1977 \| isbn=0-521-21212-X \| zbl=0376.10020 }} *{{dlmf\|id=23.15.E6\|title=Elliptic Modular Function\|first= W. P. \|last=Reinhardt\|first2=P. L.\|last2= Walker}} {{DEFAULTSORT:Modular Lambda Function}} [[Category:Modular forms]] [[Category:Elliptic functions]]
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