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 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} \ . }

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

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

Semantic latex: j(\tau) = \frac{256(1 - \modularlambdatau@{1 - \lambda})^3}{(\modularlambdatau@{1 - \lambda})^2} = \frac{256(1-\lambda+\lambda^2)^3}{\lambda^2 (1-\lambda)^2}

Confidence: 0.6805

Mathematica

Translation: j[\[Tau]] == Divide[256*(1 - ModularLambda[1 - \[Lambda]])^(3),(ModularLambda[1 - \[Lambda]])^(2)] == Divide[256*(1 - \[Lambda]+ \[Lambda]^(2))^(3),\[Lambda]^(2)*(1 - \[Lambda])^(2)]

Information

Sub Equations

j[\[Tau]] = Divide[256*(1 - ModularLambda[1 - \[Lambda]])^(3),(ModularLambda[1 - \[Lambda]])^(2)]

Divide[256*(1 - ModularLambda[1 - \[Lambda]])^(3),(ModularLambda[1 - \[Lambda]])^(2)] = Divide[256*(1 - \[Lambda]+ \[Lambda]^(2))^(3),\[Lambda]^(2)*(1 - \[Lambda])^(2)]

Free variables

\[Lambda]

\[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

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Test expression: (j*(\[Tau]))-(Divide[256*(1 - ModularLambda[1 - \[Lambda]])^(3),(ModularLambda[1 - \[Lambda]])^(2)])

ERROR:

{
    "result": "ERROR",
    "testTitle": "Simple",
    "testExpression": null,
    "resultExpression": null,
    "wasAborted": false,
    "conditionallySuccessful": false
}

Test expression: (Divide[256*(1 - ModularLambda[1 - \[Lambda]])^(3),(ModularLambda[1 - \[Lambda]])^(2)])-(Divide[256*(1 - \[Lambda]+ \[Lambda]^(2))^(3),\[Lambda]^(2)*(1 - \[Lambda])^(2)])

ERROR:

{
    "result": "ERROR",
    "testTitle": "Simple",
    "testExpression": null,
    "resultExpression": null,
    "wasAborted": false,
    "conditionallySuccessful": false
}

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

$\tau$

$\lambda$

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

$j$

$\lambda (\tau )$

Is part of

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

Complete translation information:

{
  "id" : "FORMULA_48b2456af7541e275f2039a6b92a4579",
  "formula" : "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}",
  "semanticFormula" : "j(\\tau) = \\frac{256(1 - \\modularlambdatau@{1 - \\lambda})^3}{(\\modularlambdatau@{1 - \\lambda})^2} = \\frac{256(1-\\lambda+\\lambda^2)^3}{\\lambda^2 (1-\\lambda)^2}",
  "confidence" : 0.6805,
  "translations" : {
    "Mathematica" : {
      "translation" : "j[\\[Tau]] == Divide[256*(1 - ModularLambda[1 - \\[Lambda]])^(3),(ModularLambda[1 - \\[Lambda]])^(2)] == Divide[256*(1 - \\[Lambda]+ \\[Lambda]^(2))^(3),\\[Lambda]^(2)*(1 - \\[Lambda])^(2)]",
      "translationInformation" : {
        "subEquations" : [ "j[\\[Tau]] = Divide[256*(1 - ModularLambda[1 - \\[Lambda]])^(3),(ModularLambda[1 - \\[Lambda]])^(2)]", "Divide[256*(1 - ModularLambda[1 - \\[Lambda]])^(3),(ModularLambda[1 - \\[Lambda]])^(2)] = Divide[256*(1 - \\[Lambda]+ \\[Lambda]^(2))^(3),\\[Lambda]^(2)*(1 - \\[Lambda])^(2)]" ],
        "freeVariables" : [ "\\[Lambda]", "\\[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",
          "j" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "ERROR",
        "numberOfTests" : 2,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 2,
        "crashed" : false,
        "testCalculationsGroup" : [ {
          "lhs" : "j*(\\[Tau])",
          "rhs" : "Divide[256*(1 - ModularLambda[1 - \\[Lambda]])^(3),(ModularLambda[1 - \\[Lambda]])^(2)]",
          "testExpression" : "(j*(\\[Tau]))-(Divide[256*(1 - ModularLambda[1 - \\[Lambda]])^(3),(ModularLambda[1 - \\[Lambda]])^(2)])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        }, {
          "lhs" : "Divide[256*(1 - ModularLambda[1 - \\[Lambda]])^(3),(ModularLambda[1 - \\[Lambda]])^(2)]",
          "rhs" : "Divide[256*(1 - \\[Lambda]+ \\[Lambda]^(2))^(3),\\[Lambda]^(2)*(1 - \\[Lambda])^(2)]",
          "testExpression" : "(Divide[256*(1 - ModularLambda[1 - \\[Lambda]])^(3),(ModularLambda[1 - \\[Lambda]])^(2)])-(Divide[256*(1 - \\[Lambda]+ \\[Lambda]^(2))^(3),\\[Lambda]^(2)*(1 - \\[Lambda])^(2)])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\modularlambdatau [\\modularlambdatau]"
        }
      }
    },
    "Maple" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Maple) No translation possible for given token: Cannot extract information from feature set: \\modularlambdatau [\\modularlambdatau]"
        }
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "\\tau", "\\lambda", "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", "\\lambda(\\tau)" ],
  "isPartOf" : [ "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} " ],
  "definiens" : [ ]
}

Specify your own input

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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LaTeX to CAS translator

Mathematica

Information

Tests

Symbolic

Numeric

SymPy

Information

Tests

Symbolic

Numeric

Maple

Information

Tests

Symbolic

Numeric

Dependency Graph Information

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