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 q = e^{\pi i \tau}}

${\displaystyle q=e^{\pi i\tau }}$

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

Semantic latex: q = \expe^{\cpi \iunit \tau}

Confidence: 0

Mathematica

Translation: q == Exp[Pi*I*\[Tau]]

Information

Sub Equations

q = Exp[Pi*I*\[Tau]]

Free variables

\[Tau]

q

Symbol info

Recognizes e with power as the exponential function. It was translated as a function.

Imaginary unit was translated to: I

Pi was translated to: Pi

Tests

Symbolic

Test expression: (q)-(Exp[Pi*I*\[Tau]])

ERROR:

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

Numeric

SymPy

Translation: q == exp(pi*I*Symbol('tau'))

Information

Sub Equations

q = exp(pi*I*Symbol('tau'))

Free variables

Symbol('tau')

q

Symbol info

Recognizes e with power as the exponential function. It was translated as a function.

Imaginary unit was translated to: I

Pi was translated to: pi

Tests

Symbolic

Numeric

Maple

Translation: q = exp(Pi*I*tau)

Information

Sub Equations

q = exp(Pi*I*tau)

Free variables

q

tau

Symbol info

Recognizes e with power as the exponential function. It was translated as a function.

Imaginary unit was translated to: I

Pi was translated to: Pi

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\tau$

$q$

Is part of

$q=e^{2\pi i\tau }$

Description

nome

q-expansion

term of the half-period

graded character of any element

expansion

Hauptmodul for the group

term of the Dedekind eta function

conjugacy class 4c of the monster group

function

Weierstrass 's elliptic function

fundamental pair of period

theta function

monster vertex algebra

Complete translation information:

{
  "id" : "FORMULA_337afd813b14137226c62a73aade6478",
  "formula" : "q = e^{\\pi i \\tau}",
  "semanticFormula" : "q = \\expe^{\\cpi \\iunit \\tau}",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "q == Exp[Pi*I*\\[Tau]]",
      "translationInformation" : {
        "subEquations" : [ "q = Exp[Pi*I*\\[Tau]]" ],
        "freeVariables" : [ "\\[Tau]", "q" ],
        "tokenTranslations" : {
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "\\iunit" : "Imaginary unit was translated to: I",
          "\\cpi" : "Pi was translated to: Pi"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "ERROR",
        "numberOfTests" : 1,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 1,
        "crashed" : false,
        "testCalculationsGroup" : [ {
          "lhs" : "q",
          "rhs" : "Exp[Pi*I*\\[Tau]]",
          "testExpression" : "(q)-(Exp[Pi*I*\\[Tau]])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "q == exp(pi*I*Symbol('tau'))",
      "translationInformation" : {
        "subEquations" : [ "q = exp(pi*I*Symbol('tau'))" ],
        "freeVariables" : [ "Symbol('tau')", "q" ],
        "tokenTranslations" : {
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "\\iunit" : "Imaginary unit was translated to: I",
          "\\cpi" : "Pi was translated to: pi"
        }
      }
    },
    "Maple" : {
      "translation" : "q = exp(Pi*I*tau)",
      "translationInformation" : {
        "subEquations" : [ "q = exp(Pi*I*tau)" ],
        "freeVariables" : [ "q", "tau" ],
        "tokenTranslations" : {
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "\\iunit" : "Imaginary unit was translated to: I",
          "\\cpi" : "Pi was translated to: Pi"
        }
      }
    }
  },
  "positions" : [ {
    "section" : 0,
    "sentence" : 3,
    "word" : 4
  }, {
    "section" : 2,
    "sentence" : 1,
    "word" : 20
  } ],
  "includes" : [ "\\tau", "q" ],
  "isPartOf" : [ "q=e^{2\\pi i\\tau }" ],
  "definiens" : [ {
    "definition" : "nome",
    "score" : 0.8869384888466118
  }, {
    "definition" : "q-expansion",
    "score" : 0.6687181434333315
  }, {
    "definition" : "term of the half-period",
    "score" : 0.3594939591774544
  }, {
    "definition" : "graded character of any element",
    "score" : 0.351598510491561
  }, {
    "definition" : "expansion",
    "score" : 0.3249086196240972
  }, {
    "definition" : "Hauptmodul for the group",
    "score" : 0.3249086196240972
  }, {
    "definition" : "term of the Dedekind eta function",
    "score" : 0.29297012797898336
  }, {
    "definition" : "conjugacy class 4c of the monster group",
    "score" : 0.28507467929308994
  }, {
    "definition" : "function",
    "score" : 0.28507467929308994
  }, {
    "definition" : "Weierstrass 's elliptic function",
    "score" : 0.2457129482196155
  }, {
    "definition" : "fundamental pair of period",
    "score" : 0.19699068678142098
  }, {
    "definition" : "theta function",
    "score" : 0.19699068678142098
  }, {
    "definition" : "monster vertex algebra",
    "score" : 0.14390742558173894
  } ]
}

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

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Tests

Symbolic

Numeric

SymPy

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Tests

Symbolic

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Maple

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Tests

Symbolic

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Dependency Graph Information

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