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 = \frac{e_3-e_2}{e_1-e_2} \, . }

${\displaystyle \lambda ={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}}\,.}$

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

Semantic latex: \lambda = \frac{e_3-e_2}{e_1-e_2}

Confidence: 0

Mathematica

Translation: \[Lambda] == Divide[Subscript[e, 3]- Subscript[e, 2],Subscript[e, 1]- Subscript[e, 2]]

Information

Sub Equations

\[Lambda] = Divide[Subscript[e, 3]- Subscript[e, 2],Subscript[e, 1]- Subscript[e, 2]]

Free variables

Subscript[e, 1]

Subscript[e, 2]

Subscript[e, 3]

\[Lambda]

Symbol info

You use a typical letter for a constant [the mathematical constant e == Napier's constant == 2.71828182845...].

We keep it like it is! But you should know that Mathematica uses E for this constant. If you want to translate it as a constant, use the corresponding DLMF macro \expe

Tests

Symbolic

Numeric

SymPy

Translation: Symbol('lambda') == (Symbol('{e}_{3}')- Symbol('{e}_{2}'))/(Symbol('{e}_{1}')- Symbol('{e}_{2}'))

Information

Sub Equations

Symbol('lambda') = (Symbol('{e}_{3}')- Symbol('{e}_{2}'))/(Symbol('{e}_{1}')- Symbol('{e}_{2}'))

Free variables

Symbol('lambda')

Symbol('{e}_{1}')

Symbol('{e}_{2}')

Symbol('{e}_{3}')

Symbol info

You use a typical letter for a constant [the mathematical constant e == Napier's constant == 2.71828182845...].

We keep it like it is! But you should know that SymPy uses E for this constant. If you want to translate it as a constant, use the corresponding DLMF macro \expe

Tests

Symbolic

Numeric

Maple

Translation: lambda = (e[3]- e[2])/(e[1]- e[2])

Information

Sub Equations

lambda = (e[3]- e[2])/(e[1]- e[2])

Free variables

e[1]

e[2]

e[3]

lambda

Symbol info

You use a typical letter for a constant [the mathematical constant e == Napier's constant == 2.71828182845...].

We keep it like it is! But you should know that Maple uses exp(1) for this constant. If you want to translate it as a constant, use the corresponding DLMF macro \expe

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\lambda$

$\lambda ={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}}\,$

Is part of

$\lambda ={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}}\,$

Complete translation information:

{
  "id" : "FORMULA_72dbd6d0c8921ed8ecf0a8ee10e759e0",
  "formula" : "\\lambda = \\frac{e_3-e_2}{e_1-e_2}",
  "semanticFormula" : "\\lambda = \\frac{e_3-e_2}{e_1-e_2}",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "\\[Lambda] == Divide[Subscript[e, 3]- Subscript[e, 2],Subscript[e, 1]- Subscript[e, 2]]",
      "translationInformation" : {
        "subEquations" : [ "\\[Lambda] = Divide[Subscript[e, 3]- Subscript[e, 2],Subscript[e, 1]- Subscript[e, 2]]" ],
        "freeVariables" : [ "Subscript[e, 1]", "Subscript[e, 2]", "Subscript[e, 3]", "\\[Lambda]" ],
        "tokenTranslations" : {
          "e" : "You use a typical letter for a constant [the mathematical constant e == Napier's constant ==  2.71828182845...].\nWe keep it like it is! But you should know that Mathematica uses E for this constant.\nIf you want to translate it as a constant, use the corresponding DLMF macro \\expe\n"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "SymPy" : {
      "translation" : "Symbol('lambda') == (Symbol('{e}_{3}')- Symbol('{e}_{2}'))/(Symbol('{e}_{1}')- Symbol('{e}_{2}'))",
      "translationInformation" : {
        "subEquations" : [ "Symbol('lambda') = (Symbol('{e}_{3}')- Symbol('{e}_{2}'))/(Symbol('{e}_{1}')- Symbol('{e}_{2}'))" ],
        "freeVariables" : [ "Symbol('lambda')", "Symbol('{e}_{1}')", "Symbol('{e}_{2}')", "Symbol('{e}_{3}')" ],
        "tokenTranslations" : {
          "e" : "You use a typical letter for a constant [the mathematical constant e == Napier's constant ==  2.71828182845...].\nWe keep it like it is! But you should know that SymPy uses E for this constant.\nIf you want to translate it as a constant, use the corresponding DLMF macro \\expe\n"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "Maple" : {
      "translation" : "lambda = (e[3]- e[2])/(e[1]- e[2])",
      "translationInformation" : {
        "subEquations" : [ "lambda = (e[3]- e[2])/(e[1]- e[2])" ],
        "freeVariables" : [ "e[1]", "e[2]", "e[3]", "lambda" ],
        "tokenTranslations" : {
          "e" : "You use a typical letter for a constant [the mathematical constant e == Napier's constant ==  2.71828182845...].\nWe keep it like it is! But you should know that Maple uses exp(1) for this constant.\nIf you want to translate it as a constant, use the corresponding DLMF macro \\expe\n"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "\\lambda", "\\lambda = \\frac{e_3-e_2}{e_1-e_2} \\," ],
  "isPartOf" : [ "\\lambda = \\frac{e_3-e_2}{e_1-e_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

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Tests

Symbolic

Numeric

SymPy

Information

Tests

Symbolic

Numeric

Maple

Information

Tests

Symbolic

Numeric

Dependency Graph Information

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