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 G=\frac{2}{\pi}\int_0^1\frac{dt}{\sqrt{1-t^4}}= 0.8346\ldots.}

${\displaystyle G={\frac {2}{\pi }}\int _{0}^{1}{\frac {dt}{\sqrt {1-t^{4}}}}=0.8346\ldots .}$

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

Semantic latex: G = \frac{2}{\cpi} \int_0^1 \frac{dt}{\sqrt{1-t^4}} = 0.8346 \ldots

Confidence: 0

Mathematica

Translation:

Information

Symbol info

(LaTeX -> Mathematica) The input LaTeX is invalid: Unable to retrieve free variables for limit expression.

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) The input LaTeX is invalid: Unable to retrieve free variables for limit expression.

Tests

Symbolic

Numeric

Maple

Translation:

Information

Symbol info

(LaTeX -> Maple) The input LaTeX is invalid: Unable to retrieve free variables for limit expression.

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$G$

Description

period

Gauss

function in the complex plane

Complete translation information:

{
  "id" : "FORMULA_2e962dde319af6e38a01196a40db0b86",
  "formula" : "G=\\frac{2}{\\pi}\\int_0^1\\frac{dt}{\\sqrt{1-t^4}}= 0.8346\\ldots",
  "semanticFormula" : "G = \\frac{2}{\\cpi} \\int_0^1 \\frac{dt}{\\sqrt{1-t^4}} = 0.8346 \\ldots",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Mathematica) The input LaTeX is invalid: Unable to retrieve free variables for limit expression."
        }
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) The input LaTeX is invalid: Unable to retrieve free variables for limit expression."
        }
      }
    },
    "Maple" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Maple) The input LaTeX is invalid: Unable to retrieve free variables for limit expression."
        }
      }
    }
  },
  "positions" : [ {
    "section" : 1,
    "sentence" : 2,
    "word" : 28
  } ],
  "includes" : [ "G" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "period",
    "score" : 0.6859086196238077
  }, {
    "definition" : "Gauss",
    "score" : 0.6460746792928004
  }, {
    "definition" : "function in the complex plane",
    "score" : 0.5988174995334326
  } ]
}

Specify your own input

Formula input
Wikitext context	{{Expand German\|Lemniskatischer Sinus\|date=August 2017}} In [[mathematics]], a '''lemniscatic elliptic function''' is an [[elliptic function]] related to the arc length of a [[lemniscate of Bernoulli]] studied by [[Giulio Carlo de' Toschi di Fagnano]] in 1718. It has a square period lattice and is closely related to the [[Weierstrass elliptic function]] when the Weierstrass invariants satisfy {{math\|''g''<sub>2</sub> {{=}} 1}} and {{math\|''g''<sub>3</sub> {{=}} 0}}. In the lemniscatic case, the minimal half period {{math\|''ω''<sub>1</sub>}} is real and equal to :<math>\frac{\Gamma^2\left(\frac{1}{4}\right)}{4\sqrt{\pi}}</math> where {{math\|Γ}} is the [[gamma function]]. The second smallest half period is pure imaginary and equal to {{math\|''iω''<sub>1</sub>}}. In more algebraic terms, the [[period lattice]] is a real multiple of the [[Gaussian integer]]s. The [[mathematical constant\|constant]]s {{math\|''e''<sub>1</sub>}}, {{math\|''e''<sub>2</sub>}}, and {{math\|''e''<sub>3</sub>}} are given by :<math>e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12. </math> The case {{math\|''g''<sub>2</sub> {{=}} ''a''}}, {{math\|''g''<sub>3</sub> {{=}} 0}} may be handled by a scaling transformation. However, this may involve complex numbers. If it is desired to remain within real numbers, there are two cases to consider: {{math\|''a'' > 0}} and {{math\|''a'' < 0}}. The period [[parallelogram]] is either a [[square]] or a [[rhombus]]. =={{anchor\|sl\|cl}}Lemniscate sine and cosine functions== The '''lemniscate sine''' (Latin: ''sinus lemniscatus'') and '''lemniscate cosine''' (Latin: ''cosinus lemniscatus'') functions '''{{math\|sinlemn}}''' aka '''{{math\|sl}}''' and '''{{math\|coslemn}}''' aka '''{{math\|cl}}''' are analogues of the usual [[sine]] and [[cosine]] functions, with a circle replaced by a [[lemniscate]]. They are defined by :<math>\operatorname{sl}(r)=s</math> where :<math> r=\int_0^s\frac{dt}{\sqrt{1-t^4}}</math> and :<math>\operatorname{cl}(r)=c</math> where :<math> r=\int_c^1\frac{dt}{\sqrt{1-t^4}}.</math> They are doubly periodic (or elliptic) functions in the complex plane, with periods {{math\|2{{pi}}''G''}} and {{math\|2{{pi}}''iG''}}, where [[Gauss's constant]] {{math\|''G''}} is given by :<math>G=\frac{2}{\pi}\int_0^1\frac{dt}{\sqrt{1-t^4}}= 0.8346\ldots.</math> ===Arclength of lemniscate=== [[Image:Lemniscate of Bernoulli.svg\|thumb\|400px\|right\|A lemniscate of Bernoulli and its two foci]] The [[lemniscate of Bernoulli]] :<math>\left(x^2+y^2\right)^2=x^2-y^2</math> consists of the points such that the product of their distances from the two points {{math\|({{sfrac\|1\|{{sqrt\|2}}}}, 0)}}, {{math\|(−{{sfrac\|1\|{{sqrt\|2}}}}, 0)}} is the constant {{sfrac\|1\|2}}. The length {{math\|''r''}} of the arc from the origin to a point at distance {{math\|''s''}} from the origin is given by :<math> r=\int_0^s\frac{dt}{\sqrt{1-t^4}}.</math> In other words, the sine lemniscatic function gives the distance from the origin as a function of the arc length from the origin. Similarly the cosine lemniscate function gives the distance from the origin as a function of the arc length from (1, 0). =={{anchor\|arcsl\|arccl}}Inverse functions== {{Empty section\|date=August 2017\|reason=Inverse functions are named arcsl and arccl and should be discussed here.}} ==See also== [[Gauss's constant]] [[Equianharmonic\|Equianharmonic case]] ==References== {{AS ref\|18\|658}} {{dlmf \|id=23.5.iii \|title=Lemniscate lattice \|first1=W. P. \|last1=Reinhardt \|first2=P. L. \|last2=Walker}} {{cite journal \|mr=0257326 \|author-last=Siegel \|author-first=C. L. \|title=Topics in complex function theory. Vol. I: Elliptic functions and uniformization theory \|series=Interscience Tracts in Pure and Applied Mathematics \|volume=25 \|publisher=Wiley-Interscience A Division of John Wiley & Sons \|place=New York-London-Sydney \|date=1969 \|isbn=0-471-60844-0}} ==External links== {{springer\|title=Lemniscate functions\|id=p/l058120}} [[Category:Modular forms]] [[Category:Elliptic curves]] [[Category:Lemniscatic elliptic functions\| ]]
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LaTeX to CAS translator

Mathematica

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SymPy

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Tests

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Numeric

Maple

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Tests

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Numeric

Dependency Graph Information

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