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 \frac{1+zf(z)}{1-zf(z)}=F(z)=\int\frac{e^{i\theta}+z}{e^{i\theta}-z}d\mu.}

${\displaystyle {\frac {1+zf(z)}{1-zf(z)}}=F(z)=\int {\frac {e^{i\theta }+z}{e^{i\theta }-z}}d\mu .}$

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

Semantic latex: \frac{1+zf(z)}{1-zf(z)} = F(z) = \int \frac{\expe^{\iunit \theta} + z}{\expe^{\iunit \theta} - z} \diff{\mu}

Confidence: 0

Mathematica

Translation: Divide[1 + zf[z],1 - zf[z]] == F[z] == Integrate[Divide[Exp[I*\[Theta]]+ z,Exp[I*\[Theta]]- z], \[Mu], GenerateConditions->None]

Information

Sub Equations

Divide[1 + zf[z],1 - zf[z]] = F[z]

F[z] = Integrate[Divide[Exp[I*\[Theta]]+ z,Exp[I*\[Theta]]- z], \[Mu], GenerateConditions->None]

Free variables

\[Theta]

z

Symbol info

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

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

Imaginary unit was translated to: I

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

Tests

Symbolic

Test expression: (Divide[1 + z*f*(z),1 - z*f*(z)])-(F*(z))

ERROR:

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

Test expression: (F*(z))-(Integrate[Divide[Exp[I*\[Theta]]+ z,Exp[I*\[Theta]]- z], \[Mu], GenerateConditions->None])

ERROR:

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

Numeric

SymPy

Translation: (1 + zf(z))/(1 - zf(z)) == F(z) == integrate((exp(I*Symbol('theta'))+ z)/(exp(I*Symbol('theta'))- z), Symbol('mu'))

Information

Sub Equations

(1 + zf(z))/(1 - zf(z)) = F(z)

F(z) = integrate((exp(I*Symbol('theta'))+ z)/(exp(I*Symbol('theta'))- z), Symbol('mu'))

Free variables

Symbol('theta')

z

Symbol info

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

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

Imaginary unit was translated to: I

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

Tests

Symbolic

Numeric

Maple

Translation: (1 + zf(z))/(1 - zf(z)) = F(z) = int((exp(I*theta)+ z)/(exp(I*theta)- z), mu)

Information

Sub Equations

(1 + zf(z))/(1 - zf(z)) = F(z)

F(z) = int((exp(I*theta)+ z)/(exp(I*theta)- z), mu)

Free variables

theta

z

Symbol info

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

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

Imaginary unit was translated to: I

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

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\mu$

Description

equation

Schur parameter of the function

Verblunsky coefficient of the measure

Geronimus 's theorem

Complete translation information:

{
  "id" : "FORMULA_252126e9db9ad369a9b611c47cd6107a",
  "formula" : "\\frac{1+zf(z)}{1-zf(z)}=F(z)=\\int\\frac{e^{i\\theta}+z}{e^{i\\theta}-z}d\\mu",
  "semanticFormula" : "\\frac{1+zf(z)}{1-zf(z)} = F(z) = \\int \\frac{\\expe^{\\iunit \\theta} + z}{\\expe^{\\iunit \\theta} - z} \\diff{\\mu}",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "Divide[1 + zf[z],1 - zf[z]] == F[z] == Integrate[Divide[Exp[I*\\[Theta]]+ z,Exp[I*\\[Theta]]- z], \\[Mu], GenerateConditions->None]",
      "translationInformation" : {
        "subEquations" : [ "Divide[1 + zf[z],1 - zf[z]] = F[z]", "F[z] = Integrate[Divide[Exp[I*\\[Theta]]+ z,Exp[I*\\[Theta]]- z], \\[Mu], GenerateConditions->None]" ],
        "freeVariables" : [ "\\[Theta]", "z" ],
        "tokenTranslations" : {
          "zf" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "\\iunit" : "Imaginary unit was translated to: I",
          "F" : "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" : "Divide[1 + z*f*(z),1 - z*f*(z)]",
          "rhs" : "F*(z)",
          "testExpression" : "(Divide[1 + z*f*(z),1 - z*f*(z)])-(F*(z))",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        }, {
          "lhs" : "F*(z)",
          "rhs" : "Integrate[Divide[Exp[I*\\[Theta]]+ z,Exp[I*\\[Theta]]- z], \\[Mu], GenerateConditions->None]",
          "testExpression" : "(F*(z))-(Integrate[Divide[Exp[I*\\[Theta]]+ z,Exp[I*\\[Theta]]- z], \\[Mu], GenerateConditions->None])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "(1 + zf(z))/(1 - zf(z)) == F(z) == integrate((exp(I*Symbol('theta'))+ z)/(exp(I*Symbol('theta'))- z), Symbol('mu'))",
      "translationInformation" : {
        "subEquations" : [ "(1 + zf(z))/(1 - zf(z)) = F(z)", "F(z) = integrate((exp(I*Symbol('theta'))+ z)/(exp(I*Symbol('theta'))- z), Symbol('mu'))" ],
        "freeVariables" : [ "Symbol('theta')", "z" ],
        "tokenTranslations" : {
          "zf" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "\\iunit" : "Imaginary unit was translated to: I",
          "F" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    },
    "Maple" : {
      "translation" : "(1 + zf(z))/(1 - zf(z)) = F(z) = int((exp(I*theta)+ z)/(exp(I*theta)- z), mu)",
      "translationInformation" : {
        "subEquations" : [ "(1 + zf(z))/(1 - zf(z)) = F(z)", "F(z) = int((exp(I*theta)+ z)/(exp(I*theta)- z), mu)" ],
        "freeVariables" : [ "theta", "z" ],
        "tokenTranslations" : {
          "zf" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "\\iunit" : "Imaginary unit was translated to: I",
          "F" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    }
  },
  "positions" : [ {
    "section" : 4,
    "sentence" : 0,
    "word" : 24
  } ],
  "includes" : [ "\\mu" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "equation",
    "score" : 0.722
  }, {
    "definition" : "Schur parameter of the function",
    "score" : 0.6460746792928004
  }, {
    "definition" : "Verblunsky coefficient of the measure",
    "score" : 0.5988174995334326
  }, {
    "definition" : "Geronimus 's theorem",
    "score" : 0.5049074255814494
  } ]
}

Specify your own input

Formula input
Wikitext context	In mathematics, '''orthogonal polynomials on the unit circle''' are families of [[Orthogonal polynomials\|polynomials that are orthogonal with respect to integration]] over the [[unit circle]] in the [[complex plane]], for some [[probability measure]] on the unit circle. They were introduced by {{harvs\|txt\|last=Szegő\|year1=1920\|year2=1921\|year3=1939}}. ==Definition== Suppose that <math>\mu</math> is a probability measure on the unit circle in the complex plane, whose [[support (mathematics)\|support]] is not finite. The orthogonal polynomials associated to <math>\mu</math> are the polynomials <math>\Phi_n(z)</math> with leading term <math>z^n</math> that are orthogonal with respect to the measure <math>\mu</math>. ==The Szegő recurrence == Szegő's recurrence states that :<math>\Phi_0(z) = 1</math> :<math>\Phi_{n+1}(z)=z\Phi_n(z)-\overline\alpha_n\Phi_n^(z)</math> where :<math>\Phi_n^(z)=z^n\overline{\Phi_n(1/\overline{z})}</math> is the polynomial with its coefficients reversed and complex conjugated, and where the '''Verblunsky coefficients''' <math>\alpha_n</math> are complex numbers with absolute values less than 1. ==Verblunsky's theorem== Verblunsky's theorem states that any sequence of complex numbers in the open unit disk is the sequence of Verblunsky coefficients for a unique probability measure on the unit circle with infinite support. ==Geronimus's theorem== Geronimus's theorem states that the Verblunsky coefficients of the measure μ are the [[Schur parameters]] of the function <math>f</math> defined by the equations :<math> \frac{1+zf(z)}{1-zf(z)}=F(z)=\int\frac{e^{i\theta}+z}{e^{i\theta}-z}d\mu.</math> ==Baxter's theorem== Baxter's theorem states that the Verblunsky coefficients form an absolutely convergent series if and only if the moments of <math>\mu</math> form an absolutely convergent series and the weight function <math>w</math> is strictly positive everywhere. ==Szegő's theorem== Szegő's theorem states that :<math>\prod_{n = 1}^\infty(1-\|\alpha_n\|^2) = \exp\big(\int_0^{2\pi}\log(w(\theta))d\theta/2\pi\big)</math> where <math>wd\theta / 2\pi</math> is the absolutely continuous part of the measure <math>\mu</math>. ==Rakhmanov's theorem== Rakhmanov's theorem states that if the absolutely continuous part <math>w</math> of the measure <math>\mu</math> is positive almost everywhere then the Verblunsky coefficients <math>\alpha_n</math> tend to 0. ==Examples== The [[Rogers–Szegő polynomials]] are an example of orthogonal polynomials on the unit circle. ==References== {{dlmf\|id=18.33\|title=Orthogonal Polynomials on the unit circle\|first=Tom H. \|last=Koornwinder\|first2=Roderick S. C.\|last2= Wong\|first3=Roelof \|last3=Koekoek\|\|first4=René F. \|last4=Swarttouw}} {{Citation \| last1=Simon \| first1=Barry \| author1-link=Barry Simon \| title=Orthogonal polynomials on the unit circle. Part 1. Classical theory \| url=https://books.google.com/books?id=d94r7kOSnKcC \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=American Mathematical Society Colloquium Publications \| isbn=978-0-8218-3446-6 \| mr=2105088 \| year=2005 \| volume=54}} {{Citation \| last1=Simon \| first1=Barry \| author1-link=Barry Simon \| title=Orthogonal polynomials on the unit circle. Part 2. Spectral theory \| url=https://books.google.com/books?id=54juCPc3ulwC \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=American Mathematical Society Colloquium Publications \| isbn=978-0-8218-3675-0 \| mr=2105089 \| year=2005 \| volume=54}} {{Citation \| last1=Szegő \| first1=Gábor \| title=Beiträge zur Theorie der Toeplitzschen Formen \| doi=10.1007/BF01199955 \| year=1920 \| journal=[[Mathematische Zeitschrift]] \| issn=0025-5874 \| volume=6 \| issue=3–4 \| pages=167–202\| s2cid=118147030 \| url=https://zenodo.org/record/1447413 }} {{Citation \| last1=Szegő \| first1=Gábor \| title=Beiträge zur Theorie der Toeplitzschen Formen \| doi=10.1007/BF01279027 \| year=1921 \| journal=[[Mathematische Zeitschrift]] \| issn=0025-5874 \| volume=9 \| issue=3–4 \| pages=167–190\| s2cid=125157848 \| url=https://zenodo.org/record/1447419 }} {{Citation \| last1=Szegő \| first1=Gábor \| title=Orthogonal Polynomials \| url=https://books.google.com/books?id=3hcW8HBh7gsC \| publisher= American Mathematical Society \| series=Colloquium Publications \| isbn=978-0-8218-1023-1 \| mr=0372517 \| year=1939 \| volume=XXIII}} [[Category:Orthogonal polynomials]]
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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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