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 \mu}

${\displaystyle \mu }$

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

Semantic latex: \mu

Confidence: 0

Mathematica

Translation: \[Mu]

Information

Sub Equations

\[Mu]

Free variables

\[Mu]

Tests

Symbolic

Numeric

SymPy

Translation: Symbol('mu')

Information

Sub Equations

Symbol('mu')

Free variables

Symbol('mu')

Tests

Symbolic

Numeric

Maple

Translation: mu

Information

Sub Equations

mu

Free variables

mu

Tests

Symbolic

Numeric

Dependency Graph Information

Is part of

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

Description

continuous part of the measure

measure

polynomial

probability measure on the unit circle

respect to the measure

Schur parameter of the function

Verblunsky coefficient of the measure

orthogonal polynomial

term

complex plane

support

equation

convergent series

moment

continuous part

Verblunsky coefficient

weight function

Geronimus 's theorem

Szegő 's theorem

Rakhmanov 's theorem

Baxter 's theorem

Complete translation information:

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  "formula" : "\\mu",
  "semanticFormula" : "\\mu",
  "confidence" : 0.0,
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      "translation" : "\\[Mu]",
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        "freeVariables" : [ "\\[Mu]" ]
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      "numericResults" : {
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        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "SymPy" : {
      "translation" : "Symbol('mu')",
      "translationInformation" : {
        "subEquations" : [ "Symbol('mu')" ],
        "freeVariables" : [ "Symbol('mu')" ]
      },
      "numericResults" : {
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        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
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        "testCalculationsGroups" : [ ]
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        "numberOfTests" : 0,
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        "testCalculationsGroup" : [ ]
      }
    },
    "Maple" : {
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      "translationInformation" : {
        "subEquations" : [ "mu" ],
        "freeVariables" : [ "mu" ]
      },
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      "symbolicResults" : {
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        "numberOfTests" : 0,
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        "testCalculationsGroup" : [ ]
      }
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  },
  "positions" : [ {
    "section" : 1,
    "sentence" : 0,
    "word" : 2
  }, {
    "section" : 1,
    "sentence" : 1,
    "word" : 5
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    "sentence" : 1,
    "word" : 22
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  } ],
  "includes" : [ ],
  "isPartOf" : [ "\\frac{1+zf(z)}{1-zf(z)}=F(z)=\\int\\frac{e^{i\\theta}+z}{e^{i\\theta}-z}d\\mu" ],
  "definiens" : [ {
    "definition" : "continuous part of the measure",
    "score" : 0.722
  }, {
    "definition" : "measure",
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  }, {
    "definition" : "polynomial",
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  }, {
    "definition" : "probability measure on the unit circle",
    "score" : 0.722
  }, {
    "definition" : "respect to the measure",
    "score" : 0.722
  }, {
    "definition" : "Schur parameter of the function",
    "score" : 0.722
  }, {
    "definition" : "Verblunsky coefficient of the measure",
    "score" : 0.722
  }, {
    "definition" : "orthogonal polynomial",
    "score" : 0.6432331635625809
  }, {
    "definition" : "term",
    "score" : 0.6432331635625809
  }, {
    "definition" : "complex plane",
    "score" : 0.6288842031023242
  }, {
    "definition" : "support",
    "score" : 0.5329047619047619
  }, {
    "definition" : "equation",
    "score" : 0.4111329742008307
  }, {
    "definition" : "convergent series",
    "score" : 0.3757874168456006
  }, {
    "definition" : "moment",
    "score" : 0.3730866057293667
  }, {
    "definition" : "continuous part",
    "score" : 0.35408542099079315
  }, {
    "definition" : "Verblunsky coefficient",
    "score" : 0.30952905234765915
  }, {
    "definition" : "weight function",
    "score" : 0.2593055947715278
  }, {
    "definition" : "Geronimus 's theorem",
    "score" : 0.2392282122960688
  }, {
    "definition" : "Szegő 's theorem",
    "score" : 0.1518028742676323
  }, {
    "definition" : "Rakhmanov 's theorem",
    "score" : 0.14639433608097108
  }, {
    "definition" : "Baxter 's theorem",
    "score" : 0.12704740271972384
  } ]
}

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

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SymPy

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Maple

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Tests

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Dependency Graph Information

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