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 w(x)=\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix).}

${\displaystyle w(x)=\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix).}$

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

Semantic latex: w(x) = \Gamma(a + \iunit x) \Gamma(b + \iunit x) \Gamma(c - \iunit x) \Gamma(d - \iunit x)

Confidence: 0

Mathematica

Translation: w[x] == \[CapitalGamma]*(a + I*x)*\[CapitalGamma]*(b + I*x)*\[CapitalGamma]*(c - I*x)*\[CapitalGamma]*(d - I*x)

Information

Sub Equations

w[x] = \[CapitalGamma]*(a + I*x)*\[CapitalGamma]*(b + I*x)*\[CapitalGamma]*(c - I*x)*\[CapitalGamma]*(d - I*x)

Free variables

\[CapitalGamma]

a

b

c

d

x

Symbol info

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: (w*(x))-(\[CapitalGamma]*(a + I*x)*\[CapitalGamma]*(b + I*x)*\[CapitalGamma]*(c - I*x)*\[CapitalGamma]*(d - I*x))

ERROR:

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

Numeric

SymPy

Translation: w(x) == Symbol('Gamma')*(a + I*x)*Symbol('Gamma')*(b + I*x)*Symbol('Gamma')*(c - I*x)*Symbol('Gamma')*(d - I*x)

Information

Sub Equations

w(x) = Symbol('Gamma')*(a + I*x)*Symbol('Gamma')*(b + I*x)*Symbol('Gamma')*(c - I*x)*Symbol('Gamma')*(d - I*x)

Free variables

Symbol('Gamma')

a

b

c

d

x

Symbol info

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: w(x) = Gamma*(a + I*x)*Gamma*(b + I*x)*Gamma*(c - I*x)*Gamma*(d - I*x)

Information

Sub Equations

w(x) = Gamma*(a + I*x)*Gamma*(b + I*x)*Gamma*(c - I*x)*Gamma*(d - I*x)

Free variables

Gamma

a

b

c

d

x

Symbol info

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

$w(x)=\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix)$

Is part of

$w(x)=\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix)$

Complete translation information:

{
  "id" : "FORMULA_b6fe1a8a5d1df8ec16e1d1f8eb1da0a9",
  "formula" : "w(x)=\\Gamma(a+ix)\\Gamma(b+ix)\\Gamma(c-ix)\\Gamma(d-ix)",
  "semanticFormula" : "w(x) = \\Gamma(a + \\iunit x) \\Gamma(b + \\iunit x) \\Gamma(c - \\iunit x) \\Gamma(d - \\iunit x)",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "w[x] == \\[CapitalGamma]*(a + I*x)*\\[CapitalGamma]*(b + I*x)*\\[CapitalGamma]*(c - I*x)*\\[CapitalGamma]*(d - I*x)",
      "translationInformation" : {
        "subEquations" : [ "w[x] = \\[CapitalGamma]*(a + I*x)*\\[CapitalGamma]*(b + I*x)*\\[CapitalGamma]*(c - I*x)*\\[CapitalGamma]*(d - I*x)" ],
        "freeVariables" : [ "\\[CapitalGamma]", "a", "b", "c", "d", "x" ],
        "tokenTranslations" : {
          "\\iunit" : "Imaginary unit was translated to: I",
          "w" : "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" : 1,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 1,
        "crashed" : false,
        "testCalculationsGroup" : [ {
          "lhs" : "w*(x)",
          "rhs" : "\\[CapitalGamma]*(a + I*x)*\\[CapitalGamma]*(b + I*x)*\\[CapitalGamma]*(c - I*x)*\\[CapitalGamma]*(d - I*x)",
          "testExpression" : "(w*(x))-(\\[CapitalGamma]*(a + I*x)*\\[CapitalGamma]*(b + I*x)*\\[CapitalGamma]*(c - I*x)*\\[CapitalGamma]*(d - I*x))",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "w(x) == Symbol('Gamma')*(a + I*x)*Symbol('Gamma')*(b + I*x)*Symbol('Gamma')*(c - I*x)*Symbol('Gamma')*(d - I*x)",
      "translationInformation" : {
        "subEquations" : [ "w(x) = Symbol('Gamma')*(a + I*x)*Symbol('Gamma')*(b + I*x)*Symbol('Gamma')*(c - I*x)*Symbol('Gamma')*(d - I*x)" ],
        "freeVariables" : [ "Symbol('Gamma')", "a", "b", "c", "d", "x" ],
        "tokenTranslations" : {
          "\\iunit" : "Imaginary unit was translated to: I",
          "w" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    },
    "Maple" : {
      "translation" : "w(x) = Gamma*(a + I*x)*Gamma*(b + I*x)*Gamma*(c - I*x)*Gamma*(d - I*x)",
      "translationInformation" : {
        "subEquations" : [ "w(x) = Gamma*(a + I*x)*Gamma*(b + I*x)*Gamma*(c - I*x)*Gamma*(d - I*x)" ],
        "freeVariables" : [ "Gamma", "a", "b", "c", "d", "x" ],
        "tokenTranslations" : {
          "\\iunit" : "Imaginary unit was translated to: I",
          "w" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "w(x)=\\Gamma(a+ix)\\,\\Gamma(b+ix)\\,\\Gamma(c-ix)\\,\\Gamma(d-ix)" ],
  "isPartOf" : [ "w(x)=\\Gamma(a+ix)\\,\\Gamma(b+ix)\\,\\Gamma(c-ix)\\,\\Gamma(d-ix)" ],
  "definiens" : [ ]
}

Specify your own input

Formula input
Wikitext context	In mathematics, the '''continuous Hahn polynomials''' are a family of [[orthogonal polynomials]] in the [[Askey scheme]] of hypergeometric orthogonal polynomials. They are defined in terms of [[generalized hypergeometric function]]s by :<math>p_n(x;a,b,c,d)= i^n\frac{(a+c)_n(a+d)_n}{n!}{}_3F_2\left( \begin{array}{c} -n, n+a+b+c+d-1, a+ix \\ a+c, a+d \end{array} ; 1\right)</math> {{harvs\|txt \| last1=Koekoek \| first1=Roelof \| last2=Lesky \| first2=Peter A. \| last3=Swarttouw \| first3=René F. \| title=Hypergeometric orthogonal polynomials and their q-analogues \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Springer Monographs in Mathematics \| isbn=978-3-642-05013-8 \| doi=10.1007/978-3-642-05014-5 \| mr=2656096 \| year=2010\|loc=14}} give a detailed list of their properties. Closely related polynomials include the [[dual Hahn polynomials]] ''R''<sub>''n''</sub>(''x'';γ,δ,''N''), the [[Hahn polynomials]] ''Q''<sub>''n''</sub>(''x'';''a'',''b'',''c''), and the [[continuous dual Hahn polynomials]] ''S''<sub>''n''</sub>(''x'';''a'',''b'',''c''). These polynomials all have ''q''-analogs with an extra parameter ''q'', such as the [[q-Hahn polynomials]] ''Q''<sub>''n''</sub>(''x'';α,β, ''N'';''q''), and so on. ==Orthogonality== The continuous Hahn polynomials ''p''<sub>''n''</sub>(''x'';''a'',''b'',''c'',''d'') are orthogonal with respect to the weight function :<math>w(x)=\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix).</math> In particular, they satisfy the orthogonality relation<ref>Koekoek, Lesky, & Swarttouw (2010), p. 200.</ref><ref>Askey, R. (1985), "Continuous Hahn polynomials", ''J. Phys. A: Math. Gen.'' '''18''': pp. L1017-L1019.</ref><ref>Andrews, Askey, & Roy (1999), p. 333.</ref> :<math>\begin{align}&\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix)\,p_m(x;a,b,c,d)\,p_n(x;a,b,c,d)\,dx\\ &\qquad\qquad=\frac{\Gamma(n+a+c)\,\Gamma(n+a+d)\,\Gamma(n+b+c)\,\Gamma(n+b+d)}{n!(2n+a+b+c+d-1)\,\Gamma(n+a+b+c+d-1)}\,\delta_{n m}\end{align}</math> for <math>\Re(a)>0</math>, <math>\Re(b)>0</math>, <math>\Re(c)>0</math>, <math>\Re(d)>0</math>, <math>c = \overline{a}</math>, <math>d = \overline{b}</math>. ==Recurrence and difference relations== The sequence of continuous Hahn polynomials satisfies the recurrence relation<ref>Koekoek, Lesky, & Swarttouw (2010), p. 201.</ref> :<math>xp_n(x)=p_{n+1}(x)+i(A_n+C_n)p_{n}(x)-A_{n-1}C_n p_{n-1}(x),</math> :<math>\begin{align} \text{where}\quad&p_n(x)=\frac{n!(n+a+b+c+d-1)!}{(2n+a+b+c+d-1)!}p_n(x;a,b,c,d),\\ &A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)},\\ \text{and}\quad&C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. \end{align}</math> ==Rodrigues formula== The continuous Hahn polynomials are given by the Rodrigues-like formula<ref>Koekoek, Lesky, & Swarttouw (2010), p. 202.</ref> :<math>\begin{align}&\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix)\,p_n(x;a,b,c,d)\\ &\qquad=\frac{(-1)^n}{n!}\frac{d^n}{dx^n}\left(\Gamma\left(a+\frac{n}{2}+ix\right)\,\Gamma\left(b+\frac{n}{2}+ix\right)\,\Gamma\left(c+\frac{n}{2}-ix\right)\,\Gamma\left(d+\frac{n}{2}-ix\right)\right).\end{align}</math> ==Generating functions== The continuous Hahn polynomials have the following generating function:<ref>Koekoek, Lesky, & Swarttouw (2010), p. 202.</ref> :<math>\begin{align}&\sum_{n=0}^{\infty}\frac{\Gamma(n+a+b+c+d)\,\Gamma(a+c+1)\,\Gamma(a+d+1)}{\Gamma(a+b+c+d)\,\Gamma(n+a+c+1)\,\Gamma(n+a+d+1)}(-it)^n p_n(x;a,b,c,d)\\ &\qquad=(1-t)^{1-a-b-c-d}{}_3F_2\left( \begin{array}{c} \frac12(a+b+c+d-1), \frac12(a+b+c+d), a+ix\\ a+c, a+d\end{array} ; -\frac{4t}{(1-t)^2} \right).\end{align}</math> A second, distinct generating function is given by :<math>\sum_{n=0}^{\infty}\frac{\Gamma(a+c+1)\,\Gamma(b+d+1)}{\Gamma(n+a+c+1)\,\Gamma(n+b+d+1)}t^n p_n(x;a,b,c,d)=\,_1F_1\left( \begin{array}{c} a + ix \\ a + c\end{array} ; -it\right)\,_1F_1\left( \begin{array}{c} d - ix \\ b + d\end{array} ; it\right).</math> ==Relation to other polynomials== * The [[Wilson polynomials]] are a generalization of the continuous Hahn polynomials. * The [[Bateman polynomials]] ''F''<sub>''n''</sub>(x) are related to the special case ''a''=''b''=''c''=''d''=1/2 of the continuous Hahn polynomials by :<math>p_n\left(x;\tfrac12,\tfrac12,\tfrac12,\tfrac12\right) = i^n n!F_n\left(2ix\right).</math> * The [[Jacobi polynomials]] ''P''<sub>''n''</sub><sup>(α,β)</sup>(x) can be obtained as a limiting case of the continuous Hahn polynomials:<ref>Koekoek, Lesky, & Swarttouw (2010), p. 203.</ref> :<math>P_n^{(\alpha,\beta)}=\lim_{t\to\infty}t^{-n}p_n\left(\tfrac12xt; \tfrac12(\alpha+1-it), \tfrac12(\beta+1+it), \tfrac12(\alpha+1+it), \tfrac12(\beta+1-it)\right).</math> ==References== {{Reflist}} {{Citation \| last1=Hahn \| first1=Wolfgang \| title=Über Orthogonalpolynome, die q-Differenzengleichungen genügen \| doi=10.1002/mana.19490020103 \| mr=0030647 \| year=1949 \| journal=Mathematische Nachrichten \| issn=0025-584X \| volume=2 \| pages=4–34}} {{Citation \| last1=Koekoek \| first1=Roelof \| last2=Lesky \| first2=Peter A. \| last3=Swarttouw \| first3=René F. \| title=Hypergeometric orthogonal polynomials and their q-analogues \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Springer Monographs in Mathematics \| isbn=978-3-642-05013-8 \| doi=10.1007/978-3-642-05014-5 \| mr=2656096 \| year=2010}} {{dlmf\|id=18.19\|title=Hahn Class: Definitions\|first=Tom H. \|last=Koornwinder\|first2=Roderick S. C.\|last2= Wong\|first3=Roelof \|last3=Koekoek\|\|first4=René F. \|last4=Swarttouw}} {{Citation \| last1=Andrews \| first1=George E. \| last2=Askey \| first2 = Richard \| last3=Roy \| first3=Ranjan \| title=Special functions \| publisher=[[Cambridge University Press]] \| location=Cambridge \| series=Encyclopedia of Mathematics and its Applications 71 \| isbn=978-0-521-62321-6 \| year=1999}} [[Category:Special hypergeometric functions]] [[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

Specify your own input

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