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; \lambda, \phi)= |\Gamma(\lambda+ix)|^2 e^{(2\phi-\pi)x}}

${\displaystyle w(x;\lambda ,\phi )=|\Gamma (\lambda +ix)|^{2}e^{(2\phi -\pi )x}}$

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

Semantic latex: w(x ; \lambda , \phi) =|\Gamma(\lambda + \iunit x)|^2 \expe^{(2 \phi - \cpi) x}

Confidence: 0

Mathematica

Translation: w[x ; \[Lambda], \[Phi]] == (Abs[\[CapitalGamma]*(\[Lambda]+ I*x)])^(2)* Exp[(2*\[Phi]- Pi)*x]

Information

Sub Equations

w[x ; \[Lambda], \[Phi]] = (Abs[\[CapitalGamma]*(\[Lambda]+ I*x)])^(2)* Exp[(2*\[Phi]- Pi)*x]

Free variables

\[CapitalGamma]

\[Lambda]

\[Phi]

x

Symbol info

Pi was translated to: Pi

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

Imaginary unit was translated to: I

Could be the golden ratio == golden mean == golden section == extreme and mean ratio == medial section == divine proportion == divine section == golden proportion == golden cut == golden number.

But this system doesn't know how to translate it as a constant. It was translated as a general letter.

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

Tests

Symbolic

Test expression: (w*(x ; \[Lambda], \[Phi]))-((Abs[\[CapitalGamma]*(\[Lambda]+ I*x)])^(2)* Exp[(2*\[Phi]- Pi)*x])

ERROR:

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

Numeric

SymPy

Translation: w(x ; Symbol('lambda'), Symbol('phi')) == (abs(Symbol('Gamma')*(Symbol('lambda')+ I*x)))**(2)* exp((2*Symbol('phi')- pi)*x)

Information

Sub Equations

w(x ; Symbol('lambda'), Symbol('phi')) = (abs(Symbol('Gamma')*(Symbol('lambda')+ I*x)))**(2)* exp((2*Symbol('phi')- pi)*x)

Free variables

Symbol('Gamma')

Symbol('lambda')

Symbol('phi')

x

Symbol info

Pi was translated to: pi

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

Imaginary unit was translated to: I

Could be the golden ratio == golden mean == golden section == extreme and mean ratio == medial section == divine proportion == divine section == golden proportion == golden cut == golden number.

But this system doesn't know how to translate it as a constant. It was translated as a general letter.

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

Tests

Symbolic

Numeric

Maple

Translation: w(x ; lambda , phi) = (abs(Gamma*(lambda + I*x)))^(2)* exp((2*phi - Pi)*x)

Information

Sub Equations

w(x ; lambda , phi) = (abs(Gamma*(lambda + I*x)))^(2)* exp((2*phi - Pi)*x)

Free variables

Gamma

lambda

phi

x

Symbol info

Pi was translated to: Pi

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

Imaginary unit was translated to: I

Could be the golden ratio == golden mean == golden section == extreme and mean ratio == medial section == divine proportion == divine section == golden proportion == golden cut == golden number.

But this system doesn't know how to translate it as a constant. It was translated as a general letter.

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

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$w(x;\lambda ,\varphi )$

Description

weight function

Meixner -- Pollaczek polynomial

real line with respect

orthogonality relation

Complete translation information:

{
  "id" : "FORMULA_0154abadac632cfa861715723cfd71ce",
  "formula" : "w(x; \\lambda, \\phi)= |\\Gamma(\\lambda+ix)|^2 e^{(2\\phi-\\pi)x}",
  "semanticFormula" : "w(x ; \\lambda , \\phi) =|\\Gamma(\\lambda + \\iunit x)|^2 \\expe^{(2 \\phi - \\cpi) x}",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "w[x ; \\[Lambda], \\[Phi]] == (Abs[\\[CapitalGamma]*(\\[Lambda]+ I*x)])^(2)* Exp[(2*\\[Phi]- Pi)*x]",
      "translationInformation" : {
        "subEquations" : [ "w[x ; \\[Lambda], \\[Phi]] = (Abs[\\[CapitalGamma]*(\\[Lambda]+ I*x)])^(2)* Exp[(2*\\[Phi]- Pi)*x]" ],
        "freeVariables" : [ "\\[CapitalGamma]", "\\[Lambda]", "\\[Phi]", "x" ],
        "tokenTranslations" : {
          "\\cpi" : "Pi was translated to: Pi",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "\\iunit" : "Imaginary unit was translated to: I",
          "\\phi" : "Could be the golden ratio == golden mean == golden section == extreme and mean ratio == medial section == divine proportion == divine section == golden proportion == golden cut == golden number.\nBut this system doesn't know how to translate it as a constant. It was translated as a general letter.\n",
          "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 ; \\[Lambda], \\[Phi])",
          "rhs" : "(Abs[\\[CapitalGamma]*(\\[Lambda]+ I*x)])^(2)* Exp[(2*\\[Phi]- Pi)*x]",
          "testExpression" : "(w*(x ; \\[Lambda], \\[Phi]))-((Abs[\\[CapitalGamma]*(\\[Lambda]+ I*x)])^(2)* Exp[(2*\\[Phi]- Pi)*x])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "w(x ; Symbol('lambda'), Symbol('phi')) == (abs(Symbol('Gamma')*(Symbol('lambda')+ I*x)))**(2)* exp((2*Symbol('phi')- pi)*x)",
      "translationInformation" : {
        "subEquations" : [ "w(x ; Symbol('lambda'), Symbol('phi')) = (abs(Symbol('Gamma')*(Symbol('lambda')+ I*x)))**(2)* exp((2*Symbol('phi')- pi)*x)" ],
        "freeVariables" : [ "Symbol('Gamma')", "Symbol('lambda')", "Symbol('phi')", "x" ],
        "tokenTranslations" : {
          "\\cpi" : "Pi was translated to: pi",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "\\iunit" : "Imaginary unit was translated to: I",
          "\\phi" : "Could be the golden ratio == golden mean == golden section == extreme and mean ratio == medial section == divine proportion == divine section == golden proportion == golden cut == golden number.\nBut this system doesn't know how to translate it as a constant. It was translated as a general letter.\n",
          "w" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    },
    "Maple" : {
      "translation" : "w(x ; lambda , phi) = (abs(Gamma*(lambda + I*x)))^(2)* exp((2*phi - Pi)*x)",
      "translationInformation" : {
        "subEquations" : [ "w(x ; lambda , phi) = (abs(Gamma*(lambda + I*x)))^(2)* exp((2*phi - Pi)*x)" ],
        "freeVariables" : [ "Gamma", "lambda", "phi", "x" ],
        "tokenTranslations" : {
          "\\cpi" : "Pi was translated to: Pi",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "\\iunit" : "Imaginary unit was translated to: I",
          "\\phi" : "Could be the golden ratio == golden mean == golden section == extreme and mean ratio == medial section == divine proportion == divine section == golden proportion == golden cut == golden number.\nBut this system doesn't know how to translate it as a constant. It was translated as a general letter.\n",
          "w" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    }
  },
  "positions" : [ {
    "section" : 2,
    "sentence" : 0,
    "word" : 18
  } ],
  "includes" : [ "w(x;\\lambda,\\varphi)" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "weight function",
    "score" : 0.722
  }, {
    "definition" : "Meixner -- Pollaczek polynomial",
    "score" : 0.6859086196238077
  }, {
    "definition" : "real line with respect",
    "score" : 0.6859086196238077
  }, {
    "definition" : "orthogonality relation",
    "score" : 0.6460746792928004
  } ]
}

Specify your own input

Formula input
Wikitext context	{{distinguish\|Meixner polynomials}} {{More citations needed\|date=March 2016}} In mathematics, the '''Meixner–Pollaczek polynomials''' are a family of [[orthogonal polynomials]] ''P''{{su\|b=''n''\|p=(λ)}}(''x'',φ) introduced by {{harvs\|txt\|authorlink=Josef Meixner\|last=Meixner\|year=1934}}, which up to elementary changes of variables are the same as the '''Pollaczek polynomials''' ''P''{{su\|b=''n''\|p=λ}}(''x'',''a'',''b'') rediscovered by {{harvs\|txt\|authorlink=Felix Pollaczek\|last=Pollaczek\|year=1949}} in the case λ=1/2, and later generalized by him. They are defined by :<math>P_n^{(\lambda)}(x;\phi) = \frac{(2\lambda)_n}{n!}e^{in\phi}{}_2F_1\left(\begin{array}{c} -n,~\lambda+ix\\ 2\lambda \end{array}; 1-e^{-2i\phi}\right)</math> :<math>P_n^{\lambda}(\cos \phi;a,b) = \frac{(2\lambda)_n}{n!}e^{in\phi}{}_2F_1\left(\begin{array}{c}-n,~\lambda+i(a\cos \phi+b)/\sin \phi\\ 2\lambda \end{array};1-e^{-2i\phi}\right)</math> ==Examples== The first few Meixner–Pollaczek polynomials are :<math>P_0^{(\lambda)}(x;\phi)=1</math> :<math>P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi)</math> :<math>P_2^{(\lambda)}(x;\phi)=x^2+\lambda^2+(\lambda^2+\lambda-x^2)\cos(2\phi)+(1+2\lambda)x\sin(2\phi).</math> ==Properties== ===Orthogonality=== The Meixner–Pollaczek polynomials ''P''<sub>m</sub><sup>(λ)</sup>(''x'';φ) are orthogonal on the real line with respect to the weight function :<math> w(x; \lambda, \phi)= \|\Gamma(\lambda+ix)\|^2 e^{(2\phi-\pi)x}</math> and the orthogonality relation is given by<ref>Koekoek, Lesky, & Swarttouw (2010), p. 213.</ref> :<math>\int_{-\infty}^{\infty}P_n^{(\lambda)}(x;\phi)P_m^{(\lambda)}(x;\phi)w(x; \lambda, \phi)dx=\frac{2\pi\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\delta_{mn},\quad \lambda>0,\quad 0<\phi<\pi.</math> ===Recurrence relation=== The sequence of Meixner–Pollaczek polynomials satisfies the recurrence relation<ref>Koekoek, Lesky, & Swarttouw (2010), p. 213.</ref> :<math>(n+1)P_{n+1}^{(\lambda)}(x;\phi)=2\bigl(x\sin\phi + (n+\lambda)\cos\phi\bigr)P_n^{(\lambda)}(x;\phi)-(n+2\lambda-1)P_{n-1}(x;\phi).</math> ===Rodrigues formula=== The Meixner–Pollaczek polynomials are given by the Rodrigues-like formula<ref>Koekoek, Lesky, & Swarttouw (2010), p. 214.</ref> :<math>P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!\,w(x;\lambda,\phi)}\frac{d^n}{dx^n}w\left(x;\lambda+\tfrac12n,\phi\right),</math> where ''w''(''x'';λ,φ) is the weight function given above. ===Generating function=== The Meixner–Pollaczek polynomials have the generating function<ref>Koekoek, Lesky, & Swarttouw (2010), p. 215.</ref> :<math>\sum_{n=0}^{\infty}t^n P_n^{(\lambda)}(x;\phi) = (1-e^{i\phi}t)^{-\lambda+ix}(1-e^{-i\phi}t)^{-\lambda-ix}.</math> ==See also== [[Sieved Pollaczek polynomials]] ==References== {{Reflist}} {{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=18.35\|title=Pollaczek Polynomials\|first=Tom H. \|last=Koornwinder\|first2=Roderick S. C.\|last2= Wong\|first3=Roelof \|last3=Koekoek\|\|first4=René F. \|last4=Swarttouw}} {{Citation \| last1=Meixner \| first1=J. \| title=Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion \| doi=10.1112/jlms/s1-9.1.6 \| year=1934 \| journal=J. London Math. Soc. \| volume=s1-9 \| pages=6–13}} *{{Citation \| last1=Pollaczek \| first1=Félix \| title=Sur une généralisation des polynomes de Legendre \| url=http://gallica.bnf.fr/ark:/12148/bpt6k31801/f1363 \| mr=0030037 \| year=1949 \| journal=[[Les Comptes rendus de l'Académie des sciences]] \| volume=228 \| pages=1363–1365}} {{DEFAULTSORT:Meixner-Pollaczek polynomials}} [[Category:Orthogonal polynomials]]
	Purge cached results.

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

Navigation menu

Search