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 M_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu, z\right)}

${\displaystyle M_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}M\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z\right)}$

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

Semantic latex: \WhittakerconfhyperM{\kappa}{\mu}@{z} = \exp(- z / 2) z^{\mu+\tfrac{1}{2}} \KummerconfhyperM@{\mu - \kappa + \tfrac{1}{2}}{1 + 2 \mu}{z}

Confidence: 0.89530287320794

Mathematica

Translation: WhittakerM[\[Kappa], \[Mu], z] == Exp[- z/2]*(z)^(\[Mu]+Divide[1,2])* Hypergeometric1F1[\[Mu]- \[Kappa]+Divide[1,2], 1 + 2*\[Mu], z]

Information

Sub Equations

WhittakerM[\[Kappa], \[Mu], z] = Exp[- z/2]*(z)^(\[Mu]+Divide[1,2])* Hypergeometric1F1[\[Mu]- \[Kappa]+Divide[1,2], 1 + 2*\[Mu], z]

Free variables

\[Kappa]

\[Mu]

z

Symbol info

Exponential function; Example: \exp@@{z}

Will be translated to: Exp[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E19 Mathematica: https://reference.wolfram.com/language/ref/Exp.html

Kummer confluent hypergeometric functions M; Example: \KummerconfhyperM@{a}{b}{z}

Will be translated to: Hypergeometric1F1[$0, $1, $2] Relevant links to definitions: DLMF: http://dlmf.nist.gov/13.2#E2 Mathematica:

Whittaker confluent hypergeometric function; Example: \WhittakerconfhyperM{\kappa}{\mu}@{z}

Will be translated to: WhittakerM[$0, $1, $2] Relevant links to definitions: DLMF: http://dlmf.nist.gov/13.14#E2 Mathematica: https://reference.wolfram.com/language/ref/WhittakerM.html

Tests

Symbolic

Test expression: (WhittakerM[\[Kappa], \[Mu], z])-(Exp[- z/2]*(z)^(\[Mu]+Divide[1,2])* Hypergeometric1F1[\[Mu]- \[Kappa]+Divide[1,2], 1 + 2*\[Mu], z])

ERROR:

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

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \WhittakerconfhyperM [\WhittakerconfhyperM]

Tests

Symbolic

Numeric

Maple

Translation: WhittakerM(kappa, mu, z) = exp(- z/2)*(z)^(mu +(1)/(2))* KummerM(mu - kappa +(1)/(2), 1 + 2*mu, z)

Information

Sub Equations

WhittakerM(kappa, mu, z) = exp(- z/2)*(z)^(mu +(1)/(2))* KummerM(mu - kappa +(1)/(2), 1 + 2*mu, z)

Free variables

kappa

mu

z

Symbol info

Exponential function; Example: \exp@@{z}

Will be translated to: exp($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E19 Maple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=LinearAlgebra/Trace

Kummer confluent hypergeometric functions M; Example: \KummerconfhyperM@{a}{b}{z}

Will be translated to: KummerM($0, $1, $2) Relevant links to definitions: DLMF: http://dlmf.nist.gov/13.2#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=KummerK

Whittaker confluent hypergeometric function; Example: \WhittakerconfhyperM{\kappa}{\mu}@{z}

Will be translated to: WhittakerM($0, $1, $2) Relevant links to definitions: DLMF: http://dlmf.nist.gov/13.14#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=WhittakerM

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$M$

$W_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}U\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z\right)$

$M_{\kappa ,\mu }(z)$

$W_{\kappa ,\mu }(z)$

$\mu$

$\kappa$

$z$

Description

term of Kummer 's confluent hypergeometric function

Whittaker function

solution

Complete translation information:

{
  "id" : "FORMULA_cd855d957af2fb52b11b2b69e16a5be3",
  "formula" : "M_{\\kappa,\\mu}\\left(z\\right) = \\exp\\left(-z/2\\right)z^{\\mu+\\tfrac{1}{2}}M\\left(\\mu-\\kappa+\\tfrac{1}{2}, 1+2\\mu, z\\right)",
  "semanticFormula" : "\\WhittakerconfhyperM{\\kappa}{\\mu}@{z} = \\exp(- z / 2) z^{\\mu+\\tfrac{1}{2}} \\KummerconfhyperM@{\\mu - \\kappa + \\tfrac{1}{2}}{1 + 2 \\mu}{z}",
  "confidence" : 0.8953028732079359,
  "translations" : {
    "Mathematica" : {
      "translation" : "WhittakerM[\\[Kappa], \\[Mu], z] == Exp[- z/2]*(z)^(\\[Mu]+Divide[1,2])* Hypergeometric1F1[\\[Mu]- \\[Kappa]+Divide[1,2], 1 + 2*\\[Mu], z]",
      "translationInformation" : {
        "subEquations" : [ "WhittakerM[\\[Kappa], \\[Mu], z] = Exp[- z/2]*(z)^(\\[Mu]+Divide[1,2])* Hypergeometric1F1[\\[Mu]- \\[Kappa]+Divide[1,2], 1 + 2*\\[Mu], z]" ],
        "freeVariables" : [ "\\[Kappa]", "\\[Mu]", "z" ],
        "tokenTranslations" : {
          "\\exp" : "Exponential function; Example: \\exp@@{z}\nWill be translated to: Exp[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/4.2#E19\nMathematica:  https://reference.wolfram.com/language/ref/Exp.html",
          "\\KummerconfhyperM" : "Kummer confluent hypergeometric functions M; Example: \\KummerconfhyperM@{a}{b}{z}\nWill be translated to: Hypergeometric1F1[$0, $1, $2]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/13.2#E2\nMathematica:  ",
          "\\WhittakerconfhyperM" : "Whittaker confluent hypergeometric function; Example: \\WhittakerconfhyperM{\\kappa}{\\mu}@{z}\nWill be translated to: WhittakerM[$0, $1, $2]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/13.14#E2\nMathematica:  https://reference.wolfram.com/language/ref/WhittakerM.html"
        }
      },
      "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" : "WhittakerM[\\[Kappa], \\[Mu], z]",
          "rhs" : "Exp[- z/2]*(z)^(\\[Mu]+Divide[1,2])* Hypergeometric1F1[\\[Mu]- \\[Kappa]+Divide[1,2], 1 + 2*\\[Mu], z]",
          "testExpression" : "(WhittakerM[\\[Kappa], \\[Mu], z])-(Exp[- z/2]*(z)^(\\[Mu]+Divide[1,2])* Hypergeometric1F1[\\[Mu]- \\[Kappa]+Divide[1,2], 1 + 2*\\[Mu], z])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\WhittakerconfhyperM [\\WhittakerconfhyperM]"
        }
      }
    },
    "Maple" : {
      "translation" : "WhittakerM(kappa, mu, z) = exp(- z/2)*(z)^(mu +(1)/(2))* KummerM(mu - kappa +(1)/(2), 1 + 2*mu, z)",
      "translationInformation" : {
        "subEquations" : [ "WhittakerM(kappa, mu, z) = exp(- z/2)*(z)^(mu +(1)/(2))* KummerM(mu - kappa +(1)/(2), 1 + 2*mu, z)" ],
        "freeVariables" : [ "kappa", "mu", "z" ],
        "tokenTranslations" : {
          "\\exp" : "Exponential function; Example: \\exp@@{z}\nWill be translated to: exp($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.2#E19\nMaple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=LinearAlgebra/Trace",
          "\\KummerconfhyperM" : "Kummer confluent hypergeometric functions M; Example: \\KummerconfhyperM@{a}{b}{z}\nWill be translated to: KummerM($0, $1, $2)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/13.2#E2\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=KummerK",
          "\\WhittakerconfhyperM" : "Whittaker confluent hypergeometric function; Example: \\WhittakerconfhyperM{\\kappa}{\\mu}@{z}\nWill be translated to: WhittakerM($0, $1, $2)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/13.14#E2\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=WhittakerM"
        }
      }
    }
  },
  "positions" : [ {
    "section" : 0,
    "sentence" : 4,
    "word" : 25
  } ],
  "includes" : [ "M", "W_{\\kappa,\\mu}\\left(z\\right) = \\exp\\left(-z/2\\right)z^{\\mu+\\tfrac{1}{2}}U\\left(\\mu-\\kappa+\\tfrac{1}{2}, 1+2\\mu, z\\right)", "M_{\\kappa,\\mu}(z)", "W_{\\kappa,\\mu}(z)", "\\mu", "\\kappa", "z" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "term of Kummer 's confluent hypergeometric function",
    "score" : 0.6859086196238077
  }, {
    "definition" : "Whittaker function",
    "score" : 0.6859086196238077
  }, {
    "definition" : "solution",
    "score" : 0.5988174995334326
  } ]
}

Specify your own input

Formula input
Wikitext context	{{short description\|In mathematics, a solution to a modified form of the confluent hypergeometric equation}} In mathematics, a '''Whittaker function''' is a special solution of '''Whittaker's equation''', a modified form of the [[confluent hypergeometric equation]] introduced by {{harvs\|txt\|authorlink=E. T. Whittaker\|last=Whittaker\|year=1904}} to make the formulas involving the solutions more symmetric. More generally, {{harvs\|txt\|authorlink=Hervé Jacquet\|last=Jacquet\|year1=1966\|year2=1967}} introduced Whittaker [[Function (mathematics)\|functions]] of [[reductive group]]s over [[local field]]s, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL<sub>2</sub>('''R'''). Whittaker's equation is :<math>\frac{d^2w}{dz^2}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2}\right)w=0.</math> It has a regular singular point at 0 and an irregular singular point at ∞. Two solutions are given by the '''Whittaker functions''' ''M''<sub>κ,μ</sub>(''z''), ''W''<sub>κ,μ</sub>(''z''), defined in terms of Kummer's [[confluent hypergeometric functions]] ''M'' and ''U'' by :<math>M_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu, z\right)</math> :<math>W_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}U\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu, z\right).</math> The Whittaker functions <math>M_{\kappa,\mu}(z)</math> and <math>W_{\kappa,\mu}(z)</math> are the same as those with opposite values of {{mvar\|μ}}, in other words considered as a function of {{mvar\|μ}} at fixed {{mvar\|κ}} and {{mvar\|z}} they are [[even function]]s. When {{mvar\|κ}} and {{mvar\|z}} are real, the functions give real values for real and imaginary values of {{mvar\|μ}}. These functions of {{mvar\|μ}} play a role in so-called [[Kummer space]]s.<ref>{{cite book\|title=Hilbert spaces of entire functions\|url=https://archive.org/details/hilbertspacesofe0000debr\|url-access=registration\|publisher=Prentice-Hall\|author=[[Louis de Branges]]\|asin=B0006BUXNM\|date=1968}} Sections 55-57.</ref> Whittaker functions appear as coefficients of certain representations of the group SL<sub>2</sub>('''R'''), called [[Whittaker model]]s. ==References== {{reflist}} {{AS ref \|13\|504\|14\|537}} {{citation\|first=Harry\|last=Bateman\|title=Higher transcendental functions\|volume=1\|year=1953\|publisher=McGraw-Hill\|url=http://apps.nrbook.com/bateman/Vol1.pdf}}. {{springer\|id=W/w097850\|title=Whittaker function\|first=Yu.A. \|last=Brychkov\|first2=A.P.\|last2= Prudnikov}}. {{dlmf\|first=Adri B. Olde\|last= Daalhuis\|id=13}} *{{Citation \| last1=Jacquet \| first1=Hervé \| title=Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples \| mr=0200390 \| year=1966 \| journal=Comptes Rendus de l'Académie des Sciences, Série A et B \| issn=0151-0509 \| volume=262 \| pages=A943–A945}} {{Citation \| last1=Jacquet \| first1=Hervé \| title=Fonctions de Whittaker associées aux groupes de Chevalley \| url=http://www.numdam.org/item?id=BSMF_1967__95__243_0 \| mr=0271275 \| year=1967 \| journal=Bulletin de la Société Mathématique de France \| issn=0037-9484 \| volume=95 \| pages=243–309\| doi=10.24033/bsmf.1654 }} {{springer\|id=W/w097840\|title=Whittaker equation\|first=N.Kh. \|last=Rozov}}. {{Citation \| last1=Slater \| first1=Lucy Joan \| title=Confluent hypergeometric functions \| publisher=[[Cambridge University Press]] \| mr=0107026 \| year=1960}}. {{Citation \| last1=Whittaker \| first1=Edmund T. \| title=An expression of certain known functions as generalized hypergeometric functions \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| year=1904 \| journal=Bulletin of the A.M.S. \| volume=10 \| issue=3 \| pages=125–134\| doi=10.1090/S0002-9904-1903-01077-5 }} ==Further reading== {{Cite journal\|last1=Hatamzadeh-Varmazyar\|first1=Saeed\|last2=Masouri\|first2=Zahra\|date=2012-11-01\|title=A fast numerical method for analysis of one- and two-dimensional electromagnetic scattering using a set of cardinal functions\|url=http://www.sciencedirect.com/science/article/pii/S0955799712001129\|journal=Engineering Analysis with Boundary Elements\|language=en\|volume=36\|issue=11\|pages=1631–1639\|doi=10.1016/j.enganabound.2012.04.014\|issn=0955-7997}} * {{Cite journal\|last1=Gerasimov\|first1=A. A.\|last2=Lebedev\|first2=Dmitrii R.\|last3=Oblezin\|first3=Sergei V.\|date=2012\|title=New integral representations of Whittaker functions for classical Lie groups\|url=https://iopscience.iop.org/article/10.1070/RM2012v067n01ABEH004776/meta\|journal=Russian Mathematical Surveys\|language=en\|volume=67\|issue=1\|pages=1–92\|doi=10.1070/RM2012v067n01ABEH004776\|arxiv=0705.2886\|bibcode=2012RuMaS..67....1G\|issn=0036-0279}} * {{Cite journal\|last1=Baudoin\|first1=Fabrice\|last2=O’Connell\|first2=Neil\|date=2011\|title=Exponential functionals of brownian motion and class-one Whittaker functions\|url=http://www.numdam.org/item/AIHPB_2011__47_4_1096_0/\|journal=Annales de l'I.H.P. Probabilités et statistiques\|language=en\|volume=47\|issue=4\|pages=1096–1120\|doi=10.1214/10-AIHP401\|bibcode=2011AIHPB..47.1096B\|s2cid=113388}} * {{Cite journal\|last=McKee\|first=Mark\|date=April 2009\|title=An Infinite Order Whittaker Function\|url=https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/an-infinite-order-whittaker-function/95A568D59C6D55F6E00251222479C746\|journal=Canadian Journal of Mathematics\|language=en\|volume=61\|issue=2\|pages=373–381\|doi=10.4153/CJM-2009-019-x\|issn=0008-414X}} * {{Cite journal\|last1=Mathai\|first1=A. M.\|last2=Pederzoli\|first2=Giorgio\|date=1997-03-01\|title=Some properties of matrix-variate Laplace transforms and matrix-variate Whittaker functions\|url=https://dx.doi.org/10.1016%2F0024-3795%2895%2900705-9\|journal=Linear Algebra and Its Applications\|language=en\|volume=253\|issue=1\|pages=209–226\|doi=10.1016/0024-3795(95)00705-9\|issn=0024-3795}} * {{Cite journal\|last=Whittaker\|first=J. M.\|date=May 1927\|title=On the Cardinal Function of Interpolation Theory\|url=https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/on-the-cardinal-function-of-interpolation-theory/4769EF5655E0BFBF7B10A1BA490BFD36\|journal=Proceedings of the Edinburgh Mathematical Society\|language=en\|volume=1\|issue=1\|pages=41–46\|doi=10.1017/S0013091500007318\|issn=1464-3839}} * {{Cite journal\|last=Cherednik\|first=Ivan\|date=2009\|title=Whittaker Limits of Difference Spherical Functions\|url=https://ieeexplore.ieee.org/document/8189397\|journal=International Mathematics Research Notices\|volume=2009\|issue=20\|pages=3793–3842\|doi=10.1093/imrn/rnp065\|arxiv=0807.2155\|s2cid=6253357\|issn=1687-0247}} * {{Cite journal\|last=Slater\|first=L. J.\|date=October 1954\|title=Expansions of generalized Whittaker functions\|url=https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/expansions-of-generalized-whittaker-functions/E011E597828132A26531306D6794C7C3\|journal=Mathematical Proceedings of the Cambridge Philosophical Society\|language=en\|volume=50\|issue=4\|pages=628–631\|doi=10.1017/S0305004100029765\|bibcode=1954PCPS...50..628S\|issn=1469-8064}} * {{cite arxiv\|last=Etingof\|first=Pavel\|date=1999-01-12\|title=Whittaker functions on quantum groups and q-deformed Toda operators\|eprint=math/9901053}} * {{Cite journal\|last=McNamara\|first=Peter J.\|date=2011-01-15\|title=Metaplectic Whittaker functions and crystal bases\|url=https://projecteuclid.org/euclid.dmj/1292509116\|journal=Duke Mathematical Journal\|language=EN\|volume=156\|issue=1\|pages=1–31\|doi=10.1215/00127094-2010-064\|arxiv=0907.2675\|s2cid=979197\|issn=0012-7094}} * {{Cite journal\|last1=Mathai\|first1=A. M.\|last2=Pederzoli\|first2=Giorgio\|date=1998-01-15\|title=A whittaker function of matrix argument\|url=http://www.sciencedirect.com/science/article/pii/S0024379597000591\|journal=Linear Algebra and Its Applications\|language=en\|volume=269\|issue=1\|pages=91–103\|doi=10.1016/S0024-3795(97)00059-1\|issn=0024-3795}} * {{Cite journal\|last1=Frenkel\|first1=E.\|last2=Gaitsgory\|first2=D.\|last3=Kazhdan\|first3=D.\|last4=Vilonen\|first4=K.\|date=1998\|title=Geometric realization of Whittaker functions and the Langlands conjecture\|url=https://www.ams.org/jams/1998-11-02/S0894-0347-98-00260-4/\|journal=Journal of the American Mathematical Society\|language=en\|volume=11\|issue=2\|pages=451–484\|doi=10.1090/S0894-0347-98-00260-4\|s2cid=13221400\|issn=0894-0347}} [[Category:Special hypergeometric functions]] [[Category:E. T. Whittaker]]
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LaTeX to CAS translator

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SymPy

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