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 U(a,x)=D_{-a-\tfrac12}(x),}

${\displaystyle U(a,x)=D_{-a-{\tfrac {1}{2}}}(x),}$

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

Semantic latex: \paraU@{a}{x} = \WhittakerparaD{-a-\tfrac12}@{x}

Confidence: 0.86331298726606

Mathematica

Translation: ParabolicCylinderD[- 1/2 -(a), x] == ParabolicCylinderD[- a -Divide[1,2], x]

Information

Sub Equations

ParabolicCylinderD[- 1/2 -(a), x] = ParabolicCylinderD[- a -Divide[1,2], x]

Free variables

a

x

Symbol info

Parabolic cylinder function U; Example: \paraU@{a}{z}

Will be translated to: ParabolicCylinderD[- 1/2 -($0), $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/12.4#E1 Mathematica: https://reference.wolfram.com/language/ref/ParabolicCylinderD.html

Whittaker's notation D for the parabolic cylinder function; Example: \WhittakerparaD{\nu}@{z}

Will be translated to: ParabolicCylinderD[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/12.1 Mathematica: https://reference.wolfram.com/language/ref/ParabolicCylinderD.html

Tests

Symbolic

Test expression: (ParabolicCylinderD[- 1/2 -(a), x])-(ParabolicCylinderD[- a -Divide[1,2], x])

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: \paraU [\paraU]

Tests

Symbolic

Numeric

Maple

Translation: CylinderU(a, x) = CylinderD(- a -(1)/(2), x)

Information

Sub Equations

CylinderU(a, x) = CylinderD(- a -(1)/(2), x)

Free variables

a

x

Symbol info

Parabolic cylinder function U; Example: \paraU@{a}{z}

Will be translated to: CylinderU($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/12.4#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=CylinderU

Whittaker's notation D for the parabolic cylinder function; Example: \WhittakerparaD{\nu}@{z}

Will be translated to: CylinderD($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/12.1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=CylinderD

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$a$

$f(a,z)$

$U$

$U(a,z)$

$D_{p}(x)$

$V(a,z)$

Description

parabolic cylinder

function

Abramowitz

notation

Whittaker

Stegun

Complete translation information:

{
  "id" : "FORMULA_bad8a34dee43f32a563d9f1c5d7f5719",
  "formula" : "U(a,x)=D_{-a-\\tfrac12}(x)",
  "semanticFormula" : "\\paraU@{a}{x} = \\WhittakerparaD{-a-\\tfrac12}@{x}",
  "confidence" : 0.8633129872660568,
  "translations" : {
    "Mathematica" : {
      "translation" : "ParabolicCylinderD[- 1/2 -(a), x] == ParabolicCylinderD[- a -Divide[1,2], x]",
      "translationInformation" : {
        "subEquations" : [ "ParabolicCylinderD[- 1/2 -(a), x] = ParabolicCylinderD[- a -Divide[1,2], x]" ],
        "freeVariables" : [ "a", "x" ],
        "tokenTranslations" : {
          "\\paraU" : "Parabolic cylinder function U; Example: \\paraU@{a}{z}\nWill be translated to: ParabolicCylinderD[- 1/2 -($0), $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/12.4#E1\nMathematica:  https://reference.wolfram.com/language/ref/ParabolicCylinderD.html",
          "\\WhittakerparaD" : "Whittaker's notation D for the parabolic cylinder function; Example: \\WhittakerparaD{\\nu}@{z}\nWill be translated to: ParabolicCylinderD[$0, $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/12.1\nMathematica:  https://reference.wolfram.com/language/ref/ParabolicCylinderD.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" : "ParabolicCylinderD[- 1/2 -(a), x]",
          "rhs" : "ParabolicCylinderD[- a -Divide[1,2], x]",
          "testExpression" : "(ParabolicCylinderD[- 1/2 -(a), x])-(ParabolicCylinderD[- a -Divide[1,2], x])",
          "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: \\paraU [\\paraU]"
        }
      }
    },
    "Maple" : {
      "translation" : "CylinderU(a, x) = CylinderD(- a -(1)/(2), x)",
      "translationInformation" : {
        "subEquations" : [ "CylinderU(a, x) = CylinderD(- a -(1)/(2), x)" ],
        "freeVariables" : [ "a", "x" ],
        "tokenTranslations" : {
          "\\paraU" : "Parabolic cylinder function U; Example: \\paraU@{a}{z}\nWill be translated to: CylinderU($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/12.4#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=CylinderU",
          "\\WhittakerparaD" : "Whittaker's notation D for the parabolic cylinder function; Example: \\WhittakerparaD{\\nu}@{z}\nWill be translated to: CylinderD($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/12.1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=CylinderD"
        }
      }
    }
  },
  "positions" : [ {
    "section" : 1,
    "sentence" : 7,
    "word" : 42
  } ],
  "includes" : [ "a", "f(a,z)", "U", "U(a,z)", "D_{p}(x)", "V(a,z)" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "parabolic cylinder",
    "score" : 0.6202889650070861
  }, {
    "definition" : "function",
    "score" : 0.5835050192745167
  }, {
    "definition" : "Abramowitz",
    "score" : 0.5730317852477183
  }, {
    "definition" : "notation",
    "score" : 0.5730317852477183
  }, {
    "definition" : "Whittaker",
    "score" : 0.5730317852477183
  }, {
    "definition" : "Stegun",
    "score" : 0.5243095238095238
  } ]
}

Specify your own input

Formula input
Wikitext context	[[File:Parabolic cylindrical coordinates.png\|thumb\|right\|350px\|[[Coordinate system#Coordinate surface\|Coordinate surfaces]] of parabolic cylindrical coordinates. Parabolic cylinder functions occur when [[separation of variables]] is used on [[Laplace equation\|Laplace's equation]] in these coordinates]] In [[mathematics]], the '''parabolic cylinder functions''' are [[special function]]s defined as solutions to the differential equation {{NumBlk\|:\|<math>\frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0.</math>\|{{EquationRef\|1}}}} This equation is found when the technique of [[separation of variables]] is used on [[Laplace equation\|Laplace's equation]] when expressed in [[parabolic cylindrical coordinates]]. The above equation may be brought into two distinct forms (A) and (B) by [[completing the square]] and rescaling ''z'', called [[H. F. Weber]]'s equations {{harv\|Weber\|1869}}: :<math>\frac{d^2f}{dz^2} - \left(\tfrac14z^2+a\right)f=0</math>    (A) and :<math>\frac{d^2f}{dz^2} + \left(\tfrac14z^2-a\right)f=0.</math>     (B) If :<math>f(a,z)\,</math> is a solution, then so are :<math>f(a,-z), f(-a,iz)\text{ and }f(-a,-iz).\,</math> If :<math>f(a,z)\,</math> is a solution of equation (A), then :<math>f(-ia,ze^{(1/4)\pi i})\,</math> is a solution of (B), and, by symmetry, :<math>f(-ia,-ze^{(1/4)\pi i}), f(ia,-ze^{-(1/4)\pi i})\text{ and }f(ia,ze^{-(1/4)\pi i})\,</math> are also solutions of (B). ==Solutions== There are independent even and odd solutions of the form (A). These are given by (following the notation of [[Abramowitz and Stegun]] (1965)): :<math>y_1(a;z) = \exp(-z^2/4) \;_1F_1 \left(\tfrac12a+\tfrac14; \; \tfrac12\; ; \; \frac{z^2}{2}\right)\,\,\,\,\,\, (\mathrm{even})</math> and :<math>y_2(a;z) = z\exp(-z^2/4) \;_1F_1 \left(\tfrac12a+\tfrac34; \; \tfrac32\; ; \; \frac{z^2}{2}\right)\,\,\,\,\,\, (\mathrm{odd})</math> where <math>\;_1F_1 (a;b;z)=M(a;b;z)</math> is the [[confluent hypergeometric function]]. Other pairs of independent solutions may be formed from linear combinations of the above solutions (see Abramowitz and Stegun). One such pair is based upon their behavior at infinity: :<math> U(a,z)=\frac{1}{2^\xi\sqrt{\pi}} \left[ \cos(\xi\pi)\Gamma(1/2-\xi)\,y_1(a,z) -\sqrt{2}\sin(\xi\pi)\Gamma(1-\xi)\,y_2(a,z) \right] </math> :<math> V(a,z)=\frac{1}{2^\xi\sqrt{\pi}\Gamma[1/2-a]} \left[ \sin(\xi\pi)\Gamma(1/2-\xi)\,y_1(a,z) +\sqrt{2}\cos(\xi\pi)\Gamma(1-\xi)\,y_2(a,z) \right] </math> where :<math> \xi=\frac{1}{2}a+\frac{1}{4} . </math> The function ''U''(''a'', ''z'') approaches zero for large values of z  and \|arg(''z'')\| < π/2, while ''V''(''a'', ''z'') diverges for large values of positive real ''z'' . :<math> \lim_{z\rightarrow\infty}U(a,z)/e^{-z^2/4}z^{-a-1/2}=1\,\,\,\,(\text{for}\,\|\arg(z)\|<\pi/2) </math> and :<math> \lim_{z\rightarrow\infty}V(a,z)/\sqrt{\frac{2}{\pi}}e^{z^2/4}z^{a-1/2}=1\,\,\,\,(\text{for}\,\arg(z)=0) . </math> For [[half-integer]] values of ''a'', these (that is, ''U'' and ''V'') can be re-expressed in terms of [[Hermite polynomials]]; alternatively, they can also be expressed in terms of [[Bessel function]]s. The functions ''U'' and ''V'' can also be related to the functions ''D<sub>p</sub>''(''x'') (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions (see Abramowitz and Stegun (1965)): :<math>U(a,x)=D_{-a-\tfrac12}(x),</math> :<math>V(a,x)=\frac{\Gamma(\tfrac12+a)}{\pi}[\sin( \pi a) D_{-a-\tfrac12}(x)+D_{-a-\tfrac12}(-x)] .</math> Function ''D<sub>a</sub>(z)'' was introduced by Whittaker and Watson as a solution of eq.~({{EquationNote\|1}}) with <math>\tilde a=-\frac14, \tilde b=0, \tilde c=a+\frac12</math> bounded at <math>+\infty</math>. It can be expressed in terms of confluent hypergeometric functions as :<math>D_a(z)=\frac{1}{\sqrt{\pi }}{2^{a/2} e^{-\frac{z^2}{4}} \left(\cos \left(\frac{\pi a}{2}\right) \Gamma \left(\frac{a+1}{2}\right) \, _1F_1\left(-\frac{a}{2};\frac{1}{2};\frac{z^2}{2}\right)+\sqrt{2} z \sin \left(\frac{\pi a}{2}\right) \Gamma \left(\frac{a}{2}+1\right) \, _1F_1\left(\frac{1}{2}-\frac{a}{2};\frac{3}{2};\frac{z^2}{2}\right)\right)}.</math> {{no footnotes\|date=December 2010}} == References == * {{AS ref \|19\|686}} {{springer\|id=W/w097310\|title=Weber equation\|first=N.Kh.\|last= Rozov}} {{dlmf\|id=12\|first=N. M. \|last=Temme}} Weber, H.F. (1869) "Ueber die Integration der partiellen Differentialgleichung <math>\partial^2u/\partial x^2+\partial^2u/\partial y^2+k^2u=0</math>". ''Math. Ann.'', 1, 1–36 Whittaker, E.T. (1902) "On the functions associated with the parabolic cylinder in harmonic analysis" ''Proc. London Math. Soc.''35, 417–427. *Whittaker, E. T. and Watson, G. N. "The Parabolic Cylinder Function." §16.5 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 347-348, 1990. {{DEFAULTSORT:Parabolic Cylinder Function}} [[Category:Special hypergeometric functions]]
	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