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 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] }

${\displaystyle V(a,z)={\frac {1}{2^{\xi }{\sqrt {\pi }}\Gamma [1/2-a]}}\left[\sin(\xi \pi )\Gamma (1/2-\xi )\,y_{1}(a,z)+{\sqrt {2}}\cos(\xi \pi )\Gamma (1-\xi )\,y_{2}(a,z)\right]}$

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

Semantic latex: \paraV@{a}{z} = \frac{1}{2^\xi \sqrt{\cpi} \Gamma [1 / 2 - a]} [\sin(\xi \cpi) \Gamma(1 / 2 - \xi) y_1(a , z) + \sqrt{2} \cos(\xi \cpi) \Gamma(1 - \xi) y_2(a , z)]

Confidence: 0.63503262687913

Mathematica

Translation: Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]) == Divide[1,(2)^\[Xi]*Sqrt[Pi]*\[CapitalGamma]*(1/2 - a)]*(Sin[\[Xi]*Pi]*\[CapitalGamma]*(1/2 - \[Xi])*Subscript[y, 1][a , z]+Sqrt[2]*Cos[\[Xi]*Pi]*\[CapitalGamma]*(1 - \[Xi])*Subscript[y, 2][a , z])

Information

Sub Equations

Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]) = Divide[1,(2)^\[Xi]*Sqrt[Pi]*\[CapitalGamma]*(1/2 - a)]*(Sin[\[Xi]*Pi]*\[CapitalGamma]*(1/2 - \[Xi])*Subscript[y, 1][a , z]+Sqrt[2]*Cos[\[Xi]*Pi]*\[CapitalGamma]*(1 - \[Xi])*Subscript[y, 2][a , z])

Free variables

\[CapitalGamma]

\[Xi]

a

z

Symbol info

Cosine; Example: \cos@@{z}

Will be translated to: Cos[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E2 Mathematica: https://reference.wolfram.com/language/ref/Cos.html

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

Pi was translated to: Pi

Sine; Example: \sin@@{z}

Will be translated to: Sin[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Mathematica: https://reference.wolfram.com/language/ref/Sin.html

Parabolic cylinder function V; Example: \paraV@{a}{z}

Will be translated to: Alternative translations: [Divide[GAMMA[1/2 + $0], Pi]*(Sin[Pi*($0)] * ParabolicCylinderD[-($0) - 1/2, $1] + ParabolicCylinderD[-($0) - 1/2, -($1)])]Relevant links to definitions: DLMF: http://dlmf.nist.gov/12.4#E2 Mathematica: https://reference.wolfram.com/language/ref/Divide.html

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Maple

Translation: CylinderV(a, z) = (1)/((2)^(xi)*sqrt(Pi)*Gamma*(1/2 - a))*(sin(xi*Pi)*Gamma*(1/2 - xi)*y[1](a , z)+sqrt(2)*cos(xi*Pi)*Gamma*(1 - xi)*y[2](a , z))

Information

Sub Equations

CylinderV(a, z) = (1)/((2)^(xi)*sqrt(Pi)*Gamma*(1/2 - a))*(sin(xi*Pi)*Gamma*(1/2 - xi)*y[1](a , z)+sqrt(2)*cos(xi*Pi)*Gamma*(1 - xi)*y[2](a , z))

Free variables

Gamma

a

xi

z

Symbol info

Cosine; Example: \cos@@{z}

Will be translated to: cos($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=cos

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

Pi was translated to: Pi

Sine; Example: \sin@@{z}

Will be translated to: sin($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin

Parabolic cylinder function V; Example: \paraV@{a}{z}

Will be translated to: CylinderV($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/12.4#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=CylinderV

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$a$

$z$

$V(a,z)$

$f(a,z)$

$U(a,z)$

$V$

$V(a,z)={\frac {1}{2^{\xi }{\sqrt {\pi }}\Gamma [1/2-a]}}\left[\sin(\xi \pi )\Gamma (1/2-\xi )\,y_{1}(a,z)+{\sqrt {2}}\cos(\xi \pi )\Gamma (1-\xi )\,y_{2}(a,z)\right]$

Is part of

$V(a,z)={\frac {1}{2^{\xi }{\sqrt {\pi }}\Gamma [1/2-a]}}\left[\sin(\xi \pi )\Gamma (1/2-\xi )\,y_{1}(a,z)+{\sqrt {2}}\cos(\xi \pi )\Gamma (1-\xi )\,y_{2}(a,z)\right]$

Complete translation information:

{
  "id" : "FORMULA_389030a01beef224e22eea4ee979f12a",
  "formula" : "V(a,z)=\\frac{1}{2^\\xi\\sqrt{\\pi}\\Gamma[1/2-a]}\n\\left[\n\\sin(\\xi\\pi)\\Gamma(1/2-\\xi)y_1(a,z)\n+\\sqrt{2}\\cos(\\xi\\pi)\\Gamma(1-\\xi)y_2(a,z)\n\\right]",
  "semanticFormula" : "\\paraV@{a}{z} = \\frac{1}{2^\\xi \\sqrt{\\cpi} \\Gamma [1 / 2 - a]} [\\sin(\\xi \\cpi) \\Gamma(1 / 2 - \\xi) y_1(a , z) + \\sqrt{2} \\cos(\\xi \\cpi) \\Gamma(1 - \\xi) y_2(a , z)]",
  "confidence" : 0.6350326268791293,
  "translations" : {
    "Mathematica" : {
      "translation" : "Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]) == Divide[1,(2)^\\[Xi]*Sqrt[Pi]*\\[CapitalGamma]*(1/2 - a)]*(Sin[\\[Xi]*Pi]*\\[CapitalGamma]*(1/2 - \\[Xi])*Subscript[y, 1][a , z]+Sqrt[2]*Cos[\\[Xi]*Pi]*\\[CapitalGamma]*(1 - \\[Xi])*Subscript[y, 2][a , z])",
      "translationInformation" : {
        "subEquations" : [ "Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]) = Divide[1,(2)^\\[Xi]*Sqrt[Pi]*\\[CapitalGamma]*(1/2 - a)]*(Sin[\\[Xi]*Pi]*\\[CapitalGamma]*(1/2 - \\[Xi])*Subscript[y, 1][a , z]+Sqrt[2]*Cos[\\[Xi]*Pi]*\\[CapitalGamma]*(1 - \\[Xi])*Subscript[y, 2][a , z])" ],
        "freeVariables" : [ "\\[CapitalGamma]", "\\[Xi]", "a", "z" ],
        "tokenTranslations" : {
          "\\cos" : "Cosine; Example: \\cos@@{z}\nWill be translated to: Cos[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/4.14#E2\nMathematica:  https://reference.wolfram.com/language/ref/Cos.html",
          "y" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\cpi" : "Pi was translated to: Pi",
          "\\sin" : "Sine; Example: \\sin@@{z}\nWill be translated to: Sin[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/4.14#E1\nMathematica:  https://reference.wolfram.com/language/ref/Sin.html",
          "\\paraV" : "Parabolic cylinder function V; Example: \\paraV@{a}{z}\nWill be translated to: \nAlternative translations: [Divide[GAMMA[1/2 + $0], Pi]*(Sin[Pi*($0)] * ParabolicCylinderD[-($0) - 1/2, $1] + ParabolicCylinderD[-($0) - 1/2, -($1)])]Relevant links to definitions:\nDLMF:         http://dlmf.nist.gov/12.4#E2\nMathematica:  https://reference.wolfram.com/language/ref/Divide.html"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\paraV [\\paraV]"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "Maple" : {
      "translation" : "CylinderV(a, z) = (1)/((2)^(xi)*sqrt(Pi)*Gamma*(1/2 - a))*(sin(xi*Pi)*Gamma*(1/2 - xi)*y[1](a , z)+sqrt(2)*cos(xi*Pi)*Gamma*(1 - xi)*y[2](a , z))",
      "translationInformation" : {
        "subEquations" : [ "CylinderV(a, z) = (1)/((2)^(xi)*sqrt(Pi)*Gamma*(1/2 - a))*(sin(xi*Pi)*Gamma*(1/2 - xi)*y[1](a , z)+sqrt(2)*cos(xi*Pi)*Gamma*(1 - xi)*y[2](a , z))" ],
        "freeVariables" : [ "Gamma", "a", "xi", "z" ],
        "tokenTranslations" : {
          "\\cos" : "Cosine; Example: \\cos@@{z}\nWill be translated to: cos($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.14#E2\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=cos",
          "y" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\cpi" : "Pi was translated to: Pi",
          "\\sin" : "Sine; Example: \\sin@@{z}\nWill be translated to: sin($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.14#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin",
          "\\paraV" : "Parabolic cylinder function V; Example: \\paraV@{a}{z}\nWill be translated to: CylinderV($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/12.4#E2\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=CylinderV"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "a", "z", "V(a,z)", "f(a,z)", "U(a,z)", "V", "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]" ],
  "isPartOf" : [ "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]" ],
  "definiens" : [ ]
}

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]]
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LaTeX to CAS translator

Mathematica

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SymPy

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Maple

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