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 Q_{\lambda}^{\mu}(z) = \frac{\sqrt{\pi}\ \Gamma(\lambda+\mu+1)}{2^{\lambda+1}\Gamma(\lambda+3/2)}\frac{e^{i\mu\pi}(z^2-1)^{\mu/2}}{z^{\lambda+\mu+1}} \,_2F_1 \left(\frac{\lambda+\mu+1}{2}, \frac{\lambda+\mu+2}{2}; \lambda+\frac{3}{2}; \frac{1}{z^2}\right),\qquad \text{for}\ \ |z|>1.}

${\displaystyle Q_{\lambda }^{\mu }(z)={\frac {{\sqrt {\pi }}\ \Gamma (\lambda +\mu +1)}{2^{\lambda +1}\Gamma (\lambda +3/2)}}{\frac {e^{i\mu \pi }(z^{2}-1)^{\mu /2}}{z^{\lambda +\mu +1}}}\,_{2}F_{1}\left({\frac {\lambda +\mu +1}{2}},{\frac {\lambda +\mu +2}{2}};\lambda +{\frac {3}{2}};{\frac {1}{z^{2}}}\right),\qquad {\text{for}}\ \ |z|>1.}$

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

Semantic latex: \FerrersQ[\mu]{\lambda}@{z} = \frac{\sqrt{\cpi} \EulerGamma@{\lambda + \mu + 1}}{2^{\lambda+1} \EulerGamma@{\lambda + 3 / 2}} \frac{\expe^{\iunit \mu \cpi}(z^2 - 1)^{\mu/2}}{z^{\lambda+\mu+1}}_2 F_1(\frac{\lambda+\mu+1}{2} , \frac{\lambda+\mu+2}{2} ; \lambda + \frac{3}{2} ; \frac{1}{z^2}) , \qquad{for}|z|> 1

Confidence: 0.60864552565258

Mathematica

Translation: LegendreQ[\[Lambda], \[Mu], z] == Divide[Sqrt[Pi]*Gamma[\[Lambda]+ \[Mu]+ 1],(2)^(\[Lambda]+ 1)* Gamma[\[Lambda]+ 3/2]]*Subscript[Divide[Exp[I*\[Mu]*Pi]*((z)^(2)- 1)^(\[Mu]/2),(z)^(\[Lambda]+ \[Mu]+ 1)], 2]*Subscript[F, 1][Divide[\[Lambda]+ \[Mu]+ 1,2],Divide[\[Lambda]+ \[Mu]+ 2,2]; \[Lambda]+Divide[3,2];Divide[1,(z)^(2)]]

Information

Sub Equations

LegendreQ[\[Lambda], \[Mu], z] = Divide[Sqrt[Pi]*Gamma[\[Lambda]+ \[Mu]+ 1],(2)^(\[Lambda]+ 1)* Gamma[\[Lambda]+ 3/2]]*Subscript[Divide[Exp[I*\[Mu]*Pi]*((z)^(2)- 1)^(\[Mu]/2),(z)^(\[Lambda]+ \[Mu]+ 1)], 2]*Subscript[F, 1][Divide[\[Lambda]+ \[Mu]+ 1,2],Divide[\[Lambda]+ \[Mu]+ 2,2]; \[Lambda]+Divide[3,2];Divide[1,(z)^(2)]]

Free variables

\[Lambda]

\[Mu]

f

o

r

z

Constraints

f*o*r*Abs[z] > 1

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

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

Ferrers function of second kind; Example: \FerrersQ[\mu]{\nu}@{x}

Will be translated to: LegendreQ[$1, $0, $2] Constraints: -1 < x < 1, \mu, \nu \in \Reals Relevant links to definitions: DLMF: http://dlmf.nist.gov/14.3#E2 Mathematica: https://reference.wolfram.com/language/ref/LegendreQ.html

Euler Gamma function; Example: \EulerGamma@{z}

Will be translated to: Gamma[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E1 Mathematica: https://reference.wolfram.com/language/ref/Gamma.html

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Maple

Translation: LegendreQ(lambda, mu, z) = (sqrt(Pi)*GAMMA(lambda + mu + 1))/((2)^(lambda + 1)* GAMMA(lambda + 3/2))*(exp(I*mu*Pi)*((z)^(2)- 1)^(mu/2))/((z)^(lambda + mu + 1))[2]*F[1]((lambda + mu + 1)/(2),(lambda + mu + 2)/(2); lambda +(3)/(2);(1)/((z)^(2)))

Information

Sub Equations

LegendreQ(lambda, mu, z) = (sqrt(Pi)*GAMMA(lambda + mu + 1))/((2)^(lambda + 1)* GAMMA(lambda + 3/2))*(exp(I*mu*Pi)*((z)^(2)- 1)^(mu/2))/((z)^(lambda + mu + 1))[2]*F[1]((lambda + mu + 1)/(2),(lambda + mu + 2)/(2); lambda +(3)/(2);(1)/((z)^(2)))

Free variables

f

lambda

mu

o

r

z

Constraints

f*o*r*abs(z) > 1

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

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

Ferrers function of second kind; Example: \FerrersQ[\mu]{\nu}@{x}

Will be translated to: LegendreQ($1, $0, $2) Constraints: -1 < x < 1, \mu, \nu \in \Reals Relevant links to definitions: DLMF: http://dlmf.nist.gov/14.3#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=LegendreQ

Euler Gamma function; Example: \EulerGamma@{z}

Will be translated to: GAMMA($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=GAMMA

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$Q_{\lambda }^{\mu }(z)={\frac {{\sqrt {\pi }}\ \Gamma (\lambda +\mu +1)}{2^{\lambda +1}\Gamma (\lambda +3/2)}}{\frac {e^{i\mu \pi }(z^{2}-1)^{\mu /2}}{z^{\lambda +\mu +1}}}\,_{2}F_{1}\left({\frac {\lambda +\mu +1}{2}},{\frac {\lambda +\mu +2}{2}};\lambda +{\frac {3}{2}};{\frac {1}{z^{2}}}\right),\qquad {\text{for}}\ \ |z|>1$

$z$

$Q$

$\lambda$

$\Gamma$

$\mu$

$_{2}F_{1}$

$-1$

Is part of

$Q_{\lambda }^{\mu }(z)={\frac {{\sqrt {\pi }}\ \Gamma (\lambda +\mu +1)}{2^{\lambda +1}\Gamma (\lambda +3/2)}}{\frac {e^{i\mu \pi }(z^{2}-1)^{\mu /2}}{z^{\lambda +\mu +1}}}\,_{2}F_{1}\left({\frac {\lambda +\mu +1}{2}},{\frac {\lambda +\mu +2}{2}};\lambda +{\frac {3}{2}};{\frac {1}{z^{2}}}\right),\qquad {\text{for}}\ \ |z|>1$

Complete translation information:

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  "formula" : "Q_{\\lambda}^{\\mu}(z) = \\frac{\\sqrt{\\pi}\\Gamma(\\lambda+\\mu+1)}{2^{\\lambda+1}\\Gamma(\\lambda+3/2)}\\frac{e^{i\\mu\\pi}(z^2-1)^{\\mu/2}}{z^{\\lambda+\\mu+1}} _2F_1 \\left(\\frac{\\lambda+\\mu+1}{2}, \\frac{\\lambda+\\mu+2}{2}; \\lambda+\\frac{3}{2}; \\frac{1}{z^2}\\right),\\qquad \\text{for}|z|>1",
  "semanticFormula" : "\\FerrersQ[\\mu]{\\lambda}@{z} = \\frac{\\sqrt{\\cpi} \\EulerGamma@{\\lambda + \\mu + 1}}{2^{\\lambda+1} \\EulerGamma@{\\lambda + 3 / 2}} \\frac{\\expe^{\\iunit \\mu \\cpi}(z^2 - 1)^{\\mu/2}}{z^{\\lambda+\\mu+1}}_2 F_1(\\frac{\\lambda+\\mu+1}{2} , \\frac{\\lambda+\\mu+2}{2} ; \\lambda + \\frac{3}{2} ; \\frac{1}{z^2}) , \\qquad{for}|z|> 1",
  "confidence" : 0.6086455256525842,
  "translations" : {
    "Mathematica" : {
      "translation" : "LegendreQ[\\[Lambda], \\[Mu], z] == Divide[Sqrt[Pi]*Gamma[\\[Lambda]+ \\[Mu]+ 1],(2)^(\\[Lambda]+ 1)* Gamma[\\[Lambda]+ 3/2]]*Subscript[Divide[Exp[I*\\[Mu]*Pi]*((z)^(2)- 1)^(\\[Mu]/2),(z)^(\\[Lambda]+ \\[Mu]+ 1)], 2]*Subscript[F, 1][Divide[\\[Lambda]+ \\[Mu]+ 1,2],Divide[\\[Lambda]+ \\[Mu]+ 2,2]; \\[Lambda]+Divide[3,2];Divide[1,(z)^(2)]]",
      "translationInformation" : {
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        "freeVariables" : [ "\\[Lambda]", "\\[Mu]", "f", "o", "r", "z" ],
        "constraints" : [ "f*o*r*Abs[z] > 1" ],
        "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",
          "F" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\FerrersQ1" : "Ferrers function of second kind; Example: \\FerrersQ[\\mu]{\\nu}@{x}\nWill be translated to: LegendreQ[$1, $0, $2]\nConstraints: -1 < x < 1, \\mu, \\nu \\in \\Reals\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/14.3#E2\nMathematica:  https://reference.wolfram.com/language/ref/LegendreQ.html",
          "\\EulerGamma" : "Euler Gamma function; Example: \\EulerGamma@{z}\nWill be translated to: Gamma[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/5.2#E1\nMathematica:  https://reference.wolfram.com/language/ref/Gamma.html"
        }
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        "testCalculationsGroup" : [ ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\FerrersQ [\\FerrersQ]"
        }
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      "symbolicResults" : {
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        "testCalculationsGroup" : [ ]
      }
    },
    "Maple" : {
      "translation" : "LegendreQ(lambda, mu, z) = (sqrt(Pi)*GAMMA(lambda + mu + 1))/((2)^(lambda + 1)* GAMMA(lambda + 3/2))*(exp(I*mu*Pi)*((z)^(2)- 1)^(mu/2))/((z)^(lambda + mu + 1))[2]*F[1]((lambda + mu + 1)/(2),(lambda + mu + 2)/(2); lambda +(3)/(2);(1)/((z)^(2)))",
      "translationInformation" : {
        "subEquations" : [ "LegendreQ(lambda, mu, z) = (sqrt(Pi)*GAMMA(lambda + mu + 1))/((2)^(lambda + 1)* GAMMA(lambda + 3/2))*(exp(I*mu*Pi)*((z)^(2)- 1)^(mu/2))/((z)^(lambda + mu + 1))[2]*F[1]((lambda + mu + 1)/(2),(lambda + mu + 2)/(2); lambda +(3)/(2);(1)/((z)^(2)))" ],
        "freeVariables" : [ "f", "lambda", "mu", "o", "r", "z" ],
        "constraints" : [ "f*o*r*abs(z) > 1" ],
        "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",
          "F" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\FerrersQ1" : "Ferrers function of second kind; Example: \\FerrersQ[\\mu]{\\nu}@{x}\nWill be translated to: LegendreQ($1, $0, $2)\nConstraints: -1 < x < 1, \\mu, \\nu \\in \\Reals\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/14.3#E2\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=LegendreQ",
          "\\EulerGamma" : "Euler Gamma function; Example: \\EulerGamma@{z}\nWill be translated to: GAMMA($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/5.2#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=GAMMA"
        }
      },
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  },
  "positions" : [ ],
  "includes" : [ "Q_{\\lambda}^{\\mu}(z) = \\frac{\\sqrt{\\pi}\\ \\Gamma(\\lambda+\\mu+1)}{2^{\\lambda+1}\\Gamma(\\lambda+3/2)}\\frac{e^{i\\mu\\pi}(z^2-1)^{\\mu/2}}{z^{\\lambda+\\mu+1}} \\,_2F_1 \\left(\\frac{\\lambda+\\mu+1}{2}, \\frac{\\lambda+\\mu+2}{2}; \\lambda+\\frac{3}{2}; \\frac{1}{z^2}\\right),\\qquad \\text{for}\\ \\ |z|>1", "z", "Q", "\\lambda", "\\Gamma", "\\mu", "_2F_1", "-1" ],
  "isPartOf" : [ "Q_{\\lambda}^{\\mu}(z) = \\frac{\\sqrt{\\pi}\\ \\Gamma(\\lambda+\\mu+1)}{2^{\\lambda+1}\\Gamma(\\lambda+3/2)}\\frac{e^{i\\mu\\pi}(z^2-1)^{\\mu/2}}{z^{\\lambda+\\mu+1}} \\,_2F_1 \\left(\\frac{\\lambda+\\mu+1}{2}, \\frac{\\lambda+\\mu+2}{2}; \\lambda+\\frac{3}{2}; \\frac{1}{z^2}\\right),\\qquad \\text{for}\\ \\ |z|>1" ],
  "definiens" : [ ]
}

Specify your own input

Formula input
Wikitext context	{{For\|the most common case of integer degree\|Legendre polynomials\|associated Legendre polynomials}} {{no footnotes\|date=January 2013}} In physical science and mathematics, the '''Legendre functions''' ''P''<sub>λ</sub>, ''Q''<sub>λ</sub> and '''associated Legendre functions''' ''P''{{su\|p=μ\|b=λ}}, ''Q''{{su\|p=μ\|b=λ}}, and '''Legendre functions of the second kind''', {{math\|''Q<sub>n</sub>''}}, are all solutions of Legendre's differential equation. The [[Legendre polynomials]] and the [[associated Legendre polynomials]] are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Wikipedia articles. [[File:Associated Legendre Poly.svg\|thumb\|500px\|Associated Legendre polynomial curves for l=5.]] == Legendre's differential equation == The '''general Legendre equation''' reads :<math>(1-x^2)\,y'' -2xy' + \left[\lambda(\lambda+1) - \frac{\mu^2}{1-x^2}\right]\,y = 0,\,</math> where the numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when λ is an integer (denoted n), and μ=0 are the Legendre polynomials {{math\|''P<sub>n</sub>''}}; and when λ is an integer (denoted n), and μ=m is also an integer with \|m\| < n are the associated Legendre polynomials. All other cases of λ and μ can be discussed as one, and the solutions are written ''P''{{su\|p=μ\|b=λ}}, ''Q''{{su\|p=μ\|b=λ}}. If μ=0, the superscript is omitted, and one writes just ''P''<sub>λ</sub>, ''Q''<sub>λ</sub>. However, the solution ''Q''<sub>λ</sub> when λ is an integer is often discussed separately as Legendre's function of the second kind, and denoted {{math\|''Q<sub>n</sub>''}}. This is a second order linear equation with three regular singular points (at 1, −1, and ∞). Like all such equations, it can be converted into a [[hypergeometric differential equation]] by a change of variable, and its solutions can be expressed using [[hypergeometric function]]s. == Solutions of the differential equation == Since the differential equation is linear and of second order, it has two linearly independent solutions, which can both be expressed in terms of the [[hypergeometric function]], <math> _2F_1</math>. With <math>\Gamma</math> being the [[gamma function]], the first solution is :<math>P_{\lambda}^{\mu}(z) = \frac{1}{\Gamma(1-\mu)} \left[\frac{1+z}{1-z}\right]^{\mu/2} \,_2F_1 \left(-\lambda, \lambda+1; 1-\mu; \frac{1-z}{2}\right),\qquad \text{for } \ \|1-z\|<2</math> and the second is, :<math>Q_{\lambda}^{\mu}(z) = \frac{\sqrt{\pi}\ \Gamma(\lambda+\mu+1)}{2^{\lambda+1}\Gamma(\lambda+3/2)}\frac{e^{i\mu\pi}(z^2-1)^{\mu/2}}{z^{\lambda+\mu+1}} \,_2F_1 \left(\frac{\lambda+\mu+1}{2}, \frac{\lambda+\mu+2}{2}; \lambda+\frac{3}{2}; \frac{1}{z^2}\right),\qquad \text{for}\ \ \|z\|>1.</math> These are generally known as Legendre functions of the first and second kind of noninteger degree, with the additional qualifier 'associated' if μ is non-zero. A useful relation between the ''P'' and ''Q'' solutions is [[Whipple formulae\|Whipple's formula]]. ==Legendre functions of the second kind ({{math\|''Q<sub>n</sub>''}})== The nonpolynomial solution for the special case of integer degree <math> \lambda = n \in \mathbb{N}_0 </math>, and <math> \mu = 0 </math>, is often discussed separately. It is given by :<math>Q_n(x)=\frac{n!}{1\cdot3\cdots(2n+1)}\left(x^{-(n+1)}+\frac{(n+1)(n+2)}{2(2n+3)}x^{-(n+3)}+\frac{(n+1)(n+2)(n+3)(n+4)}{2\cdot4(2n+3)(2n+5)}x^{-(n+5)}+\cdots\right)</math> This solution is necessarily singular when <math> x = \pm 1 </math>. The Legendre functions of the second kind can also be defined recursively via [[Legendre_polynomials#Definition_via_generating_function\|Bonnet's recursion formula]] :<math> Q_n(x) = \begin{cases} \frac{1}{2} \log \frac{1+x}{1-x} & n = 0 \\ P_1(x) Q_0(x) - 1 & n = 1 \\ \frac{2n-1}{n} x Q_{n-1}(x) - \frac{n-1}{n} Q_{n-2}(x) & n \geq 2 \,. \end{cases} </math> Graphs of the first five functions are given below. :[[File:Mplwp_legendreQ04.svg\|none\|Plot of the first five Legendre functions of the second kind.]] == Associated Legendre functions of the second kind == The nonpolynomial solution for the special case of integer degree <math> \lambda = n \in \mathbb{N}_0 </math>, and <math> \mu = m \in \mathbb{N}_0 </math> is given by :<math> Q_n^{m}(x) = (-1)^m (1-x^2)^\frac{m}{2} \frac{\mathrm{d}^m}{\mathrm{d}x^m}Q_n(x)\,. </math> ==Integral representations== The Legendre functions can be written as contour integrals. For example, :<math>P_\lambda(z) =P^0_\lambda(z) = \frac{1}{2\pi i} \int_{1,z} \frac{(t^2-1)^\lambda}{2^\lambda(t-z)^{\lambda+1}}dt</math> where the contour winds around the points 1 and ''z'' in the positive direction and does not wind around −1. For real x, we have :<math>P_s(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi}\left(x+\sqrt{x^2-1}\cos\theta\right)^s d\theta = \frac{1}{\pi}\int_0^1\left(x+\sqrt{x^2-1}(2t-1)\right)^s\frac{dt}{\sqrt{t(1-t)}},\qquad s\in\mathbb{C}</math> ==Legendre function as characters== The real integral representation of <math>P_s</math> are very useful in the study of harmonic analysis on <math>L^1(G//K)</math> where <math>G//K</math> is the [[Homogeneous space\|double coset space]] of <math>SL(2,\mathbb{R})</math> (see [[Zonal spherical function]]). Actually the Fourier transform on <math>L^1(G//K)</math> is given by :<math>L^1(G//K)\ni f\mapsto \hat{f}</math> where :<math>\hat{f}(s)=\int_1^\infty f(x)P_s(x)dx,\qquad -1\leq\Re(s)\leq 0 </math> == See also == * [[Ferrers function]] ==References== * {{AS ref\|8\|332}} * {{citation\|first1=Richard\|last1=Courant\|authorlink1=Richard Courant\|first2=David\|last2=Hilbert\|authorlink2=David Hilbert\|year=1953\|title=Methods of Mathematical Physics, Volume 1\|publisher=Interscience Publisher, Inc\|location=New York}}. {{dlmf\|first=T. M. \|last=Dunster\|id=14\|title=Legendre and Related Functions}} {{eom\|id=L/l058030\|first=A.B.\|last= Ivanov}} {{Citation \| last1=Snow \| first1=Chester \| title=Hypergeometric and Legendre functions with applications to integral equations of potential theory \| url=http://babel.hathitrust.org/cgi/pt?id=mdp.39015023896346 \| publisher=U. S. Government Printing Office \| location=Washington, D.C. \| series=National Bureau of Standards Applied Mathematics Series, No. 19 \| mr=0048145 \| year=1952\|origyear=1942}} {{Citation \| last1=Whittaker \| first1=E. T. \|authorlink1=E. T. Whittaker\| last2=Watson \| first2=G. N. \| authorlink2=G. N. Watson\|title=A Course in Modern Analysis \| publisher=[[Cambridge University Press]] \| isbn=978-0-521-58807-2 \| year=1963 }} ==External links== [http://functions.wolfram.com/HypergeometricFunctions/LegendrePGeneral/ Legendre function P] on the Wolfram functions site. [http://functions.wolfram.com/HypergeometricFunctions/LegendreQGeneral/ Legendre function Q] on the Wolfram functions site. [http://functions.wolfram.com/HypergeometricFunctions/LegendreP2General/ Associated Legendre function P] on the Wolfram functions site. [http://functions.wolfram.com/HypergeometricFunctions/LegendreQ2General/ Associated Legendre function Q] on the Wolfram functions site. [[Category:Hypergeometric functions]]
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