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 P_s}

${\displaystyle P_{s}}$

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

Semantic latex: \assLegendreP{s}@{z}

Confidence: 0.32468193222273

Mathematica

Translation: LegendreP[s, 0, 3, z]

Information

Sub Equations

LegendreP[s, 0, 3, z]

Free variables

s

z

Symbol info

associated Legendre polynomial of the first kind; Example: \assLegendreP{nu}@{z}

Will be translated to: LegendreP[$0, 0, 3, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/14.21#Ex1 Mathematica: https://reference.wolfram.com/language/ref/LegendreP.html

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Maple

Translation: LegendreP(s, z)

Information

Sub Equations

LegendreP(s, z)

Free variables

s

z

Symbol info

associated Legendre polynomial of the first kind; Example: \assLegendreP{nu}@{z}

Will be translated to: LegendreP($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/14.21#Ex1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=LegendreP

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$P_{n}$

$P$

Is part of

$Q_{n}(x)={\begin{cases}{\frac {1}{2}}\log {\frac {1+x}{1-x}}&n=0\\P_{1}(x)Q_{0}(x)-1&n=1\\{\frac {2n-1}{n}}xQ_{n-1}(x)-{\frac {n-1}{n}}Q_{n-2}(x)&n\geq 2\,.\end{cases}}$

$P_{n}$

$P_{\lambda }(z)=P_{\lambda }^{0}(z)={\frac {1}{2\pi i}}\int _{1,z}{\frac {(t^{2}-1)^{\lambda }}{2^{\lambda }(t-z)^{\lambda +1}}}dt$

$P_{\lambda },Q_{\lambda }$

$P_{s}(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(x+{\sqrt {x^{2}-1}}\cos \theta \right)^{s}d\theta ={\frac {1}{\pi }}\int _{0}^{1}\left(x+{\sqrt {x^{2}-1}}(2t-1)\right)^{s}{\frac {dt}{\sqrt {t(1-t)}}},\qquad s\in \mathbb {C}$

${\hat {f}}(s)=\int _{1}^{\infty }f(x)P_{s}(x)dx,\qquad -1\leq \Re (s)\leq 0$

Description

real integral representation

Fourier

double coset space

study of harmonic analysis

spherical function

Zonal

real x

example

contour

point

integer

n

wind

Legendre polynomial

positive direction

graph

function

physical science

m

polynomial solution

associated Legendre function

mathematics

solution of Legendre 's differential equation

Bonnet 's recursion formula

superscript

Legendre function of the second kind

Legendre functions of the second kind

Complete translation information:

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  "formula" : "P_s",
  "semanticFormula" : "\\assLegendreP{s}@{z}",
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      "translationInformation" : {
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          "\\assLegendreP" : "associated Legendre polynomial of the first kind; Example: \\assLegendreP{nu}@{z}\nWill be translated to: LegendreP[$0, 0, 3, $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/14.21#Ex1\nMathematica:  https://reference.wolfram.com/language/ref/LegendreP.html"
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    "Maple" : {
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        "subEquations" : [ "LegendreP(s, z)" ],
        "freeVariables" : [ "s", "z" ],
        "tokenTranslations" : {
          "\\assLegendreP" : "associated Legendre polynomial of the first kind; Example: \\assLegendreP{nu}@{z}\nWill be translated to: LegendreP($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/14.21#Ex1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=LegendreP"
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  "includes" : [ "P_{n}", "P" ],
  "isPartOf" : [ "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}", "P_{n}", "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", "P_{\\lambda},Q_{\\lambda}", "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}", "\\hat{f}(s)=\\int_1^\\infty f(x)P_s(x)dx,\\qquad -1\\leq\\Re(s)\\leq 0" ],
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}

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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