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 _2F_1}

${\displaystyle _{2}F_{1}}$

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

Semantic latex: _2F_1

Confidence: 0

Mathematica

Translation: Subscript[$0, 2]*Subscript[F, 1]

Information

Sub Equations

Subscript[$0, 2]*Subscript[F, 1]

Free variables

Subscript[F, 1]

Tests

Symbolic

Numeric

SymPy

Translation: Symbol('{$0}_{2}')*Symbol('{F}_{1}')

Information

Sub Equations

Symbol('{$0}_{2}')*Symbol('{F}_{1}')

Free variables

Symbol('{F}_{1}')

Tests

Symbolic

Numeric

Maple

Translation: $0[2]*F[1]

Information

Sub Equations

$0[2]*F[1]

Free variables

F[1]

Tests

Symbolic

Numeric

Dependency Graph Information

Is part of

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

Description

second

term of the hypergeometric function

first solution

gamma function

independent solution

differential equation

second order

Complete translation information:

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  }, {
    "definition" : "differential equation",
    "score" : 0.46655930748162855
  }, {
    "definition" : "second order",
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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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LaTeX to CAS translator

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