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 \int_{-1}^1 P_n(x)\,dx = 0 \text{ for } n\ge1,}

${\displaystyle \int _{-1}^{1}P_{n}(x)\,dx=0{\text{ for }}n\geq 1,}$

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

Semantic latex: \int_{-1}^1 \LegendrepolyP{n}@{x} \diff{x} = 0 for n \ge 1

Confidence: 0.72556733422884

Mathematica

Translation: Integrate[LegendreP[n, x], {x, - 1, 1}, GenerateConditions->None] == 0*f*o*r*n >= 1

Information

Sub Equations

Integrate[LegendreP[n, x], {x, - 1, 1}, GenerateConditions->None] = 0*f*o*r*n

0*f*o*r*n >= 1

Free variables

f

n

o

r

Symbol info

Legendre polynomial; Example: \LegendrepolyP{n}@{x}

Will be translated to: LegendreP[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.3#T1.t1.r10 Mathematica: https://reference.wolfram.com/language/ref/LegendreP.html

Tests

Symbolic

Test expression: (Integrate[LegendreP[n, x], {x, - 1, 1}, GenerateConditions->None])-(0*f*o*r*n)

ERROR:

{
    "result": "ERROR",
    "testTitle": "Simple",
    "testExpression": null,
    "resultExpression": null,
    "wasAborted": false,
    "conditionallySuccessful": false
}

Test expression: 0*f*o*r*n\geq1

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: \LegendrepolyP [\LegendrepolyP]

Tests

Symbolic

Numeric

Maple

Translation: int(LegendreP(n, x), x = - 1..1) = 0*f*o*r*n >= 1

Information

Sub Equations

int(LegendreP(n, x), x = - 1..1) = 0*f*o*r*n

0*f*o*r*n >= 1

Free variables

f

n

o

r

Symbol info

Legendre polynomial; Example: \LegendrepolyP{n}@{x}

Will be translated to: LegendreP($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.3#T1.t1.r10 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=LegendreP

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$P$

$n$

$x$

$P_{n}(x)$

$P_{n}(x)$

$P_{n}$

$Q_{n}$

$\int _{-1}^{1}P_{n}(x)\,dx=0{\text{ for }}n\geq 1$

$P_{m}$

Is part of

$\int _{-1}^{1}P_{n}(x)\,dx=0{\text{ for }}n\geq 1$

Complete translation information:

{
  "id" : "FORMULA_19d856e47dc6350e3cc44ebb62d6dbc8",
  "formula" : "\\int_{-1}^1 P_n(x)dx = 0 \\text{ for } n\\ge1",
  "semanticFormula" : "\\int_{-1}^1 \\LegendrepolyP{n}@{x} \\diff{x} = 0 for n \\ge 1",
  "confidence" : 0.7255673342288351,
  "translations" : {
    "Mathematica" : {
      "translation" : "Integrate[LegendreP[n, x], {x, - 1, 1}, GenerateConditions->None] == 0*f*o*r*n >= 1",
      "translationInformation" : {
        "subEquations" : [ "Integrate[LegendreP[n, x], {x, - 1, 1}, GenerateConditions->None] = 0*f*o*r*n", "0*f*o*r*n >= 1" ],
        "freeVariables" : [ "f", "n", "o", "r" ],
        "tokenTranslations" : {
          "\\LegendrepolyP" : "Legendre polynomial; Example: \\LegendrepolyP{n}@{x}\nWill be translated to: LegendreP[$0, $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/18.3#T1.t1.r10\nMathematica:  https://reference.wolfram.com/language/ref/LegendreP.html"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "ERROR",
        "numberOfTests" : 2,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 2,
        "crashed" : false,
        "testCalculationsGroup" : [ {
          "lhs" : "Integrate[LegendreP[n, x], {x, - 1, 1}, GenerateConditions->None]",
          "rhs" : "0*f*o*r*n",
          "testExpression" : "(Integrate[LegendreP[n, x], {x, - 1, 1}, GenerateConditions->None])-(0*f*o*r*n)",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        }, {
          "lhs" : "0*f*o*r*n",
          "rhs" : "1",
          "testExpression" : "0*f*o*r*n\\geq1",
          "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: \\LegendrepolyP [\\LegendrepolyP]"
        }
      }
    },
    "Maple" : {
      "translation" : "int(LegendreP(n, x), x = - 1..1) = 0*f*o*r*n >= 1",
      "translationInformation" : {
        "subEquations" : [ "int(LegendreP(n, x), x = - 1..1) = 0*f*o*r*n", "0*f*o*r*n >= 1" ],
        "freeVariables" : [ "f", "n", "o", "r" ],
        "tokenTranslations" : {
          "\\LegendrepolyP" : "Legendre polynomial; Example: \\LegendrepolyP{n}@{x}\nWill be translated to: LegendreP($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/18.3#T1.t1.r10\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=LegendreP"
        }
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "P", "n", "x", "P_{n}(x)", "P_n(x)", "P_n", "Q_n", "\\int_{-1}^1 P_n(x)\\,dx = 0 \\text{ for } n\\ge1", "P_m" ],
  "isPartOf" : [ "\\int_{-1}^1 P_n(x)\\,dx = 0 \\text{ for } n\\ge1" ],
  "definiens" : [ ]
}

Specify your own input

Formula input
Wikitext context	{{for\|Legendre's Homogeneous equation\|Legendre's equation}} :[[File:Legendrepolynomials6.svg\|360px\|thumb\|The six first Legendre polynomials.]] In physical science and [[mathematics]], '''Legendre polynomials''' (named after [[Adrien-Marie Legendre]], who discovered them in 1782) are a system of complete and [[orthogonal polynomial]]s, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are [[associated Legendre polynomials]], [[Legendre function]]s, Legendre functions of the second kind, and [[associated Legendre function]]s. == Definition by construction as an orthogonal system == In this approach, the polynomials are defined as an orthogonal system with respect to the weight function <math>w(x) = 1</math> over the interval <math> [-1,1]</math>. That is, <math>P_n(x)</math> is a polynomial of degree <math>n</math>, such that :<math>\int_{-1}^1 P_m(x) P_n(x)\,dx = 0 \quad \text{if } n \ne m.</math> This determines the polynomials completely up to an overall scale factor, which is fixed by the standardization <math>P_n(1) = 1</math>. That this is a constructive definition is seen thus: <math>P_0(x) = 1</math> is the only correctly standardized polynomial of degree 0. <math>P_1(x)</math> must be orthogonal to <math>P_0</math>, leading to <math>P_1(x) = x</math>, and <math>P_2(x)</math> is determined by demanding orthogonality to <math>P_0</math> and <math>P_1</math>, and so on. <math>P_n</math> is fixed by demanding orthogonality to all <math>P_m</math> with <math> m < n </math>. This gives <math> n </math> conditions, which, along with the standardization <math> P_n(1) = 1</math> fixes all <math> n+1</math> coefficients in <math> P_n(x)</math>. With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of <math>x</math> given below. This definition of the <math>P_n</math>'s is the simplest one. It does not appeal to the theory of differential equations. Second, the completeness of the polynomials follows immediately from the completeness of the powers 1, <math> x, x^2, x^3, \ldots</math>. Finally, by defining them via orthogonality with respect to the most obvious weight function on a finite interval, it sets up the Legendre polynomials as one of the three [[classical orthogonal polynomials\|classical orthogonal polynomial systems]]. The other two are the [[Laguerre polynomials]], which are orthogonal over the half line <math>[0,\infty)</math>, and the [[Hermite polynomials]], orthogonal over the full line <math>(-\infty,\infty)</math>, with weight functions that are the most natural analytic functions that ensure convergence of all integrals. == Definition via generating function == The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of <math>t</math> of the [[generating function]]<ref>{{harvnb\|Arfken\|Weber\|2005\|loc=p.743}}</ref> {{NumBlk\|:\|<math>\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^\infty P_n(x) t^n \,.</math>\|{{EquationRef\|2}}}} The coefficient of <math>t^n</math> is a polynomial in <math> x </math> of degree <math>n</math>. Expanding up to <math>t^1</math> gives :<math>P_0(x) = 1 \,,\quad P_1(x) = x.</math> Expansion to higher orders gets increasingly cumbersome, but is possible to do systematically, and again leads to one of the explicit forms given below. It is possible to obtain the higher <math>P_n</math>'s without resorting to direct expansion of the Taylor series, however. Eq. {{EquationNote\|2}} is differentiated with respect to {{math\|''t''}} on both sides and rearranged to obtain :<math>\frac{x-t}{\sqrt{1-2xt+t^2}} = \left(1-2xt+t^2\right) \sum_{n=1}^\infty n P_n(x) t^{n-1} \,.</math> Replacing the quotient of the square root with its definition in Eq. {{EquationNote\|2}}, and [[equating the coefficients]] of powers of {{math\|''t''}} in the resulting expansion gives ''Bonnet’s recursion formula'' :<math> (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)\,.</math> This relation, along with the first two polynomials {{math\|''P''<sub>0</sub>}} and {{math\|''P''<sub>1</sub>}}, allows all the rest to be generated recursively. The generating function approach is directly connected to the [[multipole expansion]] in electrostatics, as explained below, and is how the polynomials were first defined by Legendre in 1782. == Definition via differential equation == A third definition is in terms of solutions to '''Legendre's [[differential equation]]''' {{NumBlk\|:\|<math> \frac{d}{dx}\left[\left(1-x^2\right) \frac{dP_n(x)}{dx}\right] + n(n+1) P_n(x) = 0\,.</math>\|{{EquationRef\|1}}}} This differential equation has [[regular singular point]]s at {{math\|''x'' {{=}} ±1}} so if a solution is sought using the standard [[Frobenius method\|Frobenius]] or [[power series]] method, a series about the origin will only converge for {{math\|{{abs\|''x''}} < 1}} in general. When {{math\|''n''}} is an integer, the solution {{math\|''P<sub>n</sub>''(''x'')}} that is regular at {{math\|''x'' {{=}} 1}} is also regular at {{math\|''x'' {{=}} −1}}, and the series for this solution terminates (i.e. it is a polynomial). The orthogonality and completeness of these solutions is best seen from the viewpoint of [[Sturm–Liouville theory]]. We rewrite the differential equation as an eigenvalue problem, :<math>\frac{d}{dx} \left( \left(1-x^2\right) \frac{d}{dx} \right) P(x) = -\lambda P(x) \,,</math> with the eigenvalue <math>\lambda</math> in lieu of <math> n(n+1)</math>. If we demand that the solution be regular at <math>x = \pm 1</math>, the [[differential operator]] on the left is [[Hermitian]]. The eigenvalues are found to be of the form {{math\|''n''(''n'' + 1)}}, with <math>n = 0, 1, 2, \ldots</math>, and the eigenfunctions are the <math>P_n(x)</math>. The orthogonality and completeness of this set of solutions follows at once from the larger framework of Sturm–Liouville theory. The differential equation admits another, non-polynomial solution, the [[Legendre_function#Legendre_functions_of_the_second_kind_(Qn)\|Legendre functions of the second kind]] <math>Q_n</math>. A two-parameter generalization of (Eq. {{EquationNote\|1}}) is called Legendre's ''general'' differential equation, solved by the [[Associated Legendre polynomials]]. [[Legendre functions]] are solutions of Legendre's differential equation (generalized or not) with ''non-integer'' parameters. In physical settings, Legendre's differential equation arises naturally whenever one solves [[Laplace's equation]] (and related [[partial differential equation]]s) by separation of variables in [[spherical coordinates]]. From this standpoint, the eigenfunctions of the angular part of the Laplacian operator are the [[spherical harmonics]], of which the Legendre polynomials are (up to a multiplicative constant) the subset that is left invariant by rotations about the polar axis. The polynomials appear as <math>P_n(\cos\theta)</math> where <math>\theta</math> is the polar angle. This approach to the Legendre polynomials provides a deep connection to rotational symmetry. Many of their properties which are found laboriously through the methods of analysis — for example the addition theorem — are more easily found using the methods of symmetry and group theory, and acquire profound physical and geometrical meaning. == Orthonormality and completeness == The standardization <math>P_n(1) = 1</math> fixes the normalization of the Legendre polynomials (with respect to the [[L2-norm\|{{math\|''L''<sup>2</sup>}} norm]] on the interval {{math\|−1 ≤ ''x'' ≤ 1}}). Since they are also [[orthogonal function\|orthogonal]] with respect to the same norm, the two statements can be combined into the single equation, :<math>\int_{-1}^1 P_m(x) P_n(x)\,dx = \frac{2}{2n + 1} \delta_{mn},</math> (where {{math\|''δ<sub>mn</sub>''}} denotes the [[Kronecker delta]], equal to 1 if {{math\|''m'' {{=}} ''n''}} and to 0 otherwise). This normalization is most readily found by employing [[Rodrigues' formula]], given below. That the polynomials are complete means the following. Given any piecewise continuous function <math> f(x) </math> with finitely many discontinuities in the interval [−1,1], the sequence of sums :<math> f_n(x) = \sum_{\ell=0}^n a_\ell P_\ell(x)</math> converges in the mean to <math> f(x) </math> as <math> n \to \infty </math>, provided we take :<math> a_\ell = \frac{2\ell + 1}{2} \int_{-1}^1 f(x) P_\ell(x)\,dx.</math> This completeness property underlies all the expansions discussed in this article, and is often stated in the form :<math>\sum_{\ell=0}^\infty \frac{2\ell + 1}{2} P_\ell(x)P_\ell(y) = \delta(x-y), </math> with {{math\|−1 ≤ ''x'' ≤ 1}} and {{math\|−1 ≤ ''y'' ≤ 1}}. == Rodrigues' formula and other explicit formulas == An especially compact expression for the Legendre polynomials is given by [[Rodrigues' formula]]: :<math>P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 -1)^n \,.</math> This formula enables derivation of a large number of properties of the <math>P_n</math>'s. Among these are explicit representations such as :<math>\begin{align} P_n(x)&= \frac{1}{2^n} \sum_{k=0}^n \binom{n}{k}^2 (x-1)^{n-k}(x+1)^k, \\ P_n(x)&=\sum_{k=0}^n \binom{n}{k} \binom{n+k}{k} \left( \frac{x-1}{2} \right)^k, \\ P_n(x)&=\frac1{2^n}\sum_{k=0}^{[\frac n2]}(-1)^k\binom nk\binom{2n-2k}n x^{n-2k},\\ P_n(x)&= 2^n \sum_{k=0}^n x^k \binom{n}{k} \binom{\frac{n+k-1}{2}}{n}, \end{align}</math> where the last, which is also immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the [[Binomial_coefficient#Generalization_and_connection_to_the_binomial_series\|generalized form of the binomial coefficient]]. The first few Legendre polynomials are: : <math>\begin{array}{r\|r} n & P_n(x) \\ \hline 0 & 1 \\ 1 & x \\ 2 & \tfrac12 \left(3x^2-1\right) \\ 3 & \tfrac12 \left(5x^3-3x\right) \\ 4 & \tfrac18 \left(35x^4-30x^2+3\right) \\ 5 & \tfrac18 \left(63x^5-70x^3+15x\right) \\ 6 & \tfrac1{16} \left(231x^6-315x^4+105x^2-5\right) \\ 7 & \tfrac1{16} \left(429x^7-693x^5+315x^3-35x\right) \\ 8 & \tfrac1{128} \left(6435x^8-12012x^6+6930x^4-1260x^2+35\right) \\ 9 & \tfrac1{128} \left(12155x^9-25740x^7+18018x^5-4620x^3+315x\right) \\ 10 & \tfrac1{256} \left(46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63\right) \\ \hline \end{array}</math> The graphs of these polynomials (up to {{math\|''n'' {{=}} 5}}) are shown below: :[[File:Legendrepolynomials6.svg\|640px\|none\|Plot of the six first Legendre polynomials.]] ==Applications of Legendre polynomials== ===Expanding a 1/''r'' potential=== The Legendre polynomials were first introduced in 1782 by [[Adrien-Marie Legendre]]<ref>{{cite book \|first1=A.-M. \|last1=Legendre \|chapter=Recherches sur l'attraction des sphéroïdes homogènes \|title=Mémoires de Mathématiques et de Physique, présentés à l'Académie Royale des Sciences, par divers savans, et lus dans ses Assemblées \|volume=X \|pages=411–435 \|location=Paris \|date=1785 \|orig-year=1782 \|language=French \|chapter-url=http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf \|url-status=dead \|archiveurl=https://web.archive.org/web/20090920070434/http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf \|archivedate=2009-09-20 }}</ref> as the coefficients in the expansion of the [[Newtonian potential]] :<math> \frac{1}{\left\| \mathbf{x}-\mathbf{x}' \right\|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma), </math> where {{math\|''r''}} and {{math\|''r''′}} are the lengths of the vectors {{math\|'''x'''}} and {{math\|'''x'''′}} respectively and {{math\|''γ''}} is the angle between those two vectors. The series converges when {{math\|''r'' > ''r''′}}. The expression gives the [[gravitational potential]] associated to a [[point mass]] or the [[Coulomb potential]] associated to a [[point charge]]. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution. Legendre polynomials occur in the solution of [[Laplace's equation]] of the static [[electric potential\|potential]], {{math\|∇<sup>2</sup> Φ('''x''') {{=}} 0}}, in a charge-free region of space, using the method of [[separation of variables]], where the [[boundary conditions]] have axial symmetry (no dependence on an [[azimuth\|azimuthal angle]]). Where {{math\|'''ẑ'''}} is the axis of symmetry and {{math\|''θ''}} is the angle between the position of the observer and the {{math\|'''ẑ'''}} axis (the zenith angle), the solution for the potential will be :<math>\Phi(r,\theta) = \sum_{\ell=0}^\infty \left( A_\ell r^\ell + B_\ell r^{-(\ell+1)} \right) P_\ell(\cos\theta) \,.</math> {{math\|''A<sub>l</sub>''}} and {{math\|''B<sub>l</sub>''}} are to be determined according to the boundary condition of each problem.<ref>{{cite book\|last=Jackson \|first=J. D. \|title=Classical Electrodynamics \|url=https://archive.org/details/classicalelectro00jack_449 \|url-access=limited \|edition= 3rd \|location=Wiley & Sons \|date=1999 \|page=[https://archive.org/details/classicalelectro00jack_449/page/n102 103] \|isbn=978-0-471-30932-1}}</ref> They also appear when solving the [[Schrödinger equation]] in three dimensions for a central force. === Legendre polynomials in multipole expansions === [[File:Point axial multipole.svg\|right\|Diagram for the multipole expansion of electric potential.]] Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently): :<math>\frac{1}{\sqrt{1 + \eta^2 - 2\eta x}} = \sum_{k=0}^\infty \eta^k P_k(x),</math> which arise naturally in [[multipole expansion]]s. The left-hand side of the equation is the [[generating function]] for the Legendre polynomials. As an example, the [[electric potential]] {{math\|Φ(''r'',''θ'')}} (in [[spherical coordinates]]) due to a [[point charge]] located on the {{math\|''z''}}-axis at {{math\|''z'' {{=}} ''a''}} (see diagram right) varies as :<math>\Phi (r, \theta ) \propto \frac{1}{R} = \frac{1}{\sqrt{r^2 + a^2 - 2ar \cos\theta}}.</math> If the radius {{math\|''r''}} of the observation point {{math\|P}} is greater than {{math\|''a''}}, the potential may be expanded in the Legendre polynomials :<math>\Phi(r, \theta) \propto \frac{1}{r} \sum_{k=0}^\infty \left( \frac{a}{r} \right)^k P_k(\cos \theta),</math> where we have defined {{math\|''η'' {{=}} {{sfrac\|''a''\|''r''}} < 1}} and {{math\|''x'' {{=}} cos ''θ''}}. This expansion is used to develop the normal [[multipole expansion]]. Conversely, if the radius {{math\|''r''}} of the observation point {{math\|P}} is smaller than {{math\|''a''}}, the potential may still be expanded in the Legendre polynomials as above, but with {{math\|''a''}} and {{math\|''r''}} exchanged. This expansion is the basis of [[interior multipole expansion]]. === Legendre polynomials in trigonometry === The trigonometric functions {{math\|cos ''nθ''}}, also denoted as the [[Chebyshev polynomials]] {{math\|''T<sub>n</sub>''(cos ''θ'') ≡ cos ''nθ''}}, can also be multipole expanded by the Legendre polynomials {{math\|''P<sub>n</sub>''(cos ''θ'')}}. The first several orders are as follows: :<math>\begin{align} T_0(\cos\theta)&=1 &&=P_0(\cos\theta),\\[4pt] T_1(\cos\theta)&=\cos \theta&&=P_1(\cos\theta),\\[4pt] T_2(\cos\theta)&=\cos 2\theta&&=\tfrac{1}{3}\bigl(4P_2(\cos\theta)-P_0(\cos\theta)\bigr),\\[4pt] T_3(\cos\theta)&=\cos 3\theta&&=\tfrac{1}{5}\bigl(8P_3(\cos\theta)-3P_1(\cos\theta)\bigr),\\[4pt] T_4(\cos\theta)&=\cos 4\theta&&=\tfrac{1}{105}\bigl(192P_4(\cos\theta)-80P_2(\cos\theta)-7P_0(\cos\theta)\bigr),\\[4pt] T_5(\cos\theta)&=\cos 5\theta&&=\tfrac{1}{63}\bigl(128P_5(\cos\theta)-56P_3(\cos\theta)-9P_1(\cos\theta)\bigr),\\[4pt] T_6(\cos\theta)&=\cos 6\theta&&=\tfrac{1}{1155}\bigl(2560P_6(\cos\theta)-1152P_4(\cos\theta)-220P_2(\cos\theta)-33P_0(\cos\theta)\bigr). \end{align}</math> Another property is the expression for {{math\|sin (''n'' + 1)''θ''}}, which is :<math>\frac{\sin (n+1)\theta}{\sin\theta}=\sum_{\ell=0}^n P_\ell(\cos\theta) P_{n-\ell}(\cos\theta).</math> === Legendre polynomials in recurrent neural networks === A [[recurrent neural network]] that contains a {{math\|''d''}}-dimensional memory vector, <math>\mathbf{m} \in \mathbb{R}^d</math>, can be optimized such that its neural activities obey the [[linear time-invariant system]] given by the following [[state-space representation]]: :<math>\theta \dot{\mathbf{m}}(t) = A\mathbf{m}(t) + Bu(t)</math> :<math>\begin{align} A &= \left[ a \right]_{ij} \in \mathbb{R}^{d \times d} \text{,} \quad && a_{ij} = \left(2i + 1\right) \begin{cases} -1 & i < j \\ (-1)^{i-j+1} & i \ge j \end{cases} \\ B &= \left[ b \right]_i \in \mathbb{R}^{d \times 1} \text{,} \quad && b_i = (2i + 1) (-1)^i \text{.} \end{align}</math> In this case, the sliding window of <math>u</math> across the past <math>\theta</math> units of time is [[Approximation_theory\|best approximated]] by a linear combination of the first <math>d</math> shifted Legendre polynomials, weighted together by the elements of <math>\mathbf{m}</math> at time <math>t</math>: :<math>u(t - \theta') \approx \sum_{\ell=0}^{d-1} \widetilde{P}_\ell \left(\frac{\theta'}{\theta} \right) \, m_{\ell}(t) \text{,} \quad 0 \le \theta' \le \theta \text{.}</math> When combined with [[deep learning]] methods, these networks can be trained to outperform [[long short-term memory]] units and related architectures, while using fewer computational resources.<ref>{{cite conference \|last=Voelker \|first=Aaron R. \|last2=Kajić \|first2=Ivana \|last3=Eliasmith \|first3=Chris \|title=Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks \|url=http://compneuro.uwaterloo.ca/files/publications/voelker.2019.lmu.pdf \|conference=Advances in Neural Information Processing Systems \|conferenceurl=https://neurips.cc \|year=2019 }}</ref> == Additional properties of Legendre polynomials == Legendre polynomials have definite parity. That is, they are [[Even and odd functions\|even or odd]],<ref>{{harvnb\|Arfken\|Weber\|2005\|loc=p.753}}</ref> according to :<math>P_n(-x) = (-1)^n P_n(x) \,.</math> Another useful property is :<math>\int_{-1}^1 P_n(x)\,dx = 0 \text{ for } n\ge1,</math> which follows from considering the orthogonality relation with <math>P_0(x) = 1</math>. It is convenient when a Legendre series <math>\sum_i a_i P_i</math> is used to approximate a function or experimental data: the ''average'' of the series over the interval {{math\|[−1, 1]}} is simply given by the leading expansion coefficient <math>a_0</math>. Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but the actual norm is not 1) by being scaled so that :<math>P_n(1) = 1 \,.</math> The derivative at the end point is given by :<math>P_n'(1) = \frac{n(n+1)}{2} \,. </math> The [[Askey–Gasper inequality]] for Legendre polynomials reads :<math>\sum_{j=0}^n P_j(x) \ge 0 \,\quad \text{for } x\ge -1 \,.</math> The Legendre polynomials of a [[scalar product]] of [[unit vectors]] can be expanded with [[spherical harmonics]] using :<math>P_\ell \left(r \cdot r'\right) = \frac{4\pi}{2\ell + 1} \sum_{m=-\ell}^\ell Y_{\ell m}(\theta,\varphi) Y_{\ell m}^(\theta',\varphi')\,,</math> where the unit vectors {{math\|''r''}} and {{math\|''r''′}} have [[spherical coordinates]] {{math\|(''θ'',''φ'')}} and {{math\|(''θ''′,''φ''′)}}, respectively. ===Recurrence relations=== As discussed above, the Legendre polynomials obey the three-term recurrence relation known as Bonnet’s recursion formula :<math> (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)</math> and :<math> \frac{x^2-1}{n} \frac{d}{dx} P_n(x) = xP_n(x) - P_{n-1}(x) \,</math> or, with the alternative expression, which also holds at the endpoints :<math> \frac{d}{dx} P_{n+1}(x) = (n+1)P_n(x) + x \frac{d}{dx}P_{n}(x) \,.</math> Useful for the integration of Legendre polynomials is :<math>(2n+1) P_n(x) = \frac{d}{dx} \bigl( P_{n+1}(x) - P_{n-1}(x) \bigr) \,.</math> From the above one can see also that :<math>\frac{d}{dx} P_{n+1}(x) = (2n+1) P_n(x) + \bigl(2(n-2)+1\bigr) P_{n-2}(x) + \bigl(2(n-4)+1\bigr) P_{n-4}(x) + \cdots</math> or equivalently :<math>\frac{d}{dx} P_{n+1}(x) = \frac{2 P_n(x)}{\left\\| P_n \right\\|^2} + \frac{2 P_{n-2}(x)}{\left\\| P_{n-2} \right\\|^2} + \cdots</math> where {{math\|{{norm\|''P<sub>n</sub>''}}}} is the norm over the interval {{math\|−1 ≤ ''x'' ≤ 1}} :<math>\\| P_n \\| = \sqrt{\int_{-1}^1 \bigl(P_n(x)\bigr)^2 \,dx} = \sqrt{\frac{2}{2 n + 1}} \,.</math> ===Asymptotes=== Asymptotically for <math>\ell \to \infty</math><ref>{{Cite book\|title=Orthogonal polynomials\|last=1895–1985.\|first=Szegő, Gábor\|date=1975\|publisher=American Mathematical Society\|isbn=0821810235\|edition= 4th\|location=Providence\|pages=194 (Theorem 8.21.2)\|oclc=1683237}}</ref> :<math>\begin{align} P_\ell (\cos \theta) &= \sqrt{\frac{\theta}{\sin \theta}} \, J_0((\ell+1/2)\theta) + \mathcal{O}\left(\ell^{-1}\right) \\ &= \frac{2}{\sqrt{2\pi \ell\sin\theta}}\cos\left(\left(\ell + \tfrac12\right)\theta - \frac{\pi}{4}\right) + \mathcal{O}\left(\ell^{-3/2}\right), \quad \theta \in (0,\pi), \end{align}</math> and for arguments of magnitude greater than 1 :<math>\begin{align} P_\ell \left(\frac{1}{\sqrt{1-e^2}}\right) &= I_0(\ell e) + \mathcal{O}\left(\ell^{-1}\right) \\ &= \frac{1}{\sqrt{2\pi\ell e}} \frac{(1+e)^\frac{\ell+1}{2}}{(1-e)^\frac{\ell}{2}} + \mathcal{O}\left(\ell^{-1}\right)\,, \end{align}</math> where {{math\|''J''<sub>0</sub>}} and {{math\|''I''<sub>0</sub>}} are [[Bessel functions]]. === Zeros === All <math> n</math> zeros of <math>P_n(x)</math> are real, distinct from each other, and lie in the interval <math>(-1,1)</math>. Further, if we regard them as dividing the interval <math>[-1,1]</math> into <math> n+1 </math> subintervals, each subinterval will contain exactly one zero of <math>P_{n+1}</math>. This is known as the interlacing property. Because of the parity property it is evident that if <math>x_k</math> is a zero of <math>P_n(x)</math>, so is <math>-x_k</math>. These zeros play an important role in numerical integration based on [[Gaussian quadrature]]. The specific quadrature based on the <math>P_n</math>'s is known as Gauss-Legendre quadrature. From this property and the facts that <math> P_n(\pm 1) \ne 0 </math>, it follows that <math> P_n(x) </math> has <math> n-1 </math> local minima and maxima in <math> (-1,1) </math>. Equivalently, <math> dP_n(x)/dx </math> has <math> n -1 </math> zeros in <math> (-1,1) </math>. === Pointwise evaluations=== The parity and normalization implicate the values at the boundaries <math> x=\pm 1 </math> to be :<math> P_n(1) = 1 \,, \quad P_n(-1) = \begin{cases} 1 & \text{for} \quad n = 2m \\ -1 & \text{for} \quad n = 2m+1 \,. \end{cases} </math> At the origin <math> x=0 </math> one can show that the values are given by :<math> P_n(0) = \begin{cases} \frac{(-1)^{m}}{4^m} \tbinom{2m}{m} = \frac{(-1)^{m}}{2^{2m}} \frac{(2m)!}{\left(m!\right)^2} & \text{for} \quad n = 2m \\ 0 & \text{for} \quad n = 2m+1 \,. \end{cases} </math> == Legendre polynomials with transformed argument == === Shifted Legendre polynomials === The '''shifted Legendre polynomials''' are defined as :<math>\widetilde{P}_n(x) = P_n(2x-1) \,</math>. Here the "shifting" function {{math\|''x'' ↦ 2''x'' − 1}} is an [[affine transformation]] that [[bijection\|bijectively maps]] the interval {{math\|[0,1]}} to the interval {{math\|[−1,1]}}, implying that the polynomials {{math\|''P̃<sub>n</sub>''(''x'')}} are orthogonal on {{math\|[0,1]}}: :<math>\int_0^1 \widetilde{P}_m(x) \widetilde{P}_n(x)\,dx = \frac{1}{2n + 1} \delta_{mn} \,.</math> An explicit expression for the shifted Legendre polynomials is given by :<math>\widetilde{P}_n(x) = (-1)^n \sum_{k=0}^n \binom{n}{k} \binom{n+k}{k} (-x)^k \,.</math> The analogue of [[Rodrigues' formula]] for the shifted Legendre polynomials is :<math>\widetilde{P}_n(x) = \frac{1}{n!} \frac{d^n}{dx^n} \left(x^2 -x \right)^n \,.</math> The first few shifted Legendre polynomials are: :<math>\begin{array}{r\|r} n & \widetilde{P}_n(x) \\ \hline 0 & 1 \\ 1 & 2x-1 \\ 2 & 6x^2-6x+1 \\ 3 & 20x^3-30x^2+12x-1 \\ 4 & 70x^4-140x^3+90x^2-20x+1 \\ 5 & 252x^5 -630x^4 +560x^3 - 210 x^2 + 30 x -1 \end{array}</math> === Legendre rational functions === {{main\|Legendre rational functions}} The [[Legendre rational functions]] are a sequence of [[orthogonal functions]] on [0, ∞). They are obtained by composing the [[Cayley transform]] with Legendre polynomials. A rational Legendre function of degree ''n'' is defined as: :<math>R_n(x) = \frac{\sqrt{2}}{x+1}\,P_n\left(\frac{x-1}{x+1}\right)\,.</math> They are [[eigenfunction]]s of the singular [[Sturm–Liouville problem]]: :<math>(x+1)\partial_x(x\partial_x((x+1)v(x)))+\lambda v(x)=0</math> with eigenvalues :<math>\lambda_n=n(n+1)\,.</math> ==See also== {{div col\|colwidth=20em}} [[Gaussian quadrature]] * [[Gegenbauer polynomials]] * [[Turán's inequalities]] * [[Legendre wavelet]] * [[Jacobi polynomials]] * [[Romanovski polynomials]] {{div col end}} == Notes == {{reflist\|30em}} == References == {{refbegin\|30em}} * {{Abramowitz_Stegun_ref2\|8\|332\|22\|773}} * {{cite book\|first1=George B.\|last1=Arfken\|authorlink1=George B. Arfken\|first2=Hans J.\|last2=Weber\|year=2005\|title=Mathematical Methods for Physicists\|publisher=Elsevier Academic Press\|isbn=0-12-059876-0}} * {{cite book\|last=Bayin\|first=S. S.\|year=2006\|title=Mathematical Methods in Science and Engineering\|publisher=Wiley\|isbn= 978-0-470-04142-0\|at=ch. 2}} * {{cite book\|last=Belousov\|first=S. L.\|year=1962\|title=Tables of Normalized Associated Legendre Polynomials\|series=Mathematical Tables\|volume=18\|publisher=Pergamon Press\|isbn=978-0-08-009723-7}} * {{cite book\|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 \|location=New York, NY\|isbn=978-0-471-50447-4}} {{dlmf\|first=T. M. \|last=Dunster\|id=14\|title=Legendre and Related Functions}} {{cite book\| first=Refaat \| last=El Attar \| title= Legendre Polynomials and Functions \| publisher= CreateSpace \| year=2009 \| isbn = 978-1-4414-9012-4}} {{dlmf\|id=18\|title=Orthogonal Polynomials\|first=Tom H. \|last=Koornwinder\|authorlink=Tom H. Koornwinder\|first2=Roderick S. C.\|last2= Wong\|first3=Roelof \|last3=Koekoek\|first4=René F. \|last4=Swarttouw}} {{refend}} ==External links== {{Commons category\|Legendre polynomials}} [http://www.physics.drexel.edu/~tim/open/hydrofin A quick informal derivation of the Legendre polynomial in the context of the quantum mechanics of hydrogen] {{springer\|title=Legendre polynomials\|id=p/l058050}} [http://mathworld.wolfram.com/LegendrePolynomial.html Wolfram MathWorld entry on Legendre polynomials] [https://web.archive.org/web/20060427014500/http://www.du.edu/~jcalvert/math/legendre.htm Dr James B. Calvert's article on Legendre polynomials from his personal collection of mathematics] [https://web.archive.org/web/20181009221546/http://www.morehouse.edu/facstaff/cmoore/Legendre%20Polynomials.htm The Legendre Polynomials by Carlyle E. Moore] *[http://hyperphysics.phy-astr.gsu.edu/hbase/math/legend.html Legendre Polynomials from Hyperphysics] {{Authority control}} {{DEFAULTSORT:Legendre Polynomials}} [[Category:Special hypergeometric functions]] [[Category:Orthogonal polynomials]] [[Category:Polynomials]]
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