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- Legendre_polynomials abstract "In mathematics, Legendre functions are solutions to Legendre's differential equation:They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates.The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at x = ±1 so, in general, a series solution about the origin will only converge for |x| < 1. When n is an integer, the solution Pn(x) that is regular at x = 1 is also regular at x = −1, and the series for this solution terminates (i.e. it is a polynomial).These solutions for n = 0, 1, 2, ... (with the normalization Pn(1) = 1) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial Pn(x) is an nth-degree polynomial. It may be expressed using Rodrigues' formula:That these polynomials satisfy the Legendre differential equation (1) follows by differentiating (n+1) times both sides of the identityand employing the general Leibniz rule for repeated differentiation. The Pn can also be defined as the coefficients in a Taylor series expansion:In physics, this ordinary generating function is the basis for multipole expansions.".
- Legendre_polynomials thumbnail Legendrepolynomials6.svg?width=300.
- Legendre_polynomials wikiPageExternalLink legend.html.
- Legendre_polynomials wikiPageExternalLink LegendrePolyMod.html.
- Legendre_polynomials wikiPageExternalLink LegendrePolynomial.html.
- Legendre_polynomials wikiPageExternalLink legendre.htm.
- Legendre_polynomials wikiPageExternalLink Legendre%20Polynomials.htm.
- Legendre_polynomials wikiPageExternalLink hydrofin.
- Legendre_polynomials wikiPageID "100349".
- Legendre_polynomials wikiPageRevisionID "605942608".
- Legendre_polynomials authorlink "Tom H. Koornwinder".
- Legendre_polynomials b "n".
- Legendre_polynomials first "René F.".
- Legendre_polynomials first "Roderick S. C.".
- Legendre_polynomials first "Roelof".
- Legendre_polynomials first "T. M.".
- Legendre_polynomials first "Tom H.".
- Legendre_polynomials hasPhotoCollection Legendre_polynomials.
- Legendre_polynomials id "14".
- Legendre_polynomials id "18".
- Legendre_polynomials id "p/l058050".
- Legendre_polynomials last "Dunster".
- Legendre_polynomials last "Koekoek".
- Legendre_polynomials last "Koornwinder".
- Legendre_polynomials last "Swarttouw".
- Legendre_polynomials last "Wong".
- Legendre_polynomials p "0".
- Legendre_polynomials title "Legendre and Related Functions".
- Legendre_polynomials title "Legendre polynomials".
- Legendre_polynomials title "Orthogonal Polynomials".
- Legendre_polynomials subject Category:Orthogonal_polynomials.
- Legendre_polynomials subject Category:Polynomials.
- Legendre_polynomials subject Category:Special_hypergeometric_functions.
- Legendre_polynomials type Abstraction100002137.
- Legendre_polynomials type Function113783816.
- Legendre_polynomials type MathematicalRelation113783581.
- Legendre_polynomials type OrthogonalPolynomials.
- Legendre_polynomials type Polynomial105861855.
- Legendre_polynomials type Polynomials.
- Legendre_polynomials type Relation100031921.
- Legendre_polynomials type SpecialHypergeometricFunctions.
- Legendre_polynomials comment "In mathematics, Legendre functions are solutions to Legendre's differential equation:They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates.The Legendre differential equation may be solved using the standard power series method.".
- Legendre_polynomials label "Legendre polynomials".
- Legendre_polynomials label "Legendre-Polynom".
- Legendre_polynomials label "Legendre-polynoom".
- Legendre_polynomials label "Polinomio di Legendre".
- Legendre_polynomials label "Polinomios de Legendre".
- Legendre_polynomials label "Polinômios de Legendre".
- Legendre_polynomials label "Polynôme de Legendre".
- Legendre_polynomials label "Wielomiany Legendre'a".
- Legendre_polynomials label "Многочлены Лежандра".
- Legendre_polynomials label "ルジャンドル多項式".
- Legendre_polynomials label "勒让德多项式".
- Legendre_polynomials sameAs Legendre-Polynom.
- Legendre_polynomials sameAs Polinomios_de_Legendre.
- Legendre_polynomials sameAs Polynôme_de_Legendre.
- Legendre_polynomials sameAs Polinomio_di_Legendre.
- Legendre_polynomials sameAs ルジャンドル多項式.
- Legendre_polynomials sameAs 르장드르_다항식.
- Legendre_polynomials sameAs Legendre-polynoom.
- Legendre_polynomials sameAs Wielomiany_Legendre'a.
- Legendre_polynomials sameAs Polinômios_de_Legendre.
- Legendre_polynomials sameAs m.0pkk3.
- Legendre_polynomials sameAs Q215405.
- Legendre_polynomials sameAs Q215405.
- Legendre_polynomials sameAs Legendre_polynomials.
- Legendre_polynomials wasDerivedFrom Legendre_polynomials?oldid=605942608.
- Legendre_polynomials depiction Legendrepolynomials6.svg.
- Legendre_polynomials isPrimaryTopicOf Legendre_polynomials.