Legendre polynomials
E695818
Legendre polynomials are a sequence of orthogonal polynomials that arise in solving Legendre’s differential equation, playing a central role in mathematical physics, especially in problems with spherical symmetry such as potential theory and quantum mechanics.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf | orthogonal polynomials ⓘ |
| are | polynomials with real coefficients ⓘ |
| areDefinedBy | Rodrigues formula P_n(x) = 1/(2^n n!) d^n/dx^n (x^2-1)^n ⓘ |
| areEigenfunctionsOf | Legendre differential operator NERFINISHED ⓘ |
| areOrthogonalOn | interval [-1,1] ⓘ |
| areOrthogonalWithRespectTo | weight function w(x)=1 ⓘ |
| areRelatedTo |
associated Legendre functions
ⓘ
spherical harmonics Y_l^m(θ,φ) ⓘ |
| areSpecialCaseOf | Jacobi polynomials P_n^{(0,0)}(x) ⓘ |
| areUsedIn |
Gaussian quadrature (Gauss–Legendre quadrature)
NERFINISHED
ⓘ
angular part of Schrödinger equation for central potentials ⓘ approximation of random fields on the sphere ⓘ approximation theory ⓘ cosmology for CMB power spectrum expansions ⓘ electrostatics with spherical symmetry ⓘ expansion of 1/|r-r'| in spherical coordinates ⓘ geodesy ⓘ gravity field modeling ⓘ multipole expansions ⓘ numerical analysis ⓘ partial wave analysis in quantum scattering ⓘ potential theory ⓘ quantum mechanics ⓘ scattering theory ⓘ solution of Laplace equation in spherical coordinates ⓘ solution of Poisson equation with spherical symmetry ⓘ solution of boundary value problems with axial symmetry ⓘ spectral methods for differential equations ⓘ spherical harmonics expansions ⓘ |
| areUsedToConstruct | Legendre series expansions ⓘ |
| areUsedToExpand | functions on [-1,1] in orthogonal series ⓘ |
| firstPolynomial | P_0(x) = 1 ⓘ |
| form | complete set in L2([-1,1]) with weight 1 ⓘ |
| fourthPolynomial | P_3(x) = (1/2)(5x^3-3x) ⓘ |
| haveDegree | n for P_n(x) ⓘ |
| haveDomain | real variable x in [-1,1] ⓘ |
| haveGeneratingFunction | 1/sqrt(1-2xt+t^2) = sum_{n=0}^∞ P_n(x) t^n ⓘ |
| haveNormalization | P_n(1) = 1 ⓘ |
| haveParityProperty | P_n(-x) = (-1)^n P_n(x) ⓘ |
| haveValueAtMinusOne | P_n(-1) = (-1)^n ⓘ |
| haveZeros | n distinct simple zeros in (-1,1) for P_n(x) ⓘ |
| namedAfter | Adrien-Marie Legendre NERFINISHED ⓘ |
| satisfy | (1-x^2)y'' - 2xy' + n(n+1)y = 0 ⓘ |
| satisfyDifferentialEquation | Legendre differential equation NERFINISHED ⓘ |
| satisfyOrthogonalityRelation | ∫_{-1}^1 P_m(x)P_n(x) dx = 2/(2n+1) δ_{mn} ⓘ |
| satisfyRecurrence | (n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x) ⓘ |
| secondPolynomial | P_1(x) = x ⓘ |
| thirdPolynomial | P_2(x) = (1/2)(3x^2-1) ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.