🔎 Legendre Polynomial – Definition and Explanations
Introduction
Legendre polynomials
The Legendre polynomials are options of Legendre’s differential equation, and represent the best instance of a sequence of orthogonal polynomials (In arithmetic, a sequence of orthogonal polynomials is an infinite sequence of…).
Legendre’s equation
We name Legendre’s equation the equation:
We thus outline the Legendre polynomial (The Legendre polynomials are options of the differential equation of…) P_{not} (to pay every little thing (The entire understood as the entire of what exists is usually interpreted because the world or…) entire pure (In arithmetic, a pure quantity is a optimistic (or zero) quantity that principally permits…) not):
So we have now the place means the polynomial (A polynomial, in arithmetic, is the linear mixture of the merchandise of…) by Jacobi of indexes not related to the parameters α and β.
Some polynomials
The primary polynomials are:
Different definitions
Bonnet recurrence method
and for all not>0
Components of Rodrigues (Rodrigues is the smallest of the three islands within the Mascarene archipelago.)
We outline the polynomial P_{not} (for all pure integers not) by :
Definition (A definition is a discourse that claims what a factor is or what a reputation means. Therefore the…) analytic
We will additionally outline this sequence of polynomials by its producing operate:
The theorem (A theorem is a proposition that may be mathematically demonstrated, that’s, a…) residues then provides:
the place the define (COmet Nucleus TOUR (CONTOUR) is a NASA house probe that’s a part of the Program…) surrounds the origin and is caught within the that means (SENS (Methods for Engineered Negligible Senescence) is a scientific undertaking which goals to…) trigonometric.
Definitions as a sum
We outline this polynomial in two methods within the type of a sum:
(we will deduce )
Sequence decomposition of Legendre polynomials
Decomposition of a holomorphic operate
Any operate f, holomorphic inside an ellipse of foci 1 and +1, will be written as a sequence which converges uniformly contained in the ellipse:
with
Decomposition of a Lipschitz operate
Comment the quotient of the polynomial P_{not} the couple mentioned requirements (A norm, from the Latin norma (“sq., ruler”) designates a…).
That’s F steady software on [1,1]. For all pure not posing
So subsequent is of sq. (A sq. is a daily polygon with 4 sides. Which means that its…) summable, and makes it potential to elucidate the orthogonal projection of F on :
We even have:
 with
Suppose additional that f is a Lipschitz operate. We then have the extra property:^{[réf. souhaitée]}
autrement dit, l’égalité
est vraie non seulement au sens L^{2} mais au sens de la convergence simple (La convergence simple ou ponctuelle est un critère de convergence dans un espace fonctionnel,…) sur ]1.1[.
Propriétés
Degré (Le mot degré a plusieurs significations, il est notamment employé dans les domaines…)
Le polynôme P_{n} est de degré n.
Parité
Les polynômes de Legendre suivent la parité de n. On peut exprimer cette propriété par :
(en particulier, P_{n}( − 1) = ( − 1)^{n} et P_{2n + 1}(0) = 0).
Orthogonalité (En mathématiques, l’orthogonalité est un concept d’algèbre linéaire…)
Les polynômes orthogonaux les plus simples sont les polynômes de Legendre pour lesquels l’intervalle d’orthogonalité est [−1, 1] and the operate weight (Weight is the drive of gravity, of gravitational and inertial origin, exerted by the…) is solely the fixed operate of worth 1: these polynomials are orthogonal with respect to the scalar product (In vector geometry, the dot product is an algebraic operation…) set to by the connection:

The very definition of P_{not} reveals that it’s a clear vector (In arithmetic, the idea of eigenvector is an algebraic notion making use of to a…) for the personal worth (In arithmetic, the idea of eigenvector is an algebraic notion making use of to a…) n(n+1) of the endomorphism:
However this endomorphism is symmetric for the product scalar (A real scalar is a quantity that’s unbiased of the selection of base chosen to precise the…) above, since an integration by elements reveals that
 :Requirements
The sq. of the norm, in L^{2}([1,1]), is
Certainly, for all n>1, we will set up the relation
from which we deduce (utilizing that for all ok, P‘_{ok −1} is of diploma ok2k so is orthogonal to P_{ok}and performing an integration by elements):
 P_{not}P_{not + 1} is odd and for all ok, P_{ok}(1) = 1we thus arrive at (2not + 1)   P_{not}   ^{2} = 2.
 P_{not}P_{not + 1} is odd and for all ok, P_{ok}(1) = 1we thus arrive at (2not + 1)   P_{not}   ^{2} = 2.
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 :Requirements