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In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide 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 ...
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation () + [(+)] =,or equivalently [() ()] + [(+)] =,where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively.
The general Legendre equation reads ″ ′ + [(+)] =, 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 P n; and when λ is an integer (denoted n), and μ = m is also an integer with | m | < n are the associated Legendre ...
For integrating f over [,] with Gauss–Legendre quadrature, the associated orthogonal polynomials are Legendre polynomials, denoted by P n (x). With the n-th polynomial normalized so that P n (1) = 1, the i-th Gauss node, x i, is the i-th root of P n and the weights are given by the formula [5]
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was independently introduced by Olinde Rodrigues ( 1816 ), Sir James Ivory ( 1824 ) and Carl Gustav Jacobi ( 1827 ).
Plot of the Legendre rational functions for n=0,1,2 and 3 for x between 0.01 and 100. In mathematics , the Legendre rational functions are a sequence of orthogonal functions on [0, ∞) . They are obtained by composing the Cayley transform with Legendre polynomials .
Adrien-Marie Legendre (/ l ə ˈ ʒ ɑː n d ər,-ˈ ʒ ɑː n d /; [3] French: [adʁiɛ̃ maʁi ləʒɑ̃dʁ]; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him.
In mathematics, the continuous q-Legendre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Koekoek, Lesky & Swarttouw (2010) give a detailed list of their properties.