What is shifted Legendre polynomial?
Shifted Legendre Polynomials algorithm used to the. dynamic analysis of viscoelastic pipes conveying fluid. with the variable fractional order model $
What is the mathematical form of Legendre polynomial?
y ( x ) = a 1 x + a 3 x 3 + ⋯ + a k x k . In these cases, the solution is called the Legendre polynomial of degree k. Θ ′ = d Θ d θ = d Θ d x d x d θ = − sin θ d Θ d x .
What is Rodrigues formula for Legendre polynomial?
In mathematics, Rodrigues’ formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by Olinde Rodrigues (1816), Sir James Ivory (1824) and Carl Gustav Jacobi (1827).
What is nth Legendre polynomial?
They are solutions to a very important differential equation, the Legendre equation: The polynomials may be denoted by Pn(x) , called the Legendre polynomial of order n. The polynomials are either even or odd functions of x for even or odd orders n. The first few polynomials are shown below.
What is the orthogonal property of Legendre’s polynomial?
Abstract We give a remarkable additional othogonality property of the classical Legendre polynomials on the real interval [−1, 1]: polynomials up to degree n from this family are mutually orthogonal under the arcsine measure weighted by the nor- malized degree-n Christoffel function.
Why do we use Legendre polynomials?
For example, Legendre and Associate Legendre polynomials are widely used in the determination of wave functions of electrons in the orbits of an atom [3], [4] and in the determination of potential functions in the spherically symmetric geometry [5], etc.
Why Legendre polynomial is important?
What is generating function of Legendre polynomial?
The Legendre polynomials can be alternatively given by the generating function ( 1 − 2 x z + z 2 ) − 1 / 2 = ∑ n = 0 ∞ P n ( x ) z n , but there are other generating functions.
Why do we use Rodrigues formula?
In the theory of three-dimensional rotation, Rodrigues’ rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation.
What is the difference between Legendre function and Legendre polynomial?
Definition via differential equation . A two-parameter generalization of (Eq. 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.
How do Legendre polynomials relate to multipole expansion?
Legendre polynomials in multipole expansions. Conversely, if the radius r of the observation point P is smaller than a, the potential may still be expanded in the Legendre polynomials as above, but with a and r exchanged. This expansion is the basis of interior multipole expansion .
How do the Legendre polynomials satisfy the differential equation?
The Legendre polynomials satisfy the second-order differential equation (1 − x2)P ″ ℓ − 2xP ′ ℓ + ℓ(ℓ + 1)Pℓ = 0. as you can quickly verify by expanding the first term on the left-hand side of the equation. Either form of the differential equation is known as Legendre’s equation.
Are Legendre polynomials orthogonal over the half line?
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 polynomial systems. The other two are the Laguerre polynomials, which are orthogonal over the half line
What are explicit representations of the Legendre polynomials?
Among these are explicit representations such as where the last, which is also immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the generalized form of the binomial coefficient .