What is meant by Legendre polynomial?

What is meant by Legendre polynomial?

In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications.

What is the purpose of using 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.

How do you use Legendre in Matlab?

Use the legendre function to operate on a vector and then examine the format of the output. Calculate the second-degree Legendre function values of a vector. The format of the output is such that: Each row contains the function value for different values of m (the order of the associated Legendre function)

Where do Legendre polynomials come from?

The equation takes its name from Adrien Marie Legendre (1752-1833), a French mathematician who became a professor in Paris in 1775. He made important contributions to special functions, elliptic integrals, number theory, and the calculus of variations. (Kreyszig).

What is orthogonality of Legendre polynomial?

Abstract We give a remarkable second othogonality property of the classical Legendre polynomials on the real interval [−1, 1]: Polynomials up to de- gree n from this family are mutually orthogonal under the arcsine measure weighted by the degree-n normalized Christoffel function.

Are Legendre polynomials linearly independent?

Any polynomial of degree m can be represented as a linear combination of Legendre polynomials of degree at most m. show that the legendre polynomials of degree ≤ n, are linearly independent, and thus form a basis for all polynomials of degree ≤ n.

How do you write a Legendre polynomial in Matlab?

Legendre Polynomial P ( n , x ) = 2 n − 1 n x P ( n − 1 , x ) − n − 1 n P ( n − 2 , x ) , where P ( 0 , x ) = 1 P ( 1 , x ) = x . ∫ x = − 1 x = 1 P ( n , x ) P ( m , x ) d x = { 0 if n ≠ m 1 n + 1 / 2 if n = m .

How do you do factorial in Matlab?

Description. f = factorial( n ) returns the product of all positive integers less than or equal to n , where n is a nonnegative integer value. If n is an array, then f contains the factorial of each value of n . The data type and size of f is the same as that of n .

Who invented Legendre polynomial?

What is the degree of Legendre equation?

Legendre’s differential equation has the form (1 − x2)y − 2xy + l(l + 1)y = 0, (2) where the parameter l, which is a real number, (we take l = 0,1,2,ททท), is called the degree.

Is Legendre differential equation linear?

Legendre’s differential equation This is a second order linear equation with three regular singular points (at 1, −1, and ∞).

How to find Legendre polynomials of degrees 1 and 2?

Find Legendre Polynomial with Vector and Matrix Inputs. Find the Legendre polynomials of degrees 1 and 2 by setting n = [1 2]. syms x legendreP([1 2],x) ans = [ x, (3*x^2)/2 – 1/2] legendreP acts element-wise on n to return a vector with two elements.

Why are associated Legendre polynomials called Legendre functions?

Dong and Lemus (2002) generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials. These functions may actually be defined for general complex parameters and argument: They are called the Legendre functions when defined in this more general way.

How are Legendre polynomials related to hypergeometric series?

The Legendre polynomials are closely related to hypergeometric series. In the form of spherical harmonics, they express the symmetry of the two-sphere under the action of the Lie group SO (3).

Where do Legendre polynomials occur in Laplace’s equation?

Legendre polynomials occur in the solution of Laplace’s equation of the static potential, ∇2 Φ (x) = 0, in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle ).

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