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.

Can zero be a polynomial?

A polynomial having value zero (0) is known as zero polynomial. Actually, the term 0 is itself zero polynomial. It is a constant polynomial whose all the coefficients are equal to 0.

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.

Why are orthogonal polynomials important?

Just as Fourier series provide a convenient method of expanding a periodic function in a series of linearly independent terms, orthogonal polynomials provide a natural way to solve, expand, and interpret solutions to many types of important differential equations. …

Is number 8 a polynomial?

Polynomials with 0 degrees are called zero polynomials. For example, 3, 5, or 8. Polynomials with 1 as the degree of the polynomial are called linear polynomials. For example, x+y−4.

Are there any polynomials similar to the Legendre polynomials?

The “shifted” Legendre polynomials are a set of functions analogous to the Legendre polynomials, but defined on the interval (0, 1). They obey the Orthogonality relationship.

Is the Legendre polynomial part of the spherical harmonics?

Associated polynomials are sometimes called Ferrers’ Functions (Sansone 1991, p. 246). If , they reduce to the unassociated Polynomials. The associated Legendre functions are part of the Spherical Harmonics, which are the solution of Laplace’s Equation in Spherical Coordinates.

Which is the solution to the Associated Legendre differential equation?

The associated Legendre polynomials are solutions to the associated Legendre Differential Equation , where is a Positive Integer and ., . They can be given in terms of the unassociated polynomials by

When did Adrien Marie Legendre discover the Legendre polynomials?

The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre as the coefficients in the expansion of the Newtonian potential where r and r′ are the lengths of the vectors x and x′ respectively and γ is the angle between those two vectors. The series converges when r > r′.