What is the 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.

What is the Legendre differential 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.

Are Legendre polynomials even?

One of the varieties of special functions which are encountered in the solution of physical problems is the class of functions called Legendre polynomials. They are solutions to a very important differential equation, the Legendre equation: The polynomials are either even or odd functions of x for even or odd orders n.

Why are Legendre polynomials useful?

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.

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.

What is Legendre differential equation?

Since the Legendre differential equation is a second-order ordinary differential equation, it has two linearly independent solutions. A solution which is regular at finite points is called a Legendre function of the first kind, while a solution which is singular at is called a Legendre function of the second kind.

What are the coefficients of the Legendre polynomials?

The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of t {\\displaystyle t} of the generating function. 1 1 − 2 x t + t 2 = ∑ n = 0 ∞ P n ( x ) t n .

How are Legendre polynomials generated in orthonormalization?

The Legendre polynomials can also be generated using Gram-Schmidt orthonormalization in the open interval with the weighting function 1. Normalizing so that gives the expected Legendre polynomials. The “shifted” Legendre polynomials are a set of functions analogous to the Legendre polynomials, but defined on the interval (0, 1).

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 ).

How are Legendre polynomials used in Newtonian potential expansion?

Applications of Legendre polynomials Expanding a 1/r potential 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.