# What is the orthogonality relations of the associated Legendre polynomial?

## What is the orthogonality relations of the associated Legendre polynomial?

To demonstrate orthogonality of the associated Legendre polynomials, we use a result from the theory of orthogonal polynomials. Namely, a Legendre polynomial of order l is orthogonal to any polynomial Πp of order p lower than l. In bra-ket notation: ⟨Πp|Pl⟩=0 if O(Πp)≡p

What is generating function in Legendre polynomials?

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 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). The term is also used to describe similar formulas for other orthogonal polynomials.

### Which is an orthogonal function of the Legendre polynomial?

Legendre Polynomials are a set of orthogonal functions on ( 1; 1), that is Z 1 21 P l(x)P m(x)dx = ˆ 0 if l 6= m 2 l+1if l = m (1) Orthogonality of the Legendre Polynomials: Cont’d

Is the Legendre polynomial suitable for symbolic manipulation?

Copy to clipboard. gives the Legendre polynomial . Copy to clipboard. gives the associated Legendre polynomial . Mathematical function, suitable for both symbolic and numerical manipulation. Explicit formulas are given for integers n and m. The Legendre polynomials satisfy the differential equation .

What are the types of Legendre functions in Wolfram?

LegendreP [ n, m, a, z] gives Legendre functions of type a. The default is type 1. The symbolic form of type 1 involves , of type 2 involves , and of type 3 involves . Type 1 is defined only for within the unit circle in the complex plane.