This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Legendre Polynomials http://plasma.physics.swarthmore.edu/Physics112/old_assignments.html Legendre polynomials are a complete set of orthogonal functions in spherical coordinates. This means that any function can be represented as a sum of Legendre polynomials, and all the Legendre polynomials are linearly independent. They are solutions for the Legendre equation: ) 1 ( 2 ) 1 ( 2 = + + ′- ′ ′- y l l y x y x Legendre Polynomials are defined by the Rodrigues formula: ( 29 l l l l x dx d l x P 1 ! 2 1 ) ( 2- = or by: - =---- =- = ∑ l l l l N x n l n l n n l x P n l N n l n l odd for 2 / ) 1 ( even for 2 / where )! 2 ( )! ( ! 2 )! 2 2 ( ) 1 ( ) ( 2 Since Legendre polynomials are orthogonal, they follow the same rules as any other set of orthogonal functions: l l dx x P x P l l ′ ≠ = ∫- ′ if ) ( ) ( 1 1 In this case, if ' l l = then 1 2 2 ) ( ) ( ) ( ) ( 1 1 1 1 + = = ∫ ∫-- ′ l dx x P x P dx x P x P l l l l Even numbered Legendre polynomials only have even powers in their expansion, and...
View Full Document