# legendre - Lemma used to prove Rodrigues’s formula 1...

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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lemma used to prove Rodrigues’s formula 1 * November 7, 2007 We showed that P n ( x ), the Legendre polynomial of degree n , satisfies Rodrigues’s formula: P n ( x ) = 1 2 n n ! d n dx n ( ( x 2- 1) n ) , n = 0 , 1 , 2 , . . . , provided the following Lemma is proved. Lemma The polynomial p n ( x ) = d n dx n ( ( x 2- 1) n ) ( n = 1 , 2 , . . . ) is orthogonal to any polynomial of degree less than n . Proof. Let q ( x ) be a polynomial of degree less than n and consider. integraldisplay 1 − 1 d n dx n ( ( x 2- 1) n ) q ( x ) dx. We need to show that this integral is 0 for all n = 1 , 2 , . . . . Integrating by parts once yields: d n − 1 dx n − 1 ( ( x 2- 1) n ) q ( x ) | x =1 x = − 1- integraldisplay 1 − 1 d n − 1 dx n − 1 ( ( x 2- 1) n ) q ′ ( x ) dx. Repeating this several times (in total n- 1 times) we get that this integral equals: d n − 1 dx n − 1 ( ( x 2- 1) n ) q ( x ) | x =1 x = − 1- d n − 2 dx n − 2 ( ( x 2- 1) n ) q ′ ( x ) | x =1 x = − 1 + d...
View Full Document

## This note was uploaded on 09/12/2011 for the course MAP 4305 taught by Professor Deleenheer during the Summer '06 term at University of Florida.

### Page1 / 2

legendre - Lemma used to prove Rodrigues’s formula 1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online