{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Legendre_polynomials_notes

# Legendre_polynomials_notes - USEFUL FACTS AND FORMULAE FOR...

This preview shows pages 1–2. Sign up to view the full content.

USEFUL FACTS AND FORMULAE FOR LEGENDRE POLYNOMIALS 1. The differential equation The Legendre polynomials P l ( x ), l = 0, 1, ... are a set of orthogonal polynomials over the range x [-1,1]. P l ( x ) is of degree l and is a solution of the differential equation ( ) ( ) 2 2 2 1 2 1 0, d y dy x x l l y dx dx - - + + = (1.1) which can also be written as ( ) ( ) 2 1 1 0. d dy x l l y dx dx - + + = (1.2) 2. Normalization The 'normalization' for the Legendre polynomials is that P l (1) = 1. The normalization constants are ( ) 1 2 1 2 . 2 1 l P x dx l - = + (2.1) 3. Recurrence relations The Legendre polynomials satisfy a number of recurrence relations, including ( ) ( ) 1 1 1 2 1 0, l l l l P l xP lP + - + - + + = (3.1) and ( ) 1 1 0. l l l dP dP x l P dx dx + - - + = (3.2) 4. Explicit expressions for the Legendre polynomials Explicit expressions for the Legendre polynomials ( ) 0 , l k l k k P x a x = = (4.1) can be obtained from the recurrence relation for the polynomial coefficients ( ) ( ) ( )( ) 2 1 1 . 1 2 k k k k l l a a k k + + - + = + + (4.2)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Note that if l is even then P l is an even function of x and similarly if l is odd then P l is an odd function of x , i.e. 0 1 0 if is odd, 0 if is even. a l a l = = (4.3) By using the recurrence relation (3.1), it can be shown that
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

Legendre_polynomials_notes - USEFUL FACTS AND FORMULAE FOR...

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

View Full Document
Ask a homework question - tutors are online