Legendre_polynomials_notes

Legendre_polynomials_notes - USEFUL FACTS AND FORMULAE FOR...

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

View Full Document Right Arrow Icon
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
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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 Right Arrow Icon
Ask a homework question - tutors are online