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Mathematical Functions - Legendre Polynomials

Definition of the Legendre polynomial Pn(x).

Legendre Differential Equation

Even or odd Legendre polynomials.

Even/Odd Legendre Polynomials

Rodrigues formular for Legendre polynomials.

Rodrigues Formular

The Legendre polynomials are solutions of the Legendre differential equation.

Legendre Differential Equation

Executing the derivative d/dx will result in the following differential equation:

Legendre Differential Equation

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Legendre Polynomials

P0 = 1

P1 = x

P2 = (3x2 -1)/2

P3 = (5x3 - 3x)/2

P4 = (35x4 - 20x2 + 3)/8

P5 = (63x5 - 70x3 +15x)/8

P6 = (231x6 - 315x4 + 105x2 - 5)/16

P7 = (429x7 - 693x5 + 315x3 - 35x)/16

P8 = (6435x8 - 12012x6 + 6930x4 - 1260x2 + 35)/128

P9 = (12155x9 - 25740x7 + 18018x5 - 4620x3 + 315x)/128

P10 = (46189x10 - 109395x8 + 90090x6 - 30030x4 + 3465x2-63)/256    

P11 = (88179x11 - 230945x9 + 218790x7 - 90090x5 + 15015x3 - 693x)/256

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Legendre Functions of the Second Kind

Legendre Polynomial 0 Second Kind

Legendre Polynomial 1 Second Kind

Recusive Definition:

Legendre Polynomial n Second Kind

Legendre Polynomial 0 Second Kind

Legendre Polynomial 1 Second Kind

Legendre Polynomial 2 Second Kind

Legendre Polynomial 3 Second Kind

Q4

Q5

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Associated Legendre Polynomials

The associated Legendre polynomials are defined with the help of the derivatives of the Legendre polynomials as follows:

Associated Legendre Polynomials

m = 0 .. l

The Pl(x) are the Legendre polynomials.

The Pl(0)(x) of the associated Legendre polynomials are the normal Legendre polynomials.

Inserting Rodrigues formular:

Rodrigues Formular

will result in the following expression for the associated Legendre polynomials:

Associated Legendre Polynomials

The associated Legendre polynomials for negative m are then defined by

Associated Legendre Polynomials

The value of m can now be the range -l..m..l.

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Visit the Geometric Algebra page for an overview on Geometric Algebra.

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