The Legendre polynomials Pn(x) of degree n are special orthogonal polynomial functions defined on the domain [-1, 1].
|-1∫1 W(x) Pi(x) Pj(x) dx = δij|
|W(x) = 1 (-1 < x < 1)|
They obey certain recurrence relations:
|n Pn(x) = (2n – 1) x Pn-1(x) – (n – 1) Pn-2(x)|
|P0(x) = 1|
|P1(x) = x|
We may change the domain of these polynomials to [xmin, xmax] by using the following transformation:
|x′ = (2x – (xmax + xmin)) / (xmax - xmin).|
We subsequently use Pk(x′) instead of Pk(x).
© djmw, June 20, 1999