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The Legendre polynomials Pn(x) of degree n are special orthogonal polynomial functions defined on the domain [-1, 1].
Orthogonality:
| -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 19990620