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∫ − 1 1 Px dx() = l 2 ( l 2 ( − 1) ∫ − 1 1 P () (7.107)
2
2
x dx
+
−
l
l 1
1)
Repeated applications of this formula and the use of Eq. (7.86)
yields:
∫ − 1 1 Px dx() = l 2 ( 3 + 1) ∫ − 1 1 Px dx() = l 2 ( 2 + 1) (7.108)
2
2
1
l
Direct calculations show that this is also valid for l = 0 and l = 1.
Therefore, the orthonormal basis functions are given by:

u = l + 1 P x() (7.109)
l l
2

The general theorem that summarizes the decomposition of a function into
the Legendre polynomials basis states:

THEOREM
If the real function f(x) defined over the interval [–1, 1] is piecewise smooth and if the
∫ 1 2 <∞
integral − 1 fx dx() , then the series:

∞
fx() = ∑ c P x() (7.110)
l l
l=0
where

 1
c =  l + ∫ 1 f x P x dx() () (7.111)
l  2 −1 l

converges to f(x) at every continuity point of the function.
The proof of this theorem is not given here.

Example 7.10
Find the decomposition into Legendre polynomials of the following function:

0 for −≤1 x ≤ a

fx() =  (7.112)
1 for a < x ≤ 1


© 2001 by CRC Press LLC