Example 6.1.13 Let f (x) be expressible in a series of Legendre poly-
nomials. Thus
0011
0
()()()()()
nnnn
n
fxCPxCPxCPxCPx
∞
=
==++++
∑
(6.64)
where C
n
are constants to be determined.
Then multiplying both sides by ()
n
Px and integrating w.r.t. x from
–1 to 1, we get
11
0011
11
1
2
1
1
1
()()[()()()]()
2
()
21
21
()()
2
nnnn
nnn
nn
fxPxdxCPxCPxCPxPxdx
CPxdxC
n
n
CfxPxdx
−−
−
−
=++++
==⋅
+
+
⇒=
∫∫
∫
∫
1
1
1
2
1
()()0 if
2
()if
21
mn
n
PxPxdxm
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