418 Chapter 15Orthogonal expansions. Fourier analysis
and this can be used to obtain a general formula for the coefficientsc
lin (15.3). We
multiply equation (15.3) byP
k(x)and integrate. Then
and because, by (15.7), each integral on the right is zero except for the one withl 1 = 1 k,
only that term contributes to the sum:
Therefore
(15.8)
This important formula gives the values of the coefficientsc
lin the expansion of a
finite continuous function of x, for|x| 1 ≤ 11 , in terms of the Legendre polynomials.
Then, for the power series ,
(15.9)
Replacement ofP
k(x)in the integrals by its expansion in powers of x, from tabulations
like (15.4) or from the general formula (13.19), then gives the coefficientc
kin terms
of the coefficientsa
l.
EXAMPLE 15.1Use equation (15.9) fork 1 = 1 0 to findc
0.
We haveP
01 = 1 1 so that equation (15.9) is
where
Z
−+−++=
=
+111111
2
1
xdx
x
l
l
l
llif is even
00 if is oddl
caxdx
lll00 111
2
=
= −+∑
∞Z
c
k
aPxxdx
kllkl=
= −+∑
21
2
0 11∞Z ()
fx ax
lll()=
=∑
0∞c
k
Pxfxdx
kk=
−+21
2
11Z ()()
ZZ
−+−+==
11112
21
Pxfxdx c PxPxdx
k
c
kkkk()() () ()
kkZZ
−+−+==
∑
11110Pxfxdx Px cPx
kklll()() () ()
∞
=
= −+∑
dx c P x P x dx
llkl0 11∞