Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
Parseval's identity shows a relation between the average of the square of f …x†
and the coecients in the Fourier series for f …x†:
the average of f f …x†g 2 is R L
L ‰ f …x†Š2 dx=2L;
the average of (a 0 =2† is (a 0 =2† 2 ;
the average of (a n cos nx† is a 2 n=2;
the average of (b n sin nx† is b 2 n=2.
Example 4.4
Expand f …x† ˆx; 0 < x < 2, in a half-range cosine series, then write Parseval's
identity corresponding to this Fourier cosine series.
Solution: We ®rst extend the de®nition of f …x† to that of the even function of
period 4 shown in Fig. 4.7. Then 2L ˆ 4; L ˆ 2. Thus b n ˆ 0 and
a n ˆ 2 Z L
f …x† cos nx
L 0 L dx ˆ 2 Z 2
f …x† cos nx
2 0 2 dx
2 nx 4 nx 2
ˆ x sin 1
n 2 n 2 cos
2 2
0
If n ˆ 0,
Then
ˆ 4
n 2 … cos n 1† if n 6ˆ 0:
2
a 0 ˆ
nˆ1
Z L
0
xdx ˆ 2:
f …x† ˆ1 ‡ X1 4
n 2 cos n 1
2
…
nx
†cos
2 :
We now write Parseval's identity. We ®rst compute the average of ‰ f …x†Š 2 :
the average of ‰ f …x†Š 2 ˆ 1
2
Z 2
2
ff …x† g 2 dx ˆ 1
2
Z 2
2
x 2 dx ˆ 8
3 ;
Figure 4.7.
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