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13.2 The Fourier Series of a Function 437

3
2

1
x
–3 –2 –1 0 1 2 3
0

–1
–2

–3

FIGURE 13.10 Twentieth partial sum of the
series in Example 13.7.

EXAMPLE 13.7
We will compute the Fourier series of f (x) = x on [−π,π].
Because x cos(nx) is an odd function on [−π,π] for n = 0,1,2,···, each a n = 0. We need
only compute the b n s:

1 π
b n = x sin(nx)dx
π −π
2 π
= x sin(nx)dx
π 0
π

2 2x
= sin(nx) − cos(nx)
2
n π nπ
0
2 2
=− cos(nπ) = (−1) n+1 .
n n
The Fourier series of x on [−π,π] is
∞
2
n+1
(−1) sin(nx).
n
n=1
This converges to x for −π< x <π, and to 0 at x =±π. Figure 13.10 shows the twentieth
partial sum of this Fourier series compared to the function.
EXAMPLE 13.8
4 4
We will write the Fourier series of x on [−1,1]. Since x sin(nπx) is an odd function on [−1,1]
for n = 1,2,···, each b n = 0. Compute
1 2
4
a 0 = 2 x dx =
0 5
and
1
4
a n = 2 x cos(nπx)dx
0
1
4 3
(nπx) sin(nπx) + 4(nπx) cos(nπx)
= 2
(nπ) 5
0

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October 14, 2010 14:57 THM/NEIL Page-437 27410_13_ch13_p425-464