Page 480 - Advanced engineering mathematics

460 CHAPTER 13 Fourier Series

1

0.5

0
–3 –2 –1 0 1 2 3
x

–0.5

–1

FIGURE 13.23 Graph of f in Example 13.17.

All the terms sin(nπ)=0. In −1 , replace n with −n and sum from n =1to ∞, then combine
n=−∞
the two summations from 1 to ∞ to write
∞
i
n inπx
(−1) e
nπ
n=−∞,n =0
∞
i n inπx i −n −inπx
= (−1) e + (−1) e
nπ −nπ
n=1
∞
i n inπx −inπx
= (−1) e − e
nπ
n=1
∞
2 n+1
= (−1) sin(nπx).
nπ
n=1
This is the Fourier series for f (x) = x on [−1,1].
The amplitude spectrum of a complex Fourier series of a periodic function is a graph of the
points (nω 0 ,|d n |). Sometimes this graph is also referred to as a frequency spectrum.

SECTION 13.6 PROBLEMS

In each of Problems 1 through 7, write the complex Fourier 4. f (x) = 1 − x for 0 ≤ x < 6, period 6
series of f , determine the sum of the series, and plot some
−1for 0 ≤ x < 2
points of the frequency spectrum. 5. f (x) = , f has period 4
2 for 2 ≤ x < 4
1. f (x) = 2x for 0 ≤ x < 3, period 3
6. f (x) = e −x for 0 ≤ x < 5, period 5
2
2. f (x) = x for 0 ≤ x < 2, period 2
x for 0 ≤ x < 1
7. f (x) = , f has period 2
0for 0 ≤ x < 1
3. f (x) = , f has period 4 2 − x for 1 ≤ x < 2
1for 1 ≤ x < 4

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October 14, 2010 14:57 THM/NEIL Page-460 27410_13_ch13_p425-464