Page 465 - Advanced engineering mathematics

13.4 Integration and Differentiation of Fourier Series 445

8

6

4

2

0
0 0.2 0.4 0.6 0.8 1
x

FIGURE 13.16 Fortieth partial sum of the sine
series of Example 13.10.

SECTION 13.3 PROBLEMS

In each of Problems 1 through 10, write the Fourier cosine ⎧ for 0 ≤ x < 1
⎪1
and sine series for f on the interval. Determine the sum ⎨
8. f (x) = 0 for 1 ≤ x ≤ 3
of each series. Graph some of the partial sums of these ⎪
⎩ −1for 3 < x ≤ 5
series.

1. f (x) = 4,0 ≤ x ≤ 3 x 2 for 0 ≤ x < 1
9. f (x) =
1 for 1 ≤ x ≤ 4
1 for 0 ≤ x ≤ 1
2. f (x) =
−1 for 1 < x ≤ 2 10. f (x) = 1 − x ,0 ≤ x ≤ 2
3

∞ n 2
0 for 0 ≤ x ≤ π 11. Sum the series n=1 (−1) /(4n − 1). Hint: Expand
3. f (x) = sin(x) in a cosine series on [0,π] and choose an
cos(x) for π< x ≤ 2π
appropriate value of x.
4. f (x) = 2x,0 ≤ x ≤ 1
12. Let f (x) be defined on [−L, L]. Prove that f can
2
5. f (x) = x ,0 ≤ x ≤ 2
be written as a sum of an even function and an odd
−x
6. f (x) = e ,0 ≤ x ≤ 1 function on this interval.

x for 0 ≤ x ≤ 2 13. Determine all functions on [−L, L] that are both even
7. f (x) =
2 − x for 2 < x ≤ 3 and odd.
13.4 Integration and Differentiation of Fourier Series
Term by term differentiation of a Fourier series may lead to nonsense.
EXAMPLE 13.11
Let f (x) = x for −π ≤ x ≤ π. The Fourier series is
∞
2
n+1
f (x) = x = (−1) sin(nx)
n
n=1

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October 14, 2010 14:57 THM/NEIL Page-445 27410_13_ch13_p425-464