Page 461 - Advanced engineering mathematics

13.3 Sine and Cosine Series 441

⎧
⎧
⎨ cos(x) for −2 ≤ x < 0 ⎪−2 for −4 ≤ x ≤−2
⎨
16. f (x) = 19. f (x) = 1 + x 2
⎩sin(x) for 0 ≤ x ≤ 2 for −2 < x ≤ 2
⎪
0 for 2 < x ≤ 4
⎩

−1 for −4 ≤ x < 0
17. f (x) = 20. Using Problem 14, write the Fourier series of the
1 for 0 ≤ x ≤ 4 function and plot some partial sums, pointing out the
occurrence of the Gibbs phenomenon at the points of
⎧
⎪0for −1 ≤ x < 1/2 discontinuity of the function.
⎨
18. f (x) = 1for 1/2 ≤ x ≤ 3/4
21. Carry out the program of Problem 20 for the function
⎪
2for 3/4 < x ≤ 1 of Problem 16.
⎩
13.3 Sine and Cosine Series
If f is piecewise continuous on [−L, L], we can represent f (x) at all but possibly finitely many
points of [−L, L] by its Fourier series. This series may contain just sine terms, just cosine terms,
or both sine and cosine terms. We have no control over this.
If f is defined on the half interval [0, L], we can write a Fourier cosine series (containing
just cosine terms) and a Fourier sine series (containing just sine terms) for f on [0, L].

13.3.1 Cosine Series

Suppose f (x) is defined for 0 ≤ x ≤ L. To get a pure cosine series on this interval, imagine
reflecting the graph of f across the vertical axis to obtain an even function g defined on [−L, L]
(see Figure 13.13).
Because g is even, its Fourier series on [−L, L] has only cosine terms and perhaps the
constant term. But g(x) = f (x) for 0 ≤ x ≤ L, so this gives a cosine series for f on [0, L].
Furthermore, because g is even, the coefficients in the Fourier series of g on [−L, L] are
1 L
a n = g(x)cos(nπx/L)dx
L −L
2 L
= g(x)cos(nπx/L)dx
L 0
2 L
= f (x)cos(nπx/L)dx.
L 0

y

x

FIGURE 13.13 Even extension of a
function defined on [0, L].

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:57 THM/NEIL Page-441 27410_13_ch13_p425-464